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We report a fault-tolerant storage threshold of $\\sim1.9(4)\\%$ for Kitaev's toric code subject to a 3D bit-flip channel (i.e. including imperfect syndrome measurements). This number is to be compared with the $2.9\\%$ value obtained via perfect matching \\cite{H04a}. 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Research","award":["89233218CNA000001"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Fluctuation theorems provide a correspondence between properties of quantum systems in thermal equilibrium and a work distribution arising in a non-equilibrium process that connects two quantum systems with Hamiltonians <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>V</mml:mi></mml:math>. Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> at inverse temperature <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B2;</mml:mi><mml:mo>&amp;#x2265;</mml:mo><mml:mn>0</mml:mn></mml:math> starting from a purification of the thermal state of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math>. The complexity of the quantum algorithm, given by the number of uses of certain unitaries, is <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi><mml:mo stretchy=\"false\">&amp;#x007E;</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>&amp;#x03B2;</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">&amp;#x0394;</mml:mi><mml:mspace width=\"negativethinmathspace\" /><mml:mi>A</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:msub><mml:mi>w</mml:mi><mml:mi>l</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math>, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x0394;</mml:mi><mml:mspace width=\"negativethinmathspace\" /><mml:mi>A</mml:mi></mml:math> is the free-energy difference between <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>w</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:math> is a work cutoff that depends on the properties of the work distribution and the approximation error <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03F5;</mml:mi><mml:mo>&amp;#x003E;</mml:mo><mml:mn>0</mml:mn></mml:math>. If the non-equilibrium process is trivial, this complexity is exponential in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B2;</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:mi>V</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo></mml:math>, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:mi>V</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo></mml:math> is the spectral norm of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>V</mml:mi></mml:math>. This represents a significant improvement of prior quantum algorithms that have complexity exponential in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B2;</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo></mml:math> in the regime where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:mi>V</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:mo>&amp;#x226A;</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo></mml:math>. The dependence of the complexity in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03F5;</mml:mi></mml:math> varies according to the structure of the quantum systems. It can be exponential in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi></mml:math> in general, but we show it to be sublinear in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi></mml:math> if <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> commute, or polynomial in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi></mml:math> if <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> are local spin systems. The possibility of applying a unitary that drives the system out of equilibrium allows one to increase the value of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>w</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:math> and improve the complexity even further. To this end, we analyze the complexity for preparing the thermal state of the transverse field Ising model using different non-equilibrium unitary processes and see significant complexity improvements.</jats:p>","DOI":"10.22331/q-2022-10-06-825","type":"journal-article","created":{"date-parts":[[2022,10,6]],"date-time":"2022-10-06T11:19:57Z","timestamp":1665055197000},"page":"825","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Quantum algorithms from fluctuation theorems: Thermal-state preparation","prefix":"10.22331","volume":"6","author":[{"given":"Zoe","family":"Holmes","sequence":"first","affiliation":[{"name":"Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]},{"given":"Gopikrishnan","family":"Muraleedharan","sequence":"additional","affiliation":[{"name":"Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]},{"given":"Rolando D.","family":"Somma","sequence":"additional","affiliation":[{"name":"Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]},{"given":"Yigit","family":"Subasi","sequence":"additional","affiliation":[{"name":"Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]},{"given":"Burak","family":"Şahinoğlu","sequence":"additional","affiliation":[{"name":"Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]}],"member":"9598","published-online":{"date-parts":[[2022,10,6]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-10-06-825/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,10,6]],"date-time":"2022-10-06T11:20:19Z","timestamp":1665055219000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-10-06-825/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,6]]},"references-count":96,"URL":"http://dx.doi.org/10.22331/q-2022-10-06-825","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,10,6]]},"article-number":"825","id":"doi:10.22331/q-2022-10-06-825","_hash":"b4f75eec3ea7467845aedf1fb6ca21c5b58fc6f2e03d3a25f76e5a46bc777876"},"expire":1699562166757},"doi:10.22331/q-2019-07-12-163":{"value":{"indexed":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T18:08:24Z","timestamp":1668103704620},"reference-count":51,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2019,7,12]],"date-time":"2019-07-12T00:00:00Z","timestamp":1562889600000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"<jats:p>We present the problem of approximating the time-evolution operator<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>e</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup></mml:math>to error<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>ϵ</mml:mi></mml:math>, where the Hamiltonian<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:mi>G</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>⊗</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>G</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo><mml:mo>⊗</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math>is the projection of a unitary oracle<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math>onto the state<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>G</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>created by another unitary oracle. Our algorithm solves this with a query complexity<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math>to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math>is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math>in an invariant<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mtext>SU</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math>subspace. A large class of operator functions of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math>can then be computed with optimal query complexity, of which<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>e</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup></mml:math>is a special case.</jats:p>","DOI":"10.22331/q-2019-07-12-163","type":"journal-article","created":{"date-parts":[[2019,7,12]],"date-time":"2019-07-12T02:43:01Z","timestamp":1562899381000},"page":"163","source":"Crossref","is-referenced-by-count":195,"title":"Hamiltonian Simulation by Qubitization","prefix":"10.22331","volume":"3","author":[{"ORCID":"http://orcid.org/0000-0002-6934-1052","authenticated-orcid":false,"given":"Guang Hao","family":"Low","sequence":"first","affiliation":[{"name":"Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA"}]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, Department of Physics, Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA"}]}],"member":"9598","published-online":{"date-parts":[[2019,7,12]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2019-07-12-163/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,9,23]],"date-time":"2022-09-23T18:50:42Z","timestamp":1663959042000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2019-07-12-163/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,12]]},"references-count":51,"URL":"http://dx.doi.org/10.22331/q-2019-07-12-163","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2019,7,12]]},"article-number":"163","id":"doi:10.22331/q-2019-07-12-163","_hash":"0332495050f7996a7703354032ff5f2f9f70fe07a062ba2c80217e57f8462acd"},"expire":1699562167736},"doi:10.1103/PhysRevLett.114.090502":{"value":{"indexed":{"date-parts":[[2022,11,9]],"date-time":"2022-11-09T21:50:55Z","timestamp":1668030655192},"reference-count":19,"publisher":"American Physical Society (APS)","issue":"9","license":[{"start":{"date-parts":[[2015,3,3]],"date-time":"2015-03-03T00:00:00Z","timestamp":1425340800000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"},{"start":{"date-parts":[[2016,3,2]],"date-time":"2016-03-02T00:00:00Z","timestamp":1456876800000},"content-version":"am","delay-in-days":365,"URL":"http://link.aps.org/licenses/aps-default-accepted-manuscript-license"}],"funder":[{"DOI":"10.13039/501100000923","name":"Australian Research 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Rev. Lett.","published":{"date-parts":[[2015,3,3]]},"article-number":"090502","id":"doi:10.1103/PhysRevLett.114.090502","_hash":"82c2c7b1932a9a94f40f21aa2b4325862ddfbdd421eed457eafaae65675083dc"},"expire":1699562168610},"doi:10.1103/PRXQuantum.3.010329":{"value":{"indexed":{"date-parts":[[2022,11,14]],"date-time":"2022-11-14T17:35:02Z","timestamp":1668447302618},"reference-count":118,"publisher":"American Physical Society 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Codes","prefix":"10.1103","volume":"3","author":[{"ORCID":"http://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[]},{"given":"Patricio","family":"Arrangoiz-Arriola","sequence":"additional","affiliation":[]},{"given":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-0665-7161","authenticated-orcid":true,"given":"Connor T.","family":"Hann","sequence":"additional","affiliation":[]},{"given":"Joseph","family":"Iverson","sequence":"additional","affiliation":[]},{"given":"Harald","family":"Putterman","sequence":"additional","affiliation":[]},{"given":"Thomas C.","family":"Bohdanowicz","sequence":"additional","affiliation":[]},{"given":"Steven 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Efficient quantum algorithms and protocols have been developed for performing some tasks (e.g., factoring large numbers, securely communicating over a public channel, and simulating quantum mechanical systems) that appear to be very difficult with just classical resources.  In addition to identifying the separation between classical and quantum computational power, much of the theoretical focus in this field over the last decade has been concerned with finding novel ways of encoding quantum information that are robust against errors, which is an important step toward building practical quantum information processing devices.\n\nIn this thesis I present some results on the quantum error-correcting properties of oscillator codes (also described as symplectic lattice codes) and toric codes.  Any harmonic oscillator system, such as a mode of light, can be encoded with quantum information via symplectic lattice codes that are robust against shifts in the system's continuous quantum variables.  I show the existence of lattice codes whose achievable rates match the one-shot coherent information over the Gaussian quantum channel.  Also, I construct a family of symplectic self-dual lattices and search for optimal encodings of quantum information distributed between several oscillators.\n\nToric codes provide encodings of quantum information into two-dimensional spin lattices that are robust against local clusters of errors and which require only local quantum operations for error correction.  Numerical simulations of this system under realistic error models provide a calculation of the accuracy threshold for quantum memory using toric codes, which can be related to phase transitions in particular condensed matter models.  I also present a local classical processing scheme for correcting errors on toric codes, which demonstrates that quantum information can be maintained in two dimensions by purely local quantum and classical resources.","DOI":"10.7907/AHMQ-EG82","publisher":"California Institute of Technology","title":"Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Codes","URL":"https://resolver.caltech.edu/CaltechETD:etd-05122004-113132","copyright":"No commercial reproduction, distribution, display or performance rights in this work are provided.","version":"Final","reference":[],"_hash":"347fe3088c237ff38c67e94a1ba231070cc7858231e4c4d55cf02eaf6558b350"},"expire":1701130346857},"doi:10.6028/jres.069B.013":{"value":{"indexed":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T05:36:14Z","timestamp":1669354574244},"reference-count":0,"publisher":"National Institute of Standards and Technology (NIST)","issue":"1 and 2","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[1965,1]]},"DOI":"10.6028/jres.069b.013","type":"journal-article","created":{"date-parts":[[2012,8,17]],"date-time":"2012-08-17T15:49:20Z","timestamp":1345218560000},"page":"125","source":"Crossref","is-referenced-by-count":969,"title":"Maximum matching and a polyhedron with 0,1-vertices","prefix":"10.6028","volume":"69B","author":[{"given":"Jack","family":"Edmonds","sequence":"first","affiliation":[]}],"member":"4068","container-title":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","original-title":[],"language":"en","deposited":{"date-parts":[[2018,3,6]],"date-time":"2018-03-06T09:23:28Z","timestamp":1520328208000},"score":1,"resource":{"primary":{"URL":"https://nvlpubs.nist.gov/nistpubs/jres/69B/jresv69Bn1-2p125_A1b.pdf"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1965,1]]},"references-count":0,"journal-issue":{"issue":"1 and 2","published-print":{"date-parts":[[1965,1]]}},"URL":"http://dx.doi.org/10.6028/jres.069B.013","relation":{},"ISSN":["0022-4340"],"subject":["Community and Home Care"],"container-title-short":"J. 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An edge is said to <jats:italic>join</jats:italic> its end-points.</jats:p><jats:p>A <jats:italic>matching</jats:italic> in <jats:italic>G</jats:italic> is a subset of its edges such that no two meet the same vertex. We describe an efficient algorithm for finding in a given graph a matching of maximum cardinality. This problem was posed and partly solved by C. 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Phys.","published":{"date-parts":[[1990,4]]},"id":"doi:10.1007/BF02096988","_hash":"bfa4336af1a3322d0451838d3e07f9ece23f222630418354c8e7f783e5862384"},"expire":1701130356032},"doi:10.7907/059V-MG69":{"value":{"type":"thesis","id":"doi:10.7907/059V-MG69","categories":["topological quantum codes","quantum many-body physics","quantum information","Physics","quantum computation","quantum error correction"],"language":"en","author":[{"family":"Kubica","given":"Aleksander Marek"}],"contributor":[{"literal":"Institute for Quantum Information and Matter"},{"literal":"IQIM"}],"issued":{"date-parts":[[2018]]},"abstract":"This thesis is devoted to studying a class of quantum error-correcting codes — topological quantum codes. We explore the question of how one can achieve fault- tolerant quantum computation with topological codes. We treat quantum error-correcting codes not only as a compelling ingredient needed to build a quantum computer, but also as a useful theoretical tool in other areas of physics. In particular, we explore what insights topological codes can provide into challenging questions, such as the classification of quantum phases of matter.\n\nIn this thesis, we focus on a family of topological codes — color codes, which are particularly intriguing due to the rich physics they display and their computational power. We start by introducing color codes and explaining their basic properties. Then, we show how to perform fault-tolerant universal quantum computation with three-dimensional color codes by transverse gates and code switching. We later compare the resource overhead of the code-switching approach with that of a state distillation scheme. We discuss how to perform error correction with the toric and color codes, as well as introduce local decoders for those two families of codes. By exploiting a connection between error correction and statistical mechanics we estimate the storage threshold error rates for bit-flip and phase-flip noise in the three-dimensional color code. We finish by showing that the color and toric code families in d dimensions are equivalent in a sense of local unitary transformations and explore implications of this equivalence.","DOI":"10.7907/059V-MG69","publisher":"California Institute of Technology","title":"The ABCs of the Color Code: A Study of Topological Quantum Codes as Toy Models for Fault-Tolerant Quantum Computation and Quantum Phases Of Matter","URL":"https://resolver.caltech.edu/CaltechTHESIS:05282018-173928314","copyright":"No commercial reproduction, distribution, display or performance rights in this work are provided.","version":"Final","reference":[],"_hash":"2b8fe7163dfb727bb3cf03d9c769010671607be7a2aa6a038ef7b6cd04ace1c1"},"expire":1701130356842},"doi:10.1088/1367-2630/13/4/043005":{"value":{"indexed":{"date-parts":[[2022,10,24]],"date-time":"2022-10-24T13:28:53Z","timestamp":1666618133823},"reference-count":29,"publisher":"IOP Publishing","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1088/1367-2630/13/4/043005","type":"journal-article","created":{"date-parts":[[2011,4,8]],"date-time":"2011-04-08T03:39:51Z","timestamp":1302233991000},"page":"043005","source":"Crossref","is-referenced-by-count":44,"title":"Clifford gates by code deformation","prefix":"10.1088","volume":"13","author":[{"given":"H","family":"Bombin","sequence":"first","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2011,4,7]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"deposited":{"date-parts":[[2020,4,11]],"date-time":"2020-04-11T15:15:00Z","timestamp":1586618100000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/13/4/043005"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,4,7]]},"references-count":29,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2011,4,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/13/4/043005","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. Phys.","published":{"date-parts":[[2011,4,7]]},"id":"doi:10.1088/1367-2630/13/4/043005","_hash":"2baa2c287e485d93c46be397e5e8c23db392e575bcd82c03469ec47cd2e3d29f"},"expire":1701130358040},"doi:10.1088/1367-2630/ab68fd":{"value":{"indexed":{"date-parts":[[2022,11,26]],"date-time":"2022-11-26T05:38:43Z","timestamp":1669441123411},"reference-count":59,"publisher":"IOP Publishing","issue":"2","license":[{"start":{"date-parts":[[2020,2,7]],"date-time":"2020-02-07T00:00:00Z","timestamp":1581033600000},"content-version":"vor","delay-in-days":6,"URL":"http://creativecommons.org/licenses/by/4.0/"},{"start":{"date-parts":[[2020,2,7]],"date-time":"2020-02-07T00:00:00Z","timestamp":1581033600000},"content-version":"tdm","delay-in-days":6,"URL":"https://iopscience.iop.org/info/page/text-and-data-mining"}],"funder":[{"DOI":"10.13039/100011039","name":"Intelligence Advanced Research Projects Activity","doi-asserted-by":"publisher","award":["W911NF-16-0114"]}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2020,2,1]]},"abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>The color code is a topological quantum error-correcting code supporting a variety of valuable fault-tolerant logical gates. Its two-dimensional version, the triangular color code, may soon be realized with currently available superconducting hardware despite constrained qubit connectivity. To guide this experimental effort, we study the storage threshold of the triangular color code against circuit-level depolarizing noise. First, we adapt the Restriction Decoder to the setting of the triangular color code and to phenomenological noise. Then, we propose a fault-tolerant implementation of the stabilizer measurement circuits, which incorporates flag qubits. We show how information from flag qubits can be used in an efficient and scalable way with the Restriction Decoder to maintain the effective distance of the code. We numerically estimate the threshold of the triangular color code to be 0.2%, which is competitive with the thresholds of other topological quantum codes. We also prove that 1-flag stabilizer measurement circuits are sufficient to preserve the full code distance, which may be used to find simpler syndrome extraction circuits of the color code.</jats:p>","DOI":"10.1088/1367-2630/ab68fd","type":"journal-article","created":{"date-parts":[[2020,1,8]],"date-time":"2020-01-08T22:16:52Z","timestamp":1578521812000},"page":"023019","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":34,"title":"Triangular color codes on trivalent graphs with flag qubits","prefix":"10.1088","volume":"22","author":[{"ORCID":"http://orcid.org/0000-0003-3239-5783","authenticated-orcid":false,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"given":"Aleksander","family":"Kubica","sequence":"additional","affiliation":[]},{"given":"Theodore J","family":"Yoder","sequence":"additional","affiliation":[]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2020,2,7]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd","content-type":"text/html","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,24]],"date-time":"2021-11-24T21:51:16Z","timestamp":1637790676000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,1]]},"references-count":59,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2,7]]},"published-print":{"date-parts":[[2020,2,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/ab68fd","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. Phys.","published":{"date-parts":[[2020,2,1]]},"assertion":[{"value":"Triangular color codes on trivalent graphs with flag qubits","name":"article_title","label":"Article Title"},{"value":"New Journal of Physics","name":"journal_title","label":"Journal Title"},{"value":"paper","name":"article_type","label":"Article Type"},{"value":"© 2020 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft","name":"copyright_information","label":"Copyright Information"},{"value":"2019-12-02","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-01-08","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-02-07","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/ab68fd","_hash":"24e4d1e3334fad341d085e8f1c07dd94f7f66753821de5d15c8dcac777444176"},"expire":1701130359853},"doi:10.1070/PU1982v025n04ABEH004537":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T21:08:43Z","timestamp":1670015323353},"reference-count":0,"publisher":"IOP Publishing","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[1982,4,30]]},"DOI":"10.1070/pu1982v025n04abeh004537","type":"journal-article","created":{"date-parts":[[2005,11,22]],"date-time":"2005-11-22T09:22:37Z","timestamp":1132651357000},"page":"231-256","source":"Crossref","is-referenced-by-count":976,"title":"The Jahn-Teller effect and magnetism: transition metal compounds","prefix":"10.1070","volume":"25","author":[{"given":"Kliment I","family":"Kugel'","sequence":"first","affiliation":[]},{"given":"D I","family":"Khomskiĭ","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2007,10,18]]},"container-title":"Soviet Physics Uspekhi","original-title":[],"link":[{"URL":"http://stacks.iop.org/0038-5670/25/i=4/a=R03/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2018,3,7]],"date-time":"2018-03-07T09:22:28Z","timestamp":1520414548000},"score":1,"resource":{"primary":{"URL":"http://stacks.iop.org/0038-5670/25/i=4/a=R03?key=crossref.1b66cf258ac8ee13f109957b04145a7c"},"secondary":[{"URL":"http://www.turpion.org/abstract/pu4537"}]},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,4,30]]},"references-count":0,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1982,4,30]]}},"URL":"http://dx.doi.org/10.1070/PU1982v025n04ABEH004537","relation":{},"ISSN":["0038-5670"],"container-title-short":"Sov. 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However, the two-qubit error-correction technique, known as ‘flagged’ syndrome extraction, works slowly, measuring only one syndrome at a time. This is a disadvantage in experiments with high qubit rest error rates. We extend the technique to extract multiple syndromes at once, without needing more qubits. Qubits for different syndromes can flag errors in each other. This gives equally fast and more qubit-efficient alternatives to Steane’s error-correction method, and also conforms to planar geometry constraints. We further show that Steane’s code and some others can be error-corrected with no extra qubits, provided there are at least two code blocks. The rough idea is that two seven-qubit codewords can be temporarily joined into a twelve-qubit code, freeing two qubits for flagged syndrome measurement.</jats:p>","DOI":"10.1088/2058-9565/abc6f4","type":"journal-article","created":{"date-parts":[[2020,12,4]],"date-time":"2020-12-04T13:40:28Z","timestamp":1607089228000},"page":"015007","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":12,"title":"Fault-tolerant quantum error correction for Steane’s seven-qubit color code with few or no extra qubits","prefix":"10.1088","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0002-4934-8732","authenticated-orcid":false,"given":"Ben W","family":"Reichardt","sequence":"first","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2020,11,24]]},"reference":[],"container-title":"Quantum Science and Technology","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4","content-type":"text/html","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,23]],"date-time":"2022-01-23T02:12:06Z","timestamp":1642903926000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,24]]},"references-count":20,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,11,24]]},"published-print":{"date-parts":[[2021,1,1]]}},"URL":"http://dx.doi.org/10.1088/2058-9565/abc6f4","relation":{},"ISSN":["2058-9565"],"subject":["Electrical and Electronic Engineering","Physics and Astronomy (miscellaneous)","Materials Science (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum Sci. 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We believe that the construction may be of interest for hardware in which one wants to use both long-range two-qubit gates as well as short-range gates between small clusters of qubits.</jats:p>","DOI":"10.26421/qic16.15-16-1","type":"journal-article","created":{"date-parts":[[2021,2,25]],"date-time":"2021-02-25T02:02:10Z","timestamp":1614218530000},"page":"1261-1281","source":"Crossref","is-referenced-by-count":5,"title":"Noise thresholds for the [4,2,2]-concatenated toric code","prefix":"10.26421","volume":"16","author":[{"given":"Ben","family":"Criger","sequence":"first","affiliation":[]},{"given":"Barbara","family":"Terhal","sequence":"additional","affiliation":[]}],"member":"10955","published-online":{"date-parts":[[2016,11]]},"container-title":"Quantum Information and 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Manipulation of spin quantum memory generally requires a magnetic field, which hinders the integration with superconducting qubits. Completely zero-field operation is desirable for scaling up a quantum computer based on superconducting qubits. Here we demonstrate quantum error correction to protect the nuclear spin of the nitrogen as a quantum memory in a diamond nitrogen-vacancy center with two nuclear spins of the surrounding carbon isotopes under a zero magnetic field. The quantum error correction makes quantum memory resilient against operational or environmental errors without the need for magnetic fields and opens a way toward distributed quantum computation and a quantum internet with memory-based quantum interfaces or quantum repeaters.</jats:p>","DOI":"10.1038/s42005-022-00875-6","type":"journal-article","created":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T10:03:29Z","timestamp":1651053809000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":"Quantum error correction of spin quantum memories in diamond under a zero magnetic 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Magn.","published":{"date-parts":[[2006,2]]},"id":"doi:10.1109/TMAG.2005.861043","_hash":"7d256aefd88ecd0384f5a9c12f64dd165c876cde6e0f4c1879c7f8528186ca7f"},"expire":1701130636824},"doi:10.1145/1236457.1236459":{"value":{"indexed":{"date-parts":[[2022,11,22]],"date-time":"2022-11-22T03:26:48Z","timestamp":1669087608474},"reference-count":33,"publisher":"Association for Computing Machinery (ACM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2007,6]]},"abstract":"<jats:p>The PCP theorem [Arora and Safra 1998; Arora et. al. 1998] says that every language in NP has a witness format that can be checked probabilistically by reading only a constant number of bits from the proof. The celebrated equivalence of this theorem and inapproximability of certain optimization problems, due to Feige et al. [1996], has placed the PCP theorem at the heart of the area of inapproximability.</jats:p>\n          <jats:p>In this work, we present a new proof of the PCP theorem that draws on this equivalence. We give a combinatorial proof for the NP-hardness of approximating a certain constraint satisfaction problem, which can then be reinterpreted to yield the PCP theorem.</jats:p>\n          <jats:p>\n            Our approach is to consider the\n            <jats:italic>unsat value</jats:italic>\n            of a constraint system, which is the smallest fraction of unsatisfied constraints, ranging over all possible assignments for the underlying variables. We describe a new combinatorial amplification transformation that doubles the unsat-value of a constraint-system, with only a linear blowup in the size of the system. The amplification step causes an increase in alphabet-size that is corrected by a (standard) PCP composition step. Iterative application of these two steps yields a proof for the PCP theorem.\n          </jats:p>\n          <jats:p>The amplification lemma relies on a new notion of “graph powering” that can be applied to systems of binary constraints. This powering amplifies the unsat-value of a constraint system provided that the underlying graph structure is an expander.</jats:p>\n          <jats:p>\n            We also extend our amplification lemma towards construction of assignment testers (alternatively, PCPs of Proximity) which are slightly stronger objects than PCPs. We then construct PCPs and locally-testable codes whose length is linear up to a\n            <jats:italic>polylog</jats:italic>\n            factor, and whose correctness can be probabilistically verified by making a\n            <jats:italic>constant</jats:italic>\n            number of queries. Namely, we prove\n            <jats:italic>SAT</jats:italic>\n            ∈\n            <jats:italic>PCP</jats:italic>\n            <jats:sub>1/2,1</jats:sub>\n            [log\n            <jats:sub>2</jats:sub>\n            (\n            <jats:italic>n</jats:italic>\n            ⋅poly log\n            <jats:italic>n</jats:italic>\n            ),\n            <jats:italic>O</jats:italic>\n            (1)].\n          </jats:p>","DOI":"10.1145/1236457.1236459","type":"journal-article","created":{"date-parts":[[2007,9,14]],"date-time":"2007-09-14T13:44:55Z","timestamp":1189777495000},"page":"12","source":"Crossref","is-referenced-by-count":197,"title":"The PCP theorem by gap amplification","prefix":"10.1145","volume":"54","author":[{"given":"Irit","family":"Dinur","sequence":"first","affiliation":[{"name":"The Hebrew University, Jerusalem, Israel"}]}],"member":"320","reference":[],"container-title":"Journal of the ACM","original-title":[],"language":"en","link":[{"URL":"https://dl.acm.org/doi/pdf/10.1145/1236457.1236459","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,20]],"date-time":"2021-02-20T04:41:34Z","timestamp":1613796094000},"score":1,"resource":{"primary":{"URL":"https://dl.acm.org/doi/10.1145/1236457.1236459"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,6]]},"references-count":33,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2007,6]]}},"alternative-id":["10.1145/1236457.1236459"],"URL":"http://dx.doi.org/10.1145/1236457.1236459","relation":{},"ISSN":["0004-5411","1557-735X"],"subject":["Artificial Intelligence","Hardware and Architecture","Information Systems","Control and Systems Engineering","Software"],"container-title-short":"J. 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Comput.","published":{"date-parts":[[1998,6]]},"id":"doi:10.1137/S0097539796302531","_hash":"d7bee88b3b8ac5bdcecfc446cb891b28a894ba9b06770d0f3306877217519a58"},"expire":1701130699280},"doi:10.1145/502090.502098":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T19:12:55Z","timestamp":1670008375741},"reference-count":32,"publisher":"Association for Computing Machinery (ACM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2001,7]]},"abstract":"<jats:p>\n            We prove optimal, up to an arbitrary ε &gt; 0, inapproximability results for Max-E\n            <jats:italic>k</jats:italic>\n            -Sat for\n            <jats:italic>k</jats:italic>\n            ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime\n            <jats:italic>p</jats:italic>\n            and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-di-Cut, and Vertex cover.\n          </jats:p>","DOI":"10.1145/502090.502098","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T11:26:13Z","timestamp":1027769173000},"page":"798-859","source":"Crossref","is-referenced-by-count":705,"title":"Some optimal inapproximability results","prefix":"10.1145","volume":"48","author":[{"given":"Johan","family":"Håstad","sequence":"first","affiliation":[{"name":"Royal Institute of Technology, Stockholm, Sweden"}]}],"member":"320","reference":[],"container-title":"Journal of the ACM","original-title":[],"language":"en","link":[{"URL":"https://dl.acm.org/doi/pdf/10.1145/502090.502098","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,4]],"date-time":"2021-03-04T17:21:46Z","timestamp":1614878506000},"score":1,"resource":{"primary":{"URL":"https://dl.acm.org/doi/10.1145/502090.502098"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,7]]},"references-count":32,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2001,7]]}},"alternative-id":["10.1145/502090.502098"],"URL":"http://dx.doi.org/10.1145/502090.502098","relation":{},"ISSN":["0004-5411","1557-735X"],"subject":["Artificial Intelligence","Hardware and Architecture","Information Systems","Control and Systems Engineering","Software"],"container-title-short":"J. 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Inform. Theory","published":{"date-parts":[[2018,8]]},"id":"doi:10.1109/TIT.2018.2809788","_hash":"aa0983d9ef733ab3aa7a71690eee4e2ca9051e37d39434c29d12508cb5eb407b"},"expire":1701130701368},"doi:10.1145/3051093":{"value":{"indexed":{"date-parts":[[2022,9,4]],"date-time":"2022-09-04T23:48:45Z","timestamp":1662335325255},"reference-count":69,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2018,5,25]],"date-time":"2018-05-25T00:00:00Z","timestamp":1527206400000},"content-version":"vor","delay-in-days":357,"URL":"http://www.acm.org/publications/policies/copyright_policy#Background"}],"funder":[{"DOI":"10.13039/100000001","name":"NSF","doi-asserted-by":"publisher","award":["CCF-1253886, CCF-1412958, CCF-1445755 and CCF-1350572"]},{"name":"Sloan Fellowship, Rothschild Fellowship"},{"name":"ERC","award":["239986"]},{"DOI":"10.13039/501100003977","name":"Israel Science Foundation","doi-asserted-by":"crossref","award":["460/05"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2017,6,2]]},"abstract":"<jats:p>\n            Locally correctable codes (LCCs) and locally testable codes (LTCs) are error-correcting codes that admit\n            <jats:italic>local</jats:italic>\n            algorithms for correction and detection of errors. Those algorithms are local in the sense that they only query a small number of entries of the corrupted codeword. The fundamental question about LCCs and LTCs is to determine the optimal tradeoff among their rate, distance, and query complexity.\n          </jats:p>\n          <jats:p>\n            In this work, we construct the first LCCs and LTCs with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist LCCs and LTCs with block length\n            <jats:italic>n</jats:italic>\n            , constant rate (which can even be taken arbitrarily close to 1), and constant relative distance, whose query complexity is exp(Õ(√log\n            <jats:italic>n</jats:italic>\n            )) (for LCCs) and (log\n            <jats:italic>n</jats:italic>\n            )\n            <jats:sup>\n              <jats:italic>O</jats:italic>\n              (log log\n              <jats:italic>n</jats:italic>\n              )\n            </jats:sup>\n            (for LTCs).\n          </jats:p>\n          <jats:p>\n            In addition to having small query complexity, our codes also achieve better tradeoffs between the rate and the relative distance than were previously known to be achievable by LCCs or LTCs. Specifically, over large (but constant size) alphabet, our codes approach the Singleton bound, that is, they have almost the best-possible relationship between their rate and distance. Over the binary alphabet, our codes meet the Zyablov bound. Such tradeoffs between the rate and the relative distance were previously not known for any\n            <jats:italic>o</jats:italic>\n            (\n            <jats:italic>n</jats:italic>\n            ) query complexity. Our results on LCCs also immediately give locally decodable codes with the same parameters.\n          </jats:p>","DOI":"10.1145/3051093","type":"journal-article","created":{"date-parts":[[2017,5,25]],"date-time":"2017-05-25T16:16:45Z","timestamp":1495729005000},"page":"1-42","source":"Crossref","is-referenced-by-count":16,"title":"High-Rate Locally Correctable and Locally Testable Codes with Sub-Polynomial Query Complexity","prefix":"10.1145","volume":"64","author":[{"given":"Swastik","family":"Kopparty","sequence":"first","affiliation":[{"name":"Department of Mathematics 8 Department of Computer Science, Rutgers University, Piscataway NJ, USA"}]},{"given":"Or","family":"Meir","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Haifa University, Haifa, Israel"}]},{"given":"Noga","family":"Ron-Zewi","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Ben-Gurion University, Be’er Sheva, Israel"}]},{"given":"Shubhangi","family":"Saraf","sequence":"additional","affiliation":[{"name":"Department of Mathematics 8 Department of Computer Science, Rutgers University, Piscataway NJ, USA"}]}],"member":"320","reference":[],"container-title":"Journal of the ACM","original-title":[],"language":"en","link":[{"URL":"https://dl.acm.org/doi/pdf/10.1145/3051093","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://dl.acm.org/doi/pdf/10.1145/3051093","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,2]],"date-time":"2021-03-02T23:56:12Z","timestamp":1614729372000},"score":1,"resource":{"primary":{"URL":"https://dl.acm.org/doi/10.1145/3051093"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,6,2]]},"references-count":69,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2017,6,2]]}},"alternative-id":["10.1145/3051093"],"URL":"http://dx.doi.org/10.1145/3051093","relation":{},"ISSN":["0004-5411","1557-735X"],"subject":["Artificial Intelligence","Hardware and Architecture","Information Systems","Control and Systems Engineering","Software"],"container-title-short":"J. ACM","published":{"date-parts":[[2017,6,2]]},"id":"doi:10.1145/3051093","_hash":"4b068138d8842495ac031af2d85cb046e927cb169cb9042515d2b60ce1c64b8a"},"expire":1701130702275},"doi:10.1145/1162349.1162351":{"value":{"indexed":{"date-parts":[[2022,11,22]],"date-time":"2022-11-22T03:29:30Z","timestamp":1669087770977},"reference-count":32,"publisher":"Association for Computing Machinery (ACM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2006,7]]},"abstract":"<jats:p>\n            We initiate a systematic study of locally testable codes; that is, error-correcting codes that admit very efficient membership tests. Specifically, these are codes accompanied with tests that make a constant number of (random) queries into any given word and reject non-codewords with probability proportional to their distance from the code.Locally testable codes are believed to be the combinatorial core of PCPs. However, the relation is less immediate than commonly believed. Nevertheless, we show that certain PCP systems can be modified to yield locally testable codes. On the other hand, we adapt techniques that we develop for the construction of the latter to yield new PCPs.Our main results are locally testable codes and PCPs of almost-linear length. Specifically, we prove the existence of the following constructs:---Locally testable binary (linear) codes in which\n            <jats:italic>k</jats:italic>\n            information bits are encoded by a codeword of length\n            <jats:italic>k</jats:italic>\n            ⋅ exp(Õ(√(log\n            <jats:italic>k</jats:italic>\n            ))). This improves over previous results that either yield codewords of exponential length or obtained almost quadratic length codewords for sufficiently large nonbinary alphabet.---PCP systems of almost-linear length for SAT. The length of the proof is\n            <jats:italic>n</jats:italic>\n            ⋅ exp(Õ(√(log\n            <jats:italic>n</jats:italic>\n            ))) and verification in performed by a constant number (i.e., 19) of queries, as opposed to previous results that used proof length\n            <jats:italic>n</jats:italic>\n            <jats:sup>\n              (1 +\n              <jats:italic>O</jats:italic>\n              (1/\n              <jats:italic>q</jats:italic>\n              ))\n            </jats:sup>\n            for verification by\n            <jats:italic>q</jats:italic>\n            queries.The novel techniques in use include a random projection of certain codewords and PCP-oracles that preserves local-testability, an adaptation of PCP constructions to obtain “linear PCP-oracles” for proving conjunctions of linear conditions, and design of PCPs with some new soundness properties---a direct construction of locally testable (linear) codes of subexponential length.\n          </jats:p>","DOI":"10.1145/1162349.1162351","type":"journal-article","created":{"date-parts":[[2006,10,18]],"date-time":"2006-10-18T18:11:32Z","timestamp":1161195092000},"page":"558-655","source":"Crossref","is-referenced-by-count":78,"title":"Locally testable codes and PCPs of almost-linear length","prefix":"10.1145","volume":"53","author":[{"given":"Oded","family":"Goldreich","sequence":"first","affiliation":[{"name":"Weizmann Institute of Science, Rehovot, Israel"}]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, MA"}]}],"member":"320","reference":[],"container-title":"Journal of the ACM","original-title":[],"language":"en","link":[{"URL":"https://dl.acm.org/doi/pdf/10.1145/1162349.1162351","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,20]],"date-time":"2021-02-20T06:46:19Z","timestamp":1613803579000},"score":1,"resource":{"primary":{"URL":"https://dl.acm.org/doi/10.1145/1162349.1162351"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,7]]},"references-count":32,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2006,7]]}},"alternative-id":["10.1145/1162349.1162351"],"URL":"http://dx.doi.org/10.1145/1162349.1162351","relation":{},"ISSN":["0004-5411","1557-735X"],"subject":["Artificial Intelligence","Hardware and Architecture","Information Systems","Control and Systems Engineering","Software"],"container-title-short":"J. 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The following tight spherical <jats:italic>t</jats:italic>-designs are unique: the 5-design in Ω<jats:sub>7</jats:sub>, the 7-designs in Ω<jats:sub>8</jats:sub> and Ω<jats:sub>23</jats:sub>, and the 11-design in Ω<jats:sub>24</jats:sub>. It was shown in <jats:bold>[20]</jats:bold> that the maximum number of nonoverlapping unit spheres in <jats:bold>R</jats:bold><jats:sup>8</jats:sup> (resp. <jats:bold>R</jats:bold><jats:sup>24</jats:sup>) that can touch another unit sphere is 240 (resp. 196560). Arrangements of spheres meeting these bounds can be obtained from the E<jats:sub>8</jats:sub> and Leech lattices, respectively. 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Others are given in [2<jats:sup><jats:italic>m</jats:italic></jats:sup>], <jats:italic>m</jats:italic> ⩾ 5, with the same density as those of §1.6, but in which each sphere touches fewer other spheres than in the earlier packings.</jats:p>","DOI":"10.4153/cjm-1967-017-0","type":"journal-article","created":{"date-parts":[[2010,12,7]],"date-time":"2010-12-07T23:24:44Z","timestamp":1291764284000},"page":"251-267","source":"Crossref","is-referenced-by-count":123,"title":"Notes on Sphere Packings","prefix":"10.4153","volume":"19","author":[{"given":"John","family":"Leech","sequence":"first","affiliation":[]}],"member":"2643","published-online":{"date-parts":[[2018,11,20]]},"reference":[],"container-title":"Canadian Journal of Mathematics","original-title":[],"language":"en","link":[{"URL":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00054250","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,7]],"date-time":"2019-06-07T18:58:25Z","timestamp":1559933905000},"score":1,"resource":{"primary":{"URL":"https://www.cambridge.org/core/product/identifier/S0008414X00054250/type/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1967]]},"references-count":5,"alternative-id":["S0008414X00054250"],"URL":"http://dx.doi.org/10.4153/CJM-1967-017-0","relation":{},"ISSN":["0008-414X","1496-4279"],"subject":["General Mathematics"],"container-title-short":"Can. j. math.","published":{"date-parts":[[1967]]},"id":"doi:10.4153/CJM-1967-017-0","_hash":"8f3c20b864a438e7f994eeb556b77230d92d5bf65fa6e6c9e88a9a51447ffc6b"},"expire":1701130793323},"doi:10.1017/S1446788700025064":{"value":{"indexed":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T22:54:53Z","timestamp":1668120893903},"reference-count":3,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2009,4,9]],"date-time":"2009-04-09T00:00:00Z","timestamp":1239235200000},"content-version":"unspecified","delay-in-days":18149,"URL":"https://www.cambridge.org/core/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[1959,8]]},"abstract":"<jats:p>Let <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1446788700025064_inline1\" /> be a positive definite quadratic form of determinant <jats:italic>D</jats:italic>, and let <jats:italic>M</jats:italic> be the minimum of <jats:italic>f</jats:italic>(<jats:italic>x</jats:italic>) for integral x ≠ 0. Then we set <jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1446788700025064_eqn1\" /></jats:disp-formula> and <jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1446788700025064_eqn2\" /></jats:disp-formula> the maximum being over all positive forms <jats:italic>f</jats:italic> in <jats:italic>n</jats:italic> variables. <jats:italic>f</jats:italic> is said to be extreme if <jats:italic>y</jats:italic> γ<jats:sub>n</jats:sub>(<jats:italic>f</jats:italic>) is a local maximum for varying <jats:italic>f</jats:italic>, <jats:italic>absolutely extreme</jats:italic> if <jats:italic>y</jats:italic> γ(<jats:italic>f</jats:italic>) is an absolute maximum, i.e. if <jats:italic>y</jats:italic> γ(f) = γ<jats:sub>n</jats:sub>.</jats:p>","DOI":"10.1017/s1446788700025064","type":"journal-article","created":{"date-parts":[[2009,4,14]],"date-time":"2009-04-14T09:25:31Z","timestamp":1239701131000},"page":"47-63","source":"Crossref","is-referenced-by-count":44,"title":"Some extreme forms defined in terms of Abelian groups","prefix":"10.1017","volume":"1","author":[{"given":"E. 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We give a complete description of the groups of coincidence rotations with their associated indices and encapsulate their statistics by means of generating functions.</jats:p>","DOI":"10.1515/zna-1995-0802","type":"journal-article","created":{"date-parts":[[2015,2,21]],"date-time":"2015-02-21T11:25:16Z","timestamp":1424517916000},"page":"711-717","source":"Crossref","is-referenced-by-count":7,"title":"Algebraic Solution of the Coincidence Problem in Two and Three Dimensions","prefix":"10.1515","volume":"50","author":[{"given":"Michael","family":"Baake","sequence":"first","affiliation":[{"name":"Institut für Theoretische Physik, Universität Tübingen, D-72076 Tübingen, Germany"}]},{"given":"Peter A. 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Only degenerate codes with the maximal stabilizer weight much smaller than their minimum distance are considered. It is shown that with the help of OSD-like post-processing the performance of the standard belief propagation (BP) decoder on many QLDPC codes can be improved by several orders of magnitude. Using this new BP-OSD decoder we study the performance of several known classes of degenerate QLDPC codes including hypergraph product codes, hyperbicycle codes, homological product codes, and Haah's cubic codes. We also construct several interesting examples of short generalized bicycle codes. Some of them have an additional property that their syndromes are protected by small BCH codes, which may be useful for the fault-tolerant syndrome measurement. We also propose a new large family of QLDPC codes that contains the class of hypergraph product codes, where one of the used parity-check matrices is square. 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xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>&amp;#x03C0;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>N</mml:mi></mml:math> rotations around the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math> axis for some integer <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math>. The resulting generalised stabiliser formalism – denoted the XP stabiliser formalism – allows for a wider range of states and codespaces to be represented. We describe the states which arise in the formalism, and demonstrate an equivalence between XP stabiliser states and 'weighted hypergraph states' – a generalisation of both hypergraph and weighted graph states. Given an arbitrary set of XP operators, we present algorithms for determining the codespace and logical operators for an XP code. Finally, we consider whether measurements of XP operators on XP codes can be classically simulated.</jats:p>","DOI":"10.22331/q-2022-09-22-815","type":"journal-article","created":{"date-parts":[[2022,9,22]],"date-time":"2022-09-22T15:25:34Z","timestamp":1663860334000},"page":"815","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"The XP Stabiliser Formalism: a Generalisation of the Pauli Stabiliser Formalism with Arbitrary Phases","prefix":"10.22331","volume":"6","author":[{"given":"Mark A.","family":"Webster","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia"},{"name":"Sydney Quantum Academy, Sydney, NSW, Australia"}]},{"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia"},{"name":"Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, University of Copenhagen, 2100 Copenhagen, Denmark"}]},{"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia"}]}],"member":"9598","published-online":{"date-parts":[[2022,9,22]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-09-22-815/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,9,22]],"date-time":"2022-09-22T15:25:38Z","timestamp":1663860338000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-09-22-815/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,22]]},"references-count":26,"URL":"http://dx.doi.org/10.22331/q-2022-09-22-815","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,9,22]]},"article-number":"815","id":"doi:10.22331/q-2022-09-22-815","_hash":"2b2091a06ee33bc0171c65fbc5ffeaca0d45cc9813ce098f5bdc2af69796c40e"},"expire":1701130830765},"doi:10.1103/PhysRevA.105.042616":{"value":{"indexed":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T14:09:28Z","timestamp":1651068568729},"reference-count":74,"publisher":"American Physical Society (APS)","issue":"4","license":[{"start":{"date-parts":[[2022,4,25]],"date-time":"2022-04-25T00:00:00Z","timestamp":1650844800000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/501100004063","name":"Knut och Alice Wallenbergs 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A","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevA.105.042616","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.105.042616/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,25]],"date-time":"2022-04-25T14:12:27Z","timestamp":1650895947000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.105.042616"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,25]]},"references-count":74,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,4]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.105.042616","relation":{},"ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","published":{"date-parts":[[2022,4,25]]},"article-number":"042616","id":"doi:10.1103/PhysRevA.105.042616","_hash":"ec3a9523ee1980ec3427944eb2268584f81b11521fa2f8ebc0121cb1b95d3642"},"expire":1701130831779},"doi:10.22331/q-2022-04-27-698":{"value":{"indexed":{"date-parts":[[2022,5,4]],"date-time":"2022-05-04T22:54:32Z","timestamp":1651704872137},"reference-count":67,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T00:00:00Z","timestamp":1651017600000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>We consider a topological stabilizer code on a honeycomb grid, the \"XYZ<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi /><mml:mn>2</mml:mn></mml:msup></mml:math>\" code. The code is inspired by the Kitaev honeycomb model and is a simple realization of a \"matching code\" discussed by Wootton [J. Phys. A: Math. Theor. 48, 215302 (2015)], with a specific implementation of the boundary. It utilizes weight-six (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi><mml:mi>Y</mml:mi><mml:mi>Z</mml:mi><mml:mi>X</mml:mi><mml:mi>Y</mml:mi><mml:mi>Z</mml:mi></mml:math>) plaquette stabilizers and weight-two (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi><mml:mi>X</mml:mi></mml:math>) link stabilizers on a planar hexagonal grid composed of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> qubits for code distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>, with weight-three stabilizers at the boundary, stabilizing one logical qubit. We study the properties of the code using maximum-likelihood decoding, assuming perfect stabilizer measurements. For pure <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi></mml:math>, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Y</mml:mi></mml:math>, or <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math> noise, we can solve for the logical failure rate analytically, giving a threshold of 50%. In contrast to the rotated surface code and the XZZX code, which have code distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> only for pure <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Y</mml:mi></mml:math> noise, here the code distance is <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> for both pure <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math> and pure <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Y</mml:mi></mml:math> noise. Thresholds for noise with finite <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math> bias are similar to the XZZX code, but with markedly lower sub-threshold logical failure rates. The code possesses distinctive syndrome properties with unidirectional pairs of plaquette defects along the three directions of the triangular lattice for isolated errors, which may be useful for efficient matching-based or other approximate decoding.</jats:p>","DOI":"10.22331/q-2022-04-27-698","type":"journal-article","created":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T13:48:13Z","timestamp":1651067293000},"page":"698","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"The XYZ<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi /><mml:mn>2</mml:mn></mml:msup></mml:math> hexagonal stabilizer code","prefix":"10.22331","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0002-4972-4216","authenticated-orcid":false,"given":"Basudha","family":"Srivastava","sequence":"first","affiliation":[{"name":"Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden"}]},{"ORCID":"http://orcid.org/0000-0002-2534-3021","authenticated-orcid":false,"given":"Anton","family":"Frisk Kockum","sequence":"additional","affiliation":[{"name":"Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-41296 Gothenburg, Sweden"}]},{"ORCID":"http://orcid.org/0000-0003-3185-2014","authenticated-orcid":false,"given":"Mats","family":"Granath","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden"}]}],"member":"9598","published-online":{"date-parts":[[2022,4,27]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-04-27-698/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T13:48:38Z","timestamp":1651067318000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-04-27-698/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,27]]},"references-count":67,"URL":"http://dx.doi.org/10.22331/q-2022-04-27-698","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,4,27]]},"article-number":"698","id":"doi:10.22331/q-2022-04-27-698","_hash":"40cc448b9d448b6c51f9e2b28d0aadd66f8f0bf5ef94dd4543ef701b413c7127"},"expire":1701130832614},"doi:10.1088/1751-8121/ac7a75":{"value":{"indexed":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T12:27:04Z","timestamp":1663763224619},"reference-count":24,"publisher":"IOP Publishing","issue":"29","license":[{"start":{"date-parts":[[2022,7,4]],"date-time":"2022-07-04T00:00:00Z","timestamp":1656892800000},"content-version":"vor","delay-in-days":0,"URL":"https://iopscience.iop.org/page/copyright"},{"start":{"date-parts":[[2022,7,4]],"date-time":"2022-07-04T00:00:00Z","timestamp":1656892800000},"content-version":"tdm","delay-in-days":0,"URL":"https://iopscience.iop.org/info/page/text-and-data-mining"}],"funder":[{"DOI":"10.13039/501100001711","name":"Swiss National Science Foundation","doi-asserted-by":"crossref"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2022,7,22]]},"abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Matching codes are stabilizer codes based on Kitaev’s honeycomb lattice model. The hexagonal form of these codes are particularly well-suited to the heavy-hexagon device layouts currently pursued in the hardware of IBM quantum. Here we show how the stabilizers of the code can be measured solely through two-body measurements that are native to the architecture. Though the subsystem code formed by these measurements has a trivial code space, the sequence in which they are measured allows the desired logical subspace to be preserved. This therefore achieves a result similar to the recently introduced Floquet codes, but via a completely different method. The process is then run on 27 and 65 qubit devices, to compare results with simulations for a standard error model. It is found that the results correspond well to simulations where the noise strength is similar to that found in the benchmarking of the devices. The best devices show results consistent with a noise model with an error probability of around 1.5%–2%.</jats:p>","DOI":"10.1088/1751-8121/ac7a75","type":"journal-article","created":{"date-parts":[[2022,6,20]],"date-time":"2022-06-20T22:24:54Z","timestamp":1655763894000},"page":"295302","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":"Hexagonal matching codes with two-body measurements","prefix":"10.1088","volume":"55","author":[{"ORCID":"http://orcid.org/0000-0003-1943-5306","authenticated-orcid":true,"given":"James R","family":"Wootton","sequence":"first","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2022,7,4]]},"reference":[],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,30]],"date-time":"2022-06-30T16:59:58Z","timestamp":1656608398000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,4]]},"references-count":24,"journal-issue":{"issue":"29","published-online":{"date-parts":[[2022,7,4]]},"published-print":{"date-parts":[[2022,7,22]]}},"URL":"http://dx.doi.org/10.1088/1751-8121/ac7a75","relation":{},"ISSN":["1751-8113","1751-8121"],"subject":["General Physics and Astronomy","Mathematical Physics","Modeling and Simulation","Statistics and Probability","Statistical and Nonlinear Physics"],"container-title-short":"J. Phys. A: Math. Theor.","published":{"date-parts":[[2022,7,4]]},"assertion":[{"value":"Hexagonal matching codes with two-body measurements","name":"article_title","label":"Article Title"},{"value":"Journal of Physics A: Mathematical and Theoretical","name":"journal_title","label":"Journal Title"},{"value":"paper","name":"article_type","label":"Article Type"},{"value":"© 2022 IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2022-01-26","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2022-06-20","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2022-07-04","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1751-8121/ac7a75","_hash":"f7c2e5cc374d39f344d1471a419bff1e3ac6ef092c0aea0ce7890f583a00fcd8"},"expire":1701130833682},"doi:10.1103/PhysRevLett.90.016803":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T12:22:53Z","timestamp":1669983773394},"reference-count":19,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[2003,1,10]],"date-time":"2003-01-10T00:00:00Z","timestamp":1042156800000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physrevlett.90.016803","type":"journal-article","created":{"date-parts":[[2003,1,11]],"date-time":"2003-01-11T04:18:41Z","timestamp":1042258721000},"source":"Crossref","is-referenced-by-count":245,"title":"Quantum Orders in an Exact Soluble Model","prefix":"10.1103","volume":"90","author":[{"given":"Xiao-Gang","family":"Wen","sequence":"first","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2003,1,10]]},"reference":[],"container-title":"Physical Review Letters","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevLett.90.016803","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.90.016803/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,15]],"date-time":"2017-06-15T17:18:43Z","timestamp":1497547123000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevLett.90.016803"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1,10]]},"references-count":19,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2003,1]]}},"URL":"http://dx.doi.org/10.1103/PhysRevLett.90.016803","relation":{},"ISSN":["0031-9007","1079-7114"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. Rev. Lett.","published":{"date-parts":[[2003,1,10]]},"article-number":"016803","id":"doi:10.1103/PhysRevLett.90.016803","_hash":"f429aff3ca8cd29e6a08c68b5fbb9ec1e07f89380d564410f93bd28660292862"},"expire":1701130834703},"doi:10.1103/PhysRevResearch.2.013303":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T12:20:19Z","timestamp":1669983619742},"reference-count":60,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[2020,3,12]],"date-time":"2020-03-12T00:00:00Z","timestamp":1583971200000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["CE170100009"]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevresearch.2.013303","type":"journal-article","created":{"date-parts":[[2020,3,13]],"date-time":"2020-03-13T14:41:59Z","timestamp":1584110519000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":23,"title":"Parallelized quantum error correction with fracton topological codes","prefix":"10.1103","volume":"2","author":[{"ORCID":"http://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"first","affiliation":[]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2020,3,12]]},"reference":[],"container-title":"Physical Review Research","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevResearch.2.013303","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevResearch.2.013303/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,13]],"date-time":"2020-03-13T14:42:02Z","timestamp":1584110522000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevResearch.2.013303"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,12]]},"references-count":60,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,3]]}},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.013303","relation":{},"ISSN":["2643-1564"],"subject":["General Engineering"],"container-title-short":"Phys. Rev. Research","published":{"date-parts":[[2020,3,12]]},"article-number":"013303","id":"doi:10.1103/PhysRevResearch.2.013303","_hash":"e3f03f3e4ba4a1f7bb4b760255d436e175644210f06c3956b59985e66d5285d9"},"expire":1701130835808},"doi:10.1038/s41467-022-32094-6":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T16:26:21Z","timestamp":1669998381874},"reference-count":72,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,8,9]],"date-time":"2022-08-09T00:00:00Z","timestamp":1660003200000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,8,9]],"date-time":"2022-08-09T00:00:00Z","timestamp":1660003200000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["OMA-2016136","OMA-2120757","OMA-2120757"]},{"DOI":"10.13039/100000183","name":"United States Department of Defense | United States Army | U.S. Army Research, Development and Engineering Command | Army Research Office","doi-asserted-by":"publisher","award":["W911NF-21-1-0012","W911NF-18-10215"]},{"DOI":"10.13039/100007297","name":"United States Department of Defense | United States Navy | ONR | Office of Naval Research Global","doi-asserted-by":"publisher","award":["N00014-20-1-2426"]},{"DOI":"10.13039/100000185","name":"United States Department of Defense | Defense Advanced Research Projects Agency","doi-asserted-by":"publisher","award":["W911NF-20-10021"]},{"DOI":"10.13039/100000879","name":"Alfred P. Sloan Foundation","doi-asserted-by":"publisher"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"<jats:title>Abstract</jats:title><jats:p>Executing quantum algorithms on error-corrected logical qubits is a critical step for scalable quantum computing, but the requisite numbers of qubits and physical error rates are demanding for current experimental hardware. Recently, the development of error correcting codes tailored to particular physical noise models has helped relax these requirements. In this work, we propose a qubit encoding and gate protocol for <jats:sup>171</jats:sup>Yb neutral atom qubits that converts the dominant physical errors into erasures, that is, errors in known locations. The key idea is to encode qubits in a metastable electronic level, such that gate errors predominantly result in transitions to disjoint subspaces whose populations can be continuously monitored via fluorescence. We estimate that 98% of errors can be converted into erasures. We quantify the benefit of this approach via circuit-level simulations of the surface code, finding a threshold increase from 0.937% to 4.15%. We also observe a larger code distance near the threshold, leading to a faster decrease in the logical error rate for the same number of physical qubits, which is important for near-term implementations. Erasure conversion should benefit any error correcting code, and may also be applied to design new gates and encodings in other qubit platforms.</jats:p>","DOI":"10.1038/s41467-022-32094-6","type":"journal-article","created":{"date-parts":[[2022,8,9]],"date-time":"2022-08-09T10:03:15Z","timestamp":1660039395000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":"Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays","prefix":"10.1038","volume":"13","author":[{"given":"Yue","family":"Wu","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-7095-1547","authenticated-orcid":false,"given":"Shimon","family":"Kolkowitz","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-5194-0591","authenticated-orcid":false,"given":"Shruti","family":"Puri","sequence":"additional","affiliation":[]},{"given":"Jeff D.","family":"Thompson","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,8,9]]},"reference":[],"container-title":"Nature Communications","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41467-022-32094-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-022-32094-6","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-022-32094-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T18:31:22Z","timestamp":1669314682000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41467-022-32094-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,9]]},"references-count":72,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["32094"],"URL":"http://dx.doi.org/10.1038/s41467-022-32094-6","relation":{},"ISSN":["2041-1723"],"subject":["General Physics and Astronomy","General Biochemistry, Genetics and Molecular Biology","General Chemistry","Multidisciplinary"],"container-title-short":"Nat Commun","published":{"date-parts":[[2022,8,9]]},"assertion":[{"value":"11 January 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 July 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 August 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"4657","id":"doi:10.1038/s41467-022-32094-6","_hash":"ff3ca173e53c88f92a182b4910339911fe10dba552b8f4d4770ac397cd002be0"},"expire":1701130836776},"doi:10.1103/PhysRevLett.124.130501":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T12:22:44Z","timestamp":1669983764930},"reference-count":35,"publisher":"American Physical Society (APS)","issue":"13","license":[{"start":{"date-parts":[[2020,3,30]],"date-time":"2020-03-30T00:00:00Z","timestamp":1585526400000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"funder":[{"DOI":"10.13039/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["CE170100009"]},{"DOI":"10.13039/501100001774","name":"University of Sydney","doi-asserted-by":"publisher"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevlett.124.130501","type":"journal-article","created":{"date-parts":[[2020,3,30]],"date-time":"2020-03-30T16:01:23Z","timestamp":1585584083000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":36,"title":"Fault-Tolerant Thresholds for the Surface Code in Excess of \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mn>5</mml:mn><mml:mo>%</mml:mo></mml:math>\n Under Biased Noise","prefix":"10.1103","volume":"124","author":[{"ORCID":"http://orcid.org/0000-0002-3776-2864","authenticated-orcid":true,"given":"David K.","family":"Tuckett","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4387-670X","authenticated-orcid":true,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-3975-0226","authenticated-orcid":true,"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2020,3,30]]},"reference":[],"container-title":"Physical Review Letters","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevLett.124.130501","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.124.130501/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,30]],"date-time":"2020-03-30T16:06:09Z","timestamp":1585584369000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevLett.124.130501"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,30]]},"references-count":35,"journal-issue":{"issue":"13","published-print":{"date-parts":[[2020,3]]}},"URL":"http://dx.doi.org/10.1103/PhysRevLett.124.130501","relation":{},"ISSN":["0031-9007","1079-7114"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. Rev. 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The challenge is to design practical quantum error-correcting codes that perform well against realistic noise using modest resources. Here we show that a variant of the surface code—the XZZX code—offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favourable sub-threshold resource scaling that can be obtained by specialising a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation.</jats:p>","DOI":"10.1038/s41467-021-22274-1","type":"journal-article","created":{"date-parts":[[2021,4,12]],"date-time":"2021-04-12T10:09:07Z","timestamp":1618222147000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":"The XZZX surface code","prefix":"10.1038","volume":"12","author":[{"ORCID":"http://orcid.org/0000-0001-5518-7907","authenticated-orcid":false,"given":"J. 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Call details: Consolidator Grant (CoG), PE2, ERC-2015-CoG"},{"DOI":"10.13039/100011039","name":"ODNI | Intelligence Advanced Research Projects Activity","doi-asserted-by":"publisher","award":["W911NF-16-1-0071"]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2020,12]]},"abstract":"<jats:title>Abstract</jats:title><jats:p>Leakage outside of the qubit computational subspace, present in many leading experimental platforms, constitutes a threatening error for quantum error correction (QEC) for qubits. We develop a leakage-detection scheme via Hidden Markov models (HMMs) for transmon-based implementations of the surface code. By performing realistic density-matrix simulations of the distance-3 surface code (Surface-17), we observe that leakage is sharply projected and leads to an increase in the surface-code defect probability of neighboring stabilizers. Together with the analog readout of the ancilla qubits, this increase enables the accurate detection of the time and location of leakage. We restore the logical error rate below the memory break-even point by post-selecting out leakage, discarding less than half of the data for the given noise parameters. Leakage detection via HMMs opens the prospect for near-term QEC demonstrations, targeted leakage reduction and leakage-aware decoding and is applicable to other experimental platforms.</jats:p>","DOI":"10.1038/s41534-020-00330-w","type":"journal-article","created":{"date-parts":[[2020,12,14]],"date-time":"2020-12-14T11:13:36Z","timestamp":1607944416000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":"Leakage detection for a transmon-based surface code","prefix":"10.1038","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0001-7124-8933","authenticated-orcid":false,"given":"Boris Mihailov","family":"Varbanov","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4800-2518","authenticated-orcid":false,"given":"Francesco","family":"Battistel","sequence":"additional","affiliation":[]},{"given":"Brian Michael","family":"Tarasinski","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-9892-3759","authenticated-orcid":false,"given":"Viacheslav Petrovych","family":"Ostroukh","sequence":"additional","affiliation":[]},{"given":"Thomas Eugene","family":"O’Brien","sequence":"additional","affiliation":[]},{"given":"Leonardo","family":"DiCarlo","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-0218-6614","authenticated-orcid":false,"given":"Barbara Maria","family":"Terhal","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,12,14]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"http://www.nature.com/articles/s41534-020-00330-w.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/s41534-020-00330-w","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/s41534-020-00330-w.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T16:52:59Z","timestamp":1638463979000},"score":1,"resource":{"primary":{"URL":"http://www.nature.com/articles/s41534-020-00330-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12]]},"references-count":55,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,12]]}},"alternative-id":["330"],"URL":"http://dx.doi.org/10.1038/s41534-020-00330-w","relation":{},"ISSN":["2056-6387"],"subject":["Computational Theory and Mathematics","Computer Networks and Communications","Statistical and Nonlinear Physics","Computer Science (miscellaneous)"],"container-title-short":"npj Quantum Inf","published":{"date-parts":[[2020,12]]},"assertion":[{"value":"6 March 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 October 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 December 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"102","id":"doi:10.1038/s41534-020-00330-w","_hash":"40192785a1c8ba60d1a57ef903462648dd3374155d4d987c4da0cd6aa8580913"},"expire":1701130843810},"doi:10.1038/s41534-017-0039-x":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T11:52:40Z","timestamp":1669981960917},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2017,9,25]],"date-time":"2017-09-25T00:00:00Z","timestamp":1506297600000},"content-version":"unspecified","delay-in-days":0,"URL":"http://creativecommons.org/licenses/by/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2017,12]]},"DOI":"10.1038/s41534-017-0039-x","type":"journal-article","created":{"date-parts":[[2017,9,18]],"date-time":"2017-09-18T14:28:20Z","timestamp":1505744900000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":48,"title":"Density-matrix simulation of small surface codes under current and projected experimental noise","prefix":"10.1038","volume":"3","author":[{"given":"T. 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In most state-of-the-art approaches, qubits interact locally, constrained by the connectivity associated with their fixed spatial layout. Here we demonstrate a quantum processor with dynamic, non-local connectivity, in which entangled qubits are coherently transported in a highly parallel manner across two spatial dimensions, between layers of single- and two-qubit operations. Our approach makes use of neutral atom arrays trapped and transported by optical tweezers; hyperfine states are used for robust quantum information storage, and excitation into Rydberg states is used for entanglement generation<jats:sup>3–5</jats:sup>. We use this architecture to realize programmable generation of entangled graph states, such as cluster states and a seven-qubit Steane code state<jats:sup>6,7</jats:sup>. Furthermore, we shuttle entangled ancilla arrays to realize a surface code state with thirteen data and six ancillary qubits<jats:sup>8</jats:sup> and a toric code state on a torus with sixteen data and eight ancillary qubits<jats:sup>9</jats:sup>. Finally, we use this architecture to realize a hybrid analogue–digital evolution<jats:sup>2</jats:sup> and use it for measuring entanglement entropy in quantum simulations<jats:sup>10–12</jats:sup>, experimentally observing non-monotonic entanglement dynamics associated with quantum many-body scars<jats:sup>13,14</jats:sup>. Realizing a long-standing goal, these results provide a route towards scalable quantum processing and enable applications ranging from simulation to metrology.</jats:p>","DOI":"10.1038/s41586-022-04592-6","type":"journal-article","created":{"date-parts":[[2022,4,20]],"date-time":"2022-04-20T16:03:41Z","timestamp":1650470621000},"page":"451-456","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":"A quantum processor based on coherent transport of entangled atom arrays","prefix":"10.1038","volume":"604","author":[{"given":"Dolev","family":"Bluvstein","sequence":"first","affiliation":[]},{"given":"Harry","family":"Levine","sequence":"additional","affiliation":[]},{"given":"Giulia","family":"Semeghini","sequence":"additional","affiliation":[]},{"given":"Tout T.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Sepehr","family":"Ebadi","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-0605-8791","authenticated-orcid":false,"given":"Marcin","family":"Kalinowski","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Keesling","sequence":"additional","affiliation":[]},{"given":"Nishad","family":"Maskara","sequence":"additional","affiliation":[]},{"given":"Hannes","family":"Pichler","sequence":"additional","affiliation":[]},{"given":"Markus","family":"Greiner","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-9786-0538","authenticated-orcid":false,"given":"Vladan","family":"Vuletić","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-8658-1007","authenticated-orcid":false,"given":"Mikhail D.","family":"Lukin","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,20]]},"reference":[],"container-title":"Nature","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41586-022-04592-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41586-022-04592-6","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41586-022-04592-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,21]],"date-time":"2022-04-21T13:21:12Z","timestamp":1650547272000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41586-022-04592-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,20]]},"references-count":75,"journal-issue":{"issue":"7906","published-print":{"date-parts":[[2022,4,21]]}},"alternative-id":["4592"],"URL":"http://dx.doi.org/10.1038/s41586-022-04592-6","relation":{},"ISSN":["0028-0836","1476-4687"],"subject":["Multidisciplinary"],"container-title-short":"Nature","published":{"date-parts":[[2022,4,20]]},"assertion":[{"value":"6 December 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 February 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 April 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"M.G., V.V. and M.D.L. are co-founders and shareholders of QuEra Computing. A.K. is an executive at and shareholder of QuEra Computing. All other authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"id":"doi:10.1038/s41586-022-04592-6","_hash":"797c58ffc8b7e194efc4f248f0ab36f269ac304df683cbddc6189ff28c6f87de"},"expire":1701130856132},"doi:10.1016/j.nuclphysb.2004.07.003":{"value":{"indexed":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T15:55:28Z","timestamp":1669823728488},"reference-count":22,"publisher":"Elsevier BV","issue":"3","license":[{"start":{"date-parts":[[2004,10,1]],"date-time":"2004-10-01T00:00:00Z","timestamp":1096588800000},"content-version":"tdm","delay-in-days":0,"URL":"https://www.elsevier.com/tdm/userlicense/1.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2004,10]]},"DOI":"10.1016/j.nuclphysb.2004.07.003","type":"journal-article","created":{"date-parts":[[2004,8,5]],"date-time":"2004-08-05T11:25:56Z","timestamp":1091705156000},"page":"462-480","source":"Crossref","is-referenced-by-count":38,"title":"Phase structure of the random-plaquette gauge model: accuracy threshold for a toric quantum memory","prefix":"10.1016","volume":"697","author":[{"given":"Takuya","family":"Ohno","sequence":"first","affiliation":[]},{"given":"Gaku","family":"Arakawa","sequence":"additional","affiliation":[]},{"given":"Ikuo","family":"Ichinose","sequence":"additional","affiliation":[]},{"given":"Tetsuo","family":"Matsui","sequence":"additional","affiliation":[]}],"member":"78","reference":[],"container-title":"Nuclear Physics B","original-title":[],"language":"en","link":[{"URL":"https://api.elsevier.com/content/article/PII:S055032130400481X?httpAccept=text/xml","content-type":"text/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https://api.elsevier.com/content/article/PII:S055032130400481X?httpAccept=text/plain","content-type":"text/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2019,2,4]],"date-time":"2019-02-04T22:21:45Z","timestamp":1549318905000},"score":1,"resource":{"primary":{"URL":"https://linkinghub.elsevier.com/retrieve/pii/S055032130400481X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,10]]},"references-count":22,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2004,10]]}},"alternative-id":["S055032130400481X"],"URL":"http://dx.doi.org/10.1016/j.nuclphysb.2004.07.003","relation":{},"ISSN":["0550-3213"],"subject":["Nuclear and High Energy Physics"],"container-title-short":"Nuclear Physics B","published":{"date-parts":[[2004,10]]},"id":"doi:10.1016/j.nuclphysb.2004.07.003","_hash":"1fb53b5c9e17ed07fd57a35577a4faf7c3419bbcb0adfb80fb217aa9d4268af0"},"expire":1701130856767},"doi:10.1103/PhysRevLett.102.200501":{"value":{"indexed":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T18:49:39Z","timestamp":1669315779691},"reference-count":21,"publisher":"American Physical Society (APS)","issue":"20","license":[{"start":{"date-parts":[[2009,5,18]],"date-time":"2009-05-18T00:00:00Z","timestamp":1242604800000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physrevlett.102.200501","type":"journal-article","created":{"date-parts":[[2009,5,18]],"date-time":"2009-05-18T18:24:41Z","timestamp":1242671081000},"source":"Crossref","is-referenced-by-count":77,"title":"Thresholds for Topological Codes in the Presence of Loss","prefix":"10.1103","volume":"102","author":[{"given":"Thomas M.","family":"Stace","sequence":"first","affiliation":[]},{"given":"Sean D.","family":"Barrett","sequence":"additional","affiliation":[]},{"given":"Andrew C.","family":"Doherty","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2009,5,18]]},"reference":[],"container-title":"Physical Review Letters","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevLett.102.200501","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.102.200501/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,18]],"date-time":"2017-06-18T19:59:39Z","timestamp":1497815979000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevLett.102.200501"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,5,18]]},"references-count":21,"journal-issue":{"issue":"20","published-print":{"date-parts":[[2009,5]]}},"URL":"http://dx.doi.org/10.1103/PhysRevLett.102.200501","relation":{},"ISSN":["0031-9007","1079-7114"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. 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Lett.","published":{"date-parts":[[2012,11,2]]},"article-number":"180502","id":"doi:10.1103/PhysRevLett.109.180502","_hash":"4dd839b473c728fa2f8dfe6d7a9cd6b2f6fdb80badf169673834cb48bdc4bcc9"},"expire":1701130868743},"doi:10.1103/PhysRevA.86.032324":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T22:50:14Z","timestamp":1670021414162},"reference-count":50,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2012,9,18]],"date-time":"2012-09-18T00:00:00Z","timestamp":1347926400000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.86.032324","type":"journal-article","created":{"date-parts":[[2012,9,21]],"date-time":"2012-09-21T20:23:30Z","timestamp":1348259010000},"source":"Crossref","is-referenced-by-count":1183,"title":"Surface codes: Towards practical large-scale quantum computation","prefix":"10.1103","volume":"86","author":[{"given":"Austin G.","family":"Fowler","sequence":"first","affiliation":[]},{"given":"Matteo","family":"Mariantoni","sequence":"additional","affiliation":[]},{"given":"John M.","family":"Martinis","sequence":"additional","affiliation":[]},{"given":"Andrew N.","family":"Cleland","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2012,9,18]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.86.032324","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.86.032324/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,7,4]],"date-time":"2019-07-04T04:09:16Z","timestamp":1562213356000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.86.032324"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,9,18]]},"references-count":50,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,9]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.86.032324","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. Rev. A","published":{"date-parts":[[2012,9,18]]},"article-number":"032324","id":"doi:10.1103/PhysRevA.86.032324","_hash":"74ffac12650d924b8b632407ecd96716bedf02e73798868fa56c7374d99ca7b2"},"expire":1701130869808},"doi:10.22331/q-2022-05-24-721":{"value":{"indexed":{"date-parts":[[2022,10,21]],"date-time":"2022-10-21T18:11:42Z","timestamp":1666375902060},"reference-count":50,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T00:00:00Z","timestamp":1653350400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"University College London and the Engineering and Physical Sciences Research Council","award":["EP/L015242/1"]},{"name":"Australian Research Council via the Centre of Excellence in Engineered Quantum Systems","award":["CE170100009"]},{"DOI":"10.13039/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"crossref","award":["EP/R043647/1"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>We build on recent work by B. Brown (Sci. Adv. 6, eaay4929 (2020)) to develop and simulate an explicit recipe for a just-in-time decoding scheme in three 3D surface codes, which can be used to implement a transversal (non-Clifford) <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mover><mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi></mml:mrow><mml:mo accent=\"false\">&amp;#x00AF;</mml:mo></mml:mover></mml:math> between three 2D surface codes in time linear in the code distance. We present a fully detailed set of bounded-height lattice slices through the 3D codes which retain the code distance and measurement-error detecting properties of the full 3D code and admit a dimension-jumping process which expands from/collapses to 2D surface codes supported on the boundaries of each slice. At each timestep of the procedure the slices agree on a common set of overlapping qubits on which <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi></mml:math> should be applied. We use these slices to simulate the performance of a simple JIT decoder against stochastic <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi></mml:math> and measurement errors and find evidence for a threshold <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>p</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>&amp;#x223C;</mml:mo><mml:mn>0.1</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> in all three codes. We expect that this threshold could be improved by optimisation of the decoder.</jats:p>","DOI":"10.22331/q-2022-05-24-721","type":"journal-article","created":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T12:33:35Z","timestamp":1653395615000},"page":"721","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":"Numerical Implementation of Just-In-Time Decoding in Novel Lattice Slices Through the Three-Dimensional Surface Code","prefix":"10.22331","volume":"6","author":[{"given":"T. R.","family":"Scruby","sequence":"first","affiliation":[{"name":"Okinawa Institute of Science and Technology, Okinawa, 904-0495, Japan"},{"name":"Dept. of Physics and Astronomy, University College London, London, WC1E 6BT, UK"}]},{"given":"D. E.","family":"Browne","sequence":"additional","affiliation":[{"name":"Dept. of Physics and Astronomy, University College London, London, WC1E 6BT, UK"}]},{"given":"P.","family":"Webster","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia"}]},{"given":"M.","family":"Vasmer","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada"},{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada"}]}],"member":"9598","published-online":{"date-parts":[[2022,5,24]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-05-24-721/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T12:33:45Z","timestamp":1653395625000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-05-24-721/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,24]]},"references-count":50,"URL":"http://dx.doi.org/10.22331/q-2022-05-24-721","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,5,24]]},"article-number":"721","id":"doi:10.22331/q-2022-05-24-721","_hash":"e8cecb316100746bebb54ca43a1cff72dcf29b65b8ac192e5c1094f6743b88af"},"expire":1701130870968},"doi:10.1126/sciadv.aay4929":{"value":{"indexed":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T08:31:49Z","timestamp":1669797109931},"reference-count":41,"publisher":"American Association for the Advancement of Science (AAAS)","issue":"21","funder":[{"name":"Australian Research Council via the Centre of Excellence in Engineered Quantum System","award":["CE170100009"]},{"name":"University of Sydney Fellowship Programme"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2020,5,22]]},"abstract":"<jats:p>We complete a universal set of fault-tolerant quantum logic gates for a two-dimensional surface code architecture.</jats:p>","DOI":"10.1126/sciadv.aay4929","type":"journal-article","created":{"date-parts":[[2020,5,22]],"date-time":"2020-05-22T23:08:48Z","timestamp":1590188928000},"source":"Crossref","is-referenced-by-count":27,"title":"A fault-tolerant non-Clifford gate for the surface code in two dimensions","prefix":"10.1126","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia."}]}],"member":"221","reference":[],"container-title":"Science Advances","original-title":[],"language":"en","link":[{"URL":"https://syndication.highwire.org/content/doi/10.1126/sciadv.aay4929","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,14]],"date-time":"2022-01-14T01:21:12Z","timestamp":1642123272000},"score":1,"resource":{"primary":{"URL":"https://www.science.org/doi/10.1126/sciadv.aay4929"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,22]]},"references-count":41,"journal-issue":{"issue":"21","published-print":{"date-parts":[[2020,5,22]]}},"alternative-id":["10.1126/sciadv.aay4929"],"URL":"http://dx.doi.org/10.1126/sciadv.aay4929","relation":{},"ISSN":["2375-2548"],"subject":["Multidisciplinary"],"container-title-short":"Sci. Adv.","published":{"date-parts":[[2020,5,22]]},"id":"doi:10.1126/sciadv.aay4929","_hash":"e953ac20ee6cc8a14d5fbae4d4c6914fe5e1903efe7675bcbd5e2322118b04c4"},"expire":1701130871846},"doi:10.1103/PhysRevA.94.042316":{"value":{"indexed":{"date-parts":[[2022,10,26]],"date-time":"2022-10-26T11:31:25Z","timestamp":1666783885119},"reference-count":40,"publisher":"American Physical Society (APS)","issue":"4","license":[{"start":{"date-parts":[[2016,10,12]],"date-time":"2016-10-12T00:00:00Z","timestamp":1476230400000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"},{"start":{"date-parts":[[2017,10,12]],"date-time":"2017-10-12T00:00:00Z","timestamp":1507766400000},"content-version":"am","delay-in-days":365,"URL":"http://link.aps.org/licenses/aps-default-accepted-manuscript-license"}],"funder":[{"DOI":"10.13039/100006234","name":"Sandia National Laboratories","doi-asserted-by":"publisher"},{"DOI":"10.13039/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-AC04-94AL85000"]},{"DOI":"10.13039/100006168","name":"National Nuclear Security Administration","doi-asserted-by":"publisher"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.94.042316","type":"journal-article","created":{"date-parts":[[2016,10,12]],"date-time":"2016-10-12T22:08:25Z","timestamp":1476310105000},"source":"Crossref","is-referenced-by-count":11,"title":"Transversal Clifford gates on folded surface codes","prefix":"10.1103","volume":"94","author":[{"given":"Jonathan E.","family":"Moussa","sequence":"first","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2016,10,12]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/accepted/10.1103/PhysRevA.94.042316","content-type":"application/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"http://link.aps.org/article/10.1103/PhysRevA.94.042316","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.94.042316/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,4,5]],"date-time":"2017-04-05T17:27:46Z","timestamp":1491413266000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.94.042316"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,10,12]]},"references-count":40,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2016,10]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.94.042316","relation":{},"ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","published":{"date-parts":[[2016,10,12]]},"article-number":"042316","id":"doi:10.1103/PhysRevA.94.042316","_hash":"a3fd2d44dc9830e22fa1f732603c0ab7f0a0aae1d1f39947528394c57476d102"},"expire":1701130872814},"doi:10.22331/q-2021-12-02-595":{"value":{"indexed":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T15:36:34Z","timestamp":1669822594294},"reference-count":68,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T00:00:00Z","timestamp":1638403200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>In order to build a large scale quantum computer, one must be able to correct errors extremely fast. We design a fast decoding algorithm for topological codes to correct for Pauli errors and erasure and combination of both errors and erasure. Our algorithm has a worst case complexity of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math>, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math> is the number of physical qubits and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>α</mml:mi></mml:math> is the inverse of Ackermann's function, which is very slowly growing. For all practical purposes, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≤</mml:mo><mml:mn>3</mml:mn></mml:math>. We prove that our algorithm performs optimally for errors of weight up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>d</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:math> and for loss of up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math> qubits, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> is the minimum distance of the code. Numerically, we obtain a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>9.9</mml:mn><mml:mi mathvariant=\"normal\">%</mml:mi></mml:math> for the 2d-toric code with perfect syndrome measurements and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2.6</mml:mn><mml:mi mathvariant=\"normal\">%</mml:mi></mml:math> with faulty measurements.</jats:p>","DOI":"10.22331/q-2021-12-02-595","type":"journal-article","created":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T16:07:00Z","timestamp":1638461220000},"page":"595","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":14,"title":"Almost-linear time decoding algorithm for topological codes","prefix":"10.22331","volume":"5","author":[{"given":"Nicolas","family":"Delfosse","sequence":"first","affiliation":[{"name":"IQIM, California Institute of Technology, Pasadena, CA, USA"},{"name":"Department of Physics and Astronomy, University of California, Riverside, CA, USA"},{"name":"Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Naomi H.","family":"Nickerson","sequence":"additional","affiliation":[{"name":"Quantum Optics and Laser Science, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom"}]}],"member":"9598","published-online":{"date-parts":[[2021,12,2]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2021-12-02-595/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T16:07:08Z","timestamp":1638461228000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2021-12-02-595/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,2]]},"references-count":68,"URL":"http://dx.doi.org/10.22331/q-2021-12-02-595","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2021,12,2]]},"article-number":"595","id":"doi:10.22331/q-2021-12-02-595","_hash":"05711d600aa93726e31fefce01d82984d2c9f7f2ae49d049c5b1470a3c1cf2ae"},"expire":1701130873812},"doi:10.1088/2632-2153/abc609":{"value":{"indexed":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T19:20:55Z","timestamp":1669317655037},"reference-count":69,"publisher":"IOP Publishing","issue":"2","license":[{"start":{"date-parts":[[2020,12,28]],"date-time":"2020-12-28T00:00:00Z","timestamp":1609113600000},"content-version":"vor","delay-in-days":0,"URL":"http://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2020,12,28]],"date-time":"2020-12-28T00:00:00Z","timestamp":1609113600000},"content-version":"tdm","delay-in-days":0,"URL":"https://iopscience.iop.org/info/page/text-and-data-mining"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2021,6,1]]},"abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Topological error correcting codes, and particularly the surface code, currently provide the most feasible road-map towards large-scale fault-tolerant quantum computation. As such, obtaining fast and flexible decoding algorithms for these codes, within the experimentally realistic and challenging context of faulty syndrome measurements, without requiring any final read-out of the physical qubits, is of critical importance. In this work, we show that the problem of decoding such codes can be naturally reformulated as a process of repeated interactions between a decoding agent and a code environment, to which the machinery of reinforcement learning can be applied to obtain decoding agents. 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In this paper, we discuss strategies for surface-code quantum computing on small, intermediate and large scales. They are strategies for space-time trade-offs, going from slow computations using few qubits to fast computations using many qubits. Our schemes are based on surface-code patches, which not only feature a low space cost compared to other surface-code schemes, but are also conceptually simple~--~simple enough that they can be described as a tile-based game with a small set of rules. 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Bravyi et al. (2006) showed that encoding a state in the surface code using local unitary operations requires time at least linear in the lattice size <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>L</mml:mi></mml:math>, however the most efficient known method for encoding an unknown state, introduced by Dennis et al. (2002), has <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math> time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>L</mml:mi></mml:math> time steps to encode an unknown state in a distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>L</mml:mi></mml:math> planar code. We further show how an <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> complexity local unitary encoder for the toric code can be found by enforcing locality in the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math>-depth non-local renormalisation encoder. We relate these techniques by providing an <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> local unitary circuit to convert between a toric code and a planar code, and also provide optimal encoders for the rectangular, rotated and 3D surface codes. Furthermore, we show how our encoding circuit for the planar code can be used to prepare fermionic states in the compact mapping, a recently introduced fermion to qubit mapping that has a stabiliser structure similar to that of the surface code and is particularly efficient for simulating the Fermi-Hubbard model.</jats:p>","DOI":"10.22331/q-2021-08-05-517","type":"journal-article","created":{"date-parts":[[2021,8,5]],"date-time":"2021-08-05T19:19:03Z","timestamp":1628191143000},"page":"517","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":"Optimal local unitary encoding circuits for the surface code","prefix":"10.22331","volume":"5","author":[{"given":"Oscar","family":"Higgott","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"}]},{"given":"Matthew","family":"Wilson","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"},{"name":"Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom"}]},{"given":"James","family":"Hefford","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"},{"name":"Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom"}]},{"given":"James","family":"Dborin","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"},{"name":"London Centre for Nanotechnology, University College London, Gordon St., London WC1H 0AH, United Kingdom"}]},{"given":"Farhan","family":"Hanif","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"}]},{"given":"Simon","family":"Burton","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"}]},{"given":"Dan E.","family":"Browne","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"}]}],"member":"9598","published-online":{"date-parts":[[2021,8,5]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2021-08-05-517/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2021,8,19]],"date-time":"2021-08-19T16:13:34Z","timestamp":1629389614000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2021-08-05-517/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,5]]},"references-count":57,"URL":"http://dx.doi.org/10.22331/q-2021-08-05-517","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2021,8,5]]},"article-number":"517","id":"doi:10.22331/q-2021-08-05-517","_hash":"e7fa6105f83455c0425ea135567b797934e877954c633e8d2e61cffc5a39c3d6"},"expire":1701130895827},"doi:10.1103/PhysRevLett.104.050503":{"value":{"indexed":{"date-parts":[[2022,10,26]],"date-time":"2022-10-26T11:31:09Z","timestamp":1666783869708},"reference-count":20,"publisher":"American Physical Society (APS)","issue":"5","license":[{"start":{"date-parts":[[2010,2,5]],"date-time":"2010-02-05T00:00:00Z","timestamp":1265328000000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physrevlett.104.050503","type":"journal-article","created":{"date-parts":[[2010,3,19]],"date-time":"2010-03-19T17:17:32Z","timestamp":1269019052000},"source":"Crossref","is-referenced-by-count":79,"title":"Tradeoffs for Reliable Quantum Information Storage in 2D Systems","prefix":"10.1103","volume":"104","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]},{"given":"Barbara","family":"Terhal","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2010,2,5]]},"reference":[],"container-title":"Physical Review 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Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In particular, we show that for the toric code subject to such independent coherent noise, and for minimal-weight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and fault-tolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is held constant as the code block grows, and we recount the difficulties that prevented us from extending the result to that case. Nevertheless our work supports the idea that fault-tolerant quantum computing schemes will work effectively against coherent noise, providing encouraging news for quantum hardware builders who worry about the damaging effects of control errors and coherent interactions with the environment.</jats:p>","DOI":"10.1088/1367-2630/ab8e5c","type":"journal-article","created":{"date-parts":[[2020,4,29]],"date-time":"2020-04-29T22:24:15Z","timestamp":1588199055000},"page":"073066","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":13,"title":"Coherence in logical quantum channels","prefix":"10.1088","volume":"22","author":[{"ORCID":"http://orcid.org/0000-0003-4665-8839","authenticated-orcid":false,"given":"Joseph K","family":"Iverson","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-2421-4762","authenticated-orcid":false,"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2020,7,31]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c","content-type":"text/html","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,27]],"date-time":"2021-11-27T16:44:20Z","timestamp":1638031460000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,1]]},"references-count":37,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7,31]]},"published-print":{"date-parts":[[2020,7,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/ab8e5c","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. 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Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft","name":"copyright_information","label":"Copyright Information"},{"value":"2020-02-22","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-04-29","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-07-31","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/ab8e5c","_hash":"d7cba2a02c85835b48c99a33621b6d8e5a5a5255521194295cd3eaec662d4b5a"},"expire":1701130898838},"doi:10.1103/PhysRevX.10.011022":{"value":{"indexed":{"date-parts":[[2022,11,26]],"date-time":"2022-11-26T05:28:26Z","timestamp":1669440506295},"reference-count":43,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[2020,1,31]],"date-time":"2020-01-31T00:00:00Z","timestamp":1580428800000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevx.10.011022","type":"journal-article","created":{"date-parts":[[2020,1,31]],"date-time":"2020-01-31T15:54:25Z","timestamp":1580486065000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":63,"title":"Topological and Subsystem Codes on Low-Degree Graphs with Flag Qubits","prefix":"10.1103","volume":"10","author":[{"ORCID":"http://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-3375-4445","authenticated-orcid":true,"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-9614-2836","authenticated-orcid":true,"given":"Theodore J.","family":"Yoder","sequence":"additional","affiliation":[]},{"given":"Jared B.","family":"Hertzberg","sequence":"additional","affiliation":[]},{"given":"Andrew W.","family":"Cross","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2020,1,31]]},"reference":[],"container-title":"Physical Review X","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevX.10.011022","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevX.10.011022/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,31]],"date-time":"2020-01-31T15:55:37Z","timestamp":1580486137000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevX.10.011022"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,31]]},"references-count":43,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,1]]}},"URL":"http://dx.doi.org/10.1103/PhysRevX.10.011022","relation":{},"ISSN":["2160-3308"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. 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Compared to independent depolarizing noise, leaked qubits may produce many more configurations of harmful correlated errors during error-correction. In this work, we investigate different local codes in the low-error regime of a leakage gate error model. When restricting to bare-ancilla extraction, we observe that subsystem codes are good candidates for handling leakage, as their locality can limit damaging correlated errors. As a case study, we compare subspace surface codes to the subsystem surface codes introduced by Bravyi <jats:italic>et al</jats:italic>. In contrast to depolarizing noise, subsystem surface codes outperform same-distance subspace surface codes below error rates as high as ⪅ 7.5 × 10<jats:sup>−4</jats:sup> while offering better per-qubit distance protection. Furthermore, we show that at low to intermediate distances, Bacon–Shor codes offer better per-qubit error protection against leakage in an ion-trap motivated error model below error rates as high as ⪅ 1.2 × 10<jats:sup>−3</jats:sup>. For restricted leakage models, this advantage can be extended to higher distances by relaxing to unverified two-qubit cat state extraction in the surface code. These results highlight an intrinsic benefit of subsystem code locality to error-corrective performance.</jats:p>","DOI":"10.1088/1367-2630/ab3372","type":"journal-article","created":{"date-parts":[[2019,7,18]],"date-time":"2019-07-18T17:46:25Z","timestamp":1563471985000},"page":"073055","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":15,"title":"Handling leakage with subsystem codes","prefix":"10.1088","volume":"21","author":[{"given":"Natalie C","family":"Brown","sequence":"first","affiliation":[]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[]},{"given":"Kenneth R","family":"Brown","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2019,7,30]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"link":[{"URL":"http://iopscience.iop.org/article/10.1088/1367-2630/ab3372","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"http://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372","content-type":"text/html","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,18]],"date-time":"2021-11-18T22:38:02Z","timestamp":1637275082000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,1]]},"references-count":102,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2019,7,30]]},"published-print":{"date-parts":[[2019,7,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/ab3372","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. 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Attempts to process and manipulate quantum states can destroy the encoded information. Nigg\n            <jats:italic>et al.</jats:italic>\n            encoded the quantum state of a single qubit (in this case, a trapped ion) over the global properties of a series of trapped ions. These so-called stabilizers protected the information against noise sources that can degrade the single qubit. The protocol provides a route to fault-tolerant quantum computing.\n          </jats:p>\n          <jats:p>\n            <jats:italic>Science</jats:italic>\n            , this issue p.\n            <jats:related-article xmlns:xlink=\"http://www.w3.org/1999/xlink\" ext-link-type=\"doi\" issue=\"6194\" page=\"302\" related-article-type=\"in-this-issue\" vol=\"345\" xlink:href=\"10.1126/science.1253742\">302</jats:related-article>\n          </jats:p>","DOI":"10.1126/science.1253742","type":"journal-article","created":{"date-parts":[[2014,6,13]],"date-time":"2014-06-13T07:43:13Z","timestamp":1402645393000},"page":"302-305","source":"Crossref","is-referenced-by-count":255,"title":"Quantum computations on a topologically encoded qubit","prefix":"10.1126","volume":"345","author":[{"given":"D.","family":"Nigg","sequence":"first","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."}]},{"given":"M.","family":"Müller","sequence":"additional","affiliation":[{"name":"Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain."}]},{"given":"E. A.","family":"Martinez","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."}]},{"given":"P.","family":"Schindler","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."}]},{"given":"M.","family":"Hennrich","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."}]},{"given":"T.","family":"Monz","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."}]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[{"name":"Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain."}]},{"given":"R.","family":"Blatt","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."},{"name":"Institut für Quantenoptik und Quanteninformation der Österreichischen Akademie der Wissenschaften, A-6020 Innsbruck, Austria."}]}],"member":"221","reference":[],"container-title":"Science","original-title":[],"language":"en","link":[{"URL":"https://syndication.highwire.org/content/doi/10.1126/science.1253742","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,14]],"date-time":"2022-01-14T08:38:13Z","timestamp":1642149493000},"score":1,"resource":{"primary":{"URL":"https://www.science.org/doi/10.1126/science.1253742"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,7,18]]},"references-count":28,"journal-issue":{"issue":"6194","published-print":{"date-parts":[[2014,7,18]]}},"alternative-id":["10.1126/science.1253742"],"URL":"http://dx.doi.org/10.1126/science.1253742","relation":{},"ISSN":["0036-8075","1095-9203"],"subject":["Multidisciplinary"],"container-title-short":"Science","published":{"date-parts":[[2014,7,18]]},"id":"doi:10.1126/science.1253742","_hash":"3d42a5648eb362e7ec14a2d66488f0a194a4bfb508fbffb7bd9c94aeef20eb2e"},"expire":1701130939919},"doi:10.1103/PhysRevLett.121.050502":{"value":{"indexed":{"date-parts":[[2022,12,1]],"date-time":"2022-12-01T18:26:05Z","timestamp":1669919165484},"reference-count":24,"publisher":"American Physical Society (APS)","issue":"5","license":[{"start":{"date-parts":[[2018,8,1]],"date-time":"2018-08-01T00:00:00Z","timestamp":1533081600000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"},{"start":{"date-parts":[[2019,8,1]],"date-time":"2019-08-01T00:00:00Z","timestamp":1564617600000},"content-version":"am","delay-in-days":365,"URL":"https://link.aps.org/licenses/aps-default-accepted-manuscript-license"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1254119"]},{"DOI":"10.13039/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["W911NF-12-1-0541"]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevlett.121.050502","type":"journal-article","created":{"date-parts":[[2018,8,1]],"date-time":"2018-08-01T14:01:31Z","timestamp":1533132091000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":78,"title":"Quantum Error Correction with Only Two Extra Qubits","prefix":"10.1103","volume":"121","author":[{"given":"Rui","family":"Chao","sequence":"first","affiliation":[]},{"given":"Ben W.","family":"Reichardt","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2018,8,1]]},"reference":[],"container-title":"Physical Review Letters","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/accepted/10.1103/PhysRevLett.121.050502","content-type":"application/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"https://link.aps.org/article/10.1103/PhysRevLett.121.050502","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.121.050502/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,28]],"date-time":"2022-08-28T16:00:00Z","timestamp":1661702400000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevLett.121.050502"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,8,1]]},"references-count":24,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2018,8]]}},"URL":"http://dx.doi.org/10.1103/PhysRevLett.121.050502","relation":{},"ISSN":["0031-9007","1079-7114"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. Rev. Lett.","published":{"date-parts":[[2018,8,1]]},"article-number":"050502","id":"doi:10.1103/PhysRevLett.121.050502","_hash":"88832226f861bba82985d4694b4abf513e1dbfa2102a8c91ed97549812e0a806"},"expire":1701130940936},"doi:10.22331/q-2017-04-25-2":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T12:21:28Z","timestamp":1669983688752},"reference-count":45,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2017,4,25]],"date-time":"2017-04-25T00:00:00Z","timestamp":1493078400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"<jats:p>The surface code is one of the most successful approaches to topological quantum error-correction. It boasts the smallest known syndrome extraction circuits and correspondingly largest thresholds. Defect-based logical encodings of a new variety called twists have made it possible to implement the full Clifford group without state distillation. Here we investigate a patch-based encoding involving a modified twist. In our modified formulation, the resulting codes, called triangle codes for the shape of their planar layout, have only weight-four checks and relatively simple syndrome extraction circuits that maintain a high, near surface-code-level threshold. They also use 25% fewer physical qubits per logical qubit than the surface code. Moreover, benefiting from the twist, we can implement all Clifford gates by lattice surgery without the need for state distillation. By a surgical transformation to the surface code, we also develop a scheme of doing all Clifford gates on surface code patches in an atypical planar layout, though with less qubit efficiency than the triangle code. Finally, we remark that logical qubits encoded in triangle codes are naturally amenable to logical tomography, and the smallest triangle code can demonstrate high-pseudothreshold fault-tolerance to depolarizing noise using just 13 physical qubits.</jats:p>","DOI":"10.22331/q-2017-04-25-2","type":"journal-article","created":{"date-parts":[[2017,4,25]],"date-time":"2017-04-25T08:44:49Z","timestamp":1493109889000},"page":"2","source":"Crossref","is-referenced-by-count":57,"title":"The surface code with a twist","prefix":"10.22331","volume":"1","author":[{"ORCID":"http://orcid.org/0000-0001-9614-2836","authenticated-orcid":false,"given":"Theodore J.","family":"Yoder","sequence":"first","affiliation":[{"name":"Department of Physics, Massachusetts Institute of Technology"}]},{"given":"Isaac H.","family":"Kim","sequence":"additional","affiliation":[{"name":"IBM, Thomas J. Watson Research Center"}]}],"member":"9598","published-online":{"date-parts":[[2017,4,25]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","deposited":{"date-parts":[[2017,6,7]],"date-time":"2017-06-07T12:30:04Z","timestamp":1496838604000},"score":1,"resource":{"primary":{"URL":"http://quantum-journal.org/papers/q-2017-04-25-2/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,25]]},"references-count":45,"URL":"http://dx.doi.org/10.22331/q-2017-04-25-2","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2017,4,25]]},"article-number":"2","id":"doi:10.22331/q-2017-04-25-2","_hash":"7194ed8f454d3aa458d6dc260ccd4d7b836910a84e12f13b8fc08da46921d1ce"},"expire":1701130941933},"doi:10.1103/PhysRevX.6.031039":{"value":{"indexed":{"date-parts":[[2022,7,8]],"date-time":"2022-07-08T11:11:59Z","timestamp":1657278719949},"reference-count":44,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2016,9,13]],"date-time":"2016-09-13T00:00:00Z","timestamp":1473724800000},"content-version":"vor","delay-in-days":0,"URL":"http://creativecommons.org/licenses/by/3.0/"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1111337"]},{"DOI":"10.13039/100000183","name":"Army Research Office","doi-asserted-by":"publisher"},{"DOI":"10.13039/100014037","name":"National Defense Science and Engineering Graduate","doi-asserted-by":"crossref"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physrevx.6.031039","type":"journal-article","created":{"date-parts":[[2016,9,13]],"date-time":"2016-09-13T22:08:34Z","timestamp":1473804514000},"source":"Crossref","is-referenced-by-count":33,"title":"Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes","prefix":"10.1103","volume":"6","author":[{"given":"Theodore J.","family":"Yoder","sequence":"first","affiliation":[]},{"given":"Ryuji","family":"Takagi","sequence":"additional","affiliation":[]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2016,9,13]]},"reference":[],"container-title":"Physical Review X","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevX.6.031039","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevX.6.031039/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,7,8]],"date-time":"2022-07-08T10:44:03Z","timestamp":1657277043000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevX.6.031039"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,9,13]]},"references-count":44,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2016,9]]}},"URL":"http://dx.doi.org/10.1103/PhysRevX.6.031039","relation":{},"ISSN":["2160-3308"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. Rev. X","published":{"date-parts":[[2016,9,13]]},"article-number":"031039","id":"doi:10.1103/PhysRevX.6.031039","_hash":"66678c69609fa6dd75baebe3ea8409918169e03a642fdc2f3d016847034678d9"},"expire":1701130943019},"doi:10.1038/s41586-022-04819-6":{"value":{"indexed":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T08:31:41Z","timestamp":1669797101488},"reference-count":52,"publisher":"Springer Science and Business Media LLC","issue":"7916","license":[{"start":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T00:00:00Z","timestamp":1651708800000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T00:00:00Z","timestamp":1651708800000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2022,6,30]]},"abstract":"<jats:title>Abstract</jats:title><jats:p>Solid-state spin qubits is a promising platform for quantum computation and quantum networks<jats:sup>1,2</jats:sup>. Recent experiments have demonstrated high-quality control over multi-qubit systems<jats:sup>3–8</jats:sup>, elementary quantum algorithms<jats:sup>8–11</jats:sup> and non-fault-tolerant error correction<jats:sup>12–14</jats:sup>. Large-scale systems will require using error-corrected logical qubits that are operated fault tolerantly, so that reliable computation becomes possible despite noisy operations<jats:sup>15–18</jats:sup>. Overcoming imperfections in this way remains an important outstanding challenge for quantum science<jats:sup>15,19–27</jats:sup>. Here, we demonstrate fault-tolerant operations on a logical qubit using spin qubits in diamond. Our approach is based on the five-qubit code with a recently discovered flag protocol that enables fault tolerance using a total of seven qubits<jats:sup>28–30</jats:sup>. We encode the logical qubit using a new protocol based on repeated multi-qubit measurements and show that it outperforms non-fault-tolerant encoding schemes. We then fault-tolerantly manipulate the logical qubit through a complete set of single-qubit Clifford gates. Finally, we demonstrate flagged stabilizer measurements with real-time processing of the outcomes. Such measurements are a primitive for fault-tolerant quantum error correction. Although future improvements in fidelity and the number of qubits will be required to suppress logical error rates below the physical error rates, our realization of fault-tolerant protocols on the logical-qubit level is a key step towards quantum information processing based on solid-state spins.</jats:p>","DOI":"10.1038/s41586-022-04819-6","type":"journal-article","created":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T16:08:44Z","timestamp":1651766924000},"page":"884-889","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":"Fault-tolerant operation of a logical qubit in a diamond quantum processor","prefix":"10.1038","volume":"606","author":[{"ORCID":"http://orcid.org/0000-0001-8205-8166","authenticated-orcid":false,"given":"M. H.","family":"Abobeih","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-5483-0339","authenticated-orcid":false,"given":"Y.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"J.","family":"Randall","sequence":"additional","affiliation":[]},{"given":"S. J. H.","family":"Loenen","sequence":"additional","affiliation":[]},{"given":"C. E.","family":"Bradley","sequence":"additional","affiliation":[]},{"given":"M.","family":"Markham","sequence":"additional","affiliation":[]},{"given":"D. J.","family":"Twitchen","sequence":"additional","affiliation":[]},{"given":"B. M.","family":"Terhal","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-2355-727X","authenticated-orcid":false,"given":"T. H.","family":"Taminiau","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,5,5]]},"reference":[],"container-title":"Nature","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41586-022-04819-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41586-022-04819-6","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41586-022-04819-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,29]],"date-time":"2022-06-29T16:14:20Z","timestamp":1656519260000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41586-022-04819-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,5]]},"references-count":52,"journal-issue":{"issue":"7916","published-print":{"date-parts":[[2022,6,30]]}},"alternative-id":["4819"],"URL":"http://dx.doi.org/10.1038/s41586-022-04819-6","relation":{},"ISSN":["0028-0836","1476-4687"],"subject":["Multidisciplinary"],"container-title-short":"Nature","published":{"date-parts":[[2022,5,5]]},"assertion":[{"value":"10 August 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 April 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 May 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"id":"doi:10.1038/s41586-022-04819-6","_hash":"feb8f5e8b0491c6decae984a614e15edb37fb7eb77324f14d77503509c701387"},"expire":1701130943942},"doi:10.1093/nsr/nwab011":{"value":{"indexed":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T10:27:20Z","timestamp":1669372040640},"reference-count":31,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T00:00:00Z","timestamp":1611187200000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["21931001","21871121","21971097"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2022,1,19]]},"abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error-correcting code, with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state and state decoding. To address this challenge, we experimentally realise the [5, 1, 3] code, the so-called smallest perfect code that permits corrections of generic single-qubit errors. In the experiment, having optimised the encoding circuit, we employ an array of superconducting qubits to realise the [5, 1, 3] code for several typical logical states including the magic state, an indispensable resource for realising non-Clifford gates. The encoded states are prepared with an average fidelity of $57.1(3)\\%$ while with a high fidelity of $98.6(1)\\%$ in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilisers. We further implement logical Pauli operations with a fidelity of $97.2(2)\\%$ within the code space. Finally, we realise the decoding circuit and recover the input state with an overall fidelity of $74.5(6)\\%$, in total with 92 gates. Our work demonstrates each key aspect of the [5, 1, 3] code and verifies the viability of experimental realisation of quantum error-correcting codes with superconducting qubits.</jats:p>","DOI":"10.1093/nsr/nwab011","type":"journal-article","created":{"date-parts":[[2021,1,15]],"date-time":"2021-01-15T12:20:13Z","timestamp":1610713213000},"source":"Crossref","is-referenced-by-count":12,"title":"Experimental exploration of five-qubit quantum error-correcting code with superconducting qubits","prefix":"10.1093","volume":"9","author":[{"given":"Ming","family":"Gong","sequence":"first","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Xiao","family":"Yuan","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University , Beijing 100084, China"},{"name":"Department of Materials, University of Oxford , Oxford OX1 3PH, UK"}]},{"given":"Shiyu","family":"Wang","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Yulin","family":"Wu","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Youwei","family":"Zhao","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Chen","family":"Zha","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Shaowei","family":"Li","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Zhen","family":"Zhang","sequence":"additional","affiliation":[{"name":"Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University , Beijing 100084, China"}]},{"given":"Qi","family":"Zhao","sequence":"additional","affiliation":[{"name":"Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University , Beijing 100084, China"}]},{"given":"Yunchao","family":"Liu","sequence":"additional","affiliation":[{"name":"Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University , Beijing 100084, China"}]},{"given":"Futian","family":"Liang","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Jin","family":"Lin","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum 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Many applications call for error rates as low as 10<jats:sup>−15</jats:sup> (refs. <jats:sup>2–9</jats:sup>), but state-of-the-art quantum platforms typically have physical error rates near 10<jats:sup>−3</jats:sup> (refs. <jats:sup>10–14</jats:sup>). Quantum error correction<jats:sup>15–17</jats:sup> promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device<jats:sup>18,19</jats:sup> and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. 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This error has been corrected online.","order":6,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing financial interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"id":"doi:10.1038/nature23460","_hash":"f91184c126fccbc0c2acff16893e1ad33fb8840d103e2aa9cacef7013dd42755"},"expire":1701130964064},"doi:10.1103/PhysRevA.71.022316":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T22:32:45Z","timestamp":1670020365199},"reference-count":21,"publisher":"American Physical Society (APS)","issue":"2","license":[{"start":{"date-parts":[[2005,2,22]],"date-time":"2005-02-22T00:00:00Z","timestamp":1109030400000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.71.022316","type":"journal-article","created":{"date-parts":[[2005,2,23]],"date-time":"2005-02-23T00:08:13Z","timestamp":1109117293000},"source":"Crossref","is-referenced-by-count":660,"title":"Universal quantum computation with ideal Clifford gates and noisy ancillas","prefix":"10.1103","volume":"71","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Alexei","family":"Kitaev","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2005,2,22]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.71.022316","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.71.022316/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,16]],"date-time":"2017-06-16T16:09:36Z","timestamp":1497629376000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.71.022316"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,2,22]]},"references-count":21,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2005,2]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.71.022316","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. Rev. A","published":{"date-parts":[[2005,2,22]]},"article-number":"022316","id":"doi:10.1103/PhysRevA.71.022316","_hash":"3a11db61d225b7ff0135dacc040de04605323391ead05bfc15a0c9df00fab4ad"},"expire":1701130964943},"doi:10.22331/q-2017-10-03-31":{"value":{"indexed":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T19:15:48Z","timestamp":1669317348147},"reference-count":53,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2017,10,3]],"date-time":"2017-10-03T00:00:00Z","timestamp":1506988800000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"<jats:p>We present an infinite family of protocols to distill magic states for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>-gates that has a low space overhead and uses an asymptotic number of input magic states to achieve a given target error that is conjectured to be optimal. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>-gate depth of the circuit scale linearly in the logarithm of the target error <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>δ</mml:mi></mml:math> (up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>δ</mml:mi></mml:math>). Unlike other distillation protocols, this protocol achieves this performance without concatenation and the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. The protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy, with the same asymptotic performance. The protocol relies on the construction of weakly self-dual CSS codes with many logical qubits and large distance, allowing us to implement control-SWAPs on multiple qubits. We call this code the \"inner code\". The control-SWAPs are then used to measure properties of the magic state and detect errors, using another code that we call the \"outer code\". Alternatively, we use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. We present several specific small examples of this protocol.</jats:p>","DOI":"10.22331/q-2017-10-03-31","type":"journal-article","created":{"date-parts":[[2017,10,3]],"date-time":"2017-10-03T11:34:08Z","timestamp":1507030448000},"page":"31","source":"Crossref","is-referenced-by-count":38,"title":"Magic state distillation with low space overhead and optimal asymptotic input count","prefix":"10.22331","volume":"1","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Quantum Architectures and Computation, Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"name":"Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA"},{"name":"Quantum Architectures and Computation, Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"D.","family":"Poulin","sequence":"additional","affiliation":[{"name":"Département de Physique & Institut Quantique, Université de Sherbrooke, Quebec, Canada"}]},{"given":"D.","family":"Wecker","sequence":"additional","affiliation":[{"name":"Quantum Architectures and Computation, Microsoft Research, Redmond, WA 98052, USA"}]}],"member":"9598","published-online":{"date-parts":[[2017,10,3]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","deposited":{"date-parts":[[2018,1,16]],"date-time":"2018-01-16T15:52:13Z","timestamp":1516117933000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2017-10-03-31/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,3]]},"references-count":53,"URL":"http://dx.doi.org/10.22331/q-2017-10-03-31","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2017,10,3]]},"article-number":"31","id":"doi:10.22331/q-2017-10-03-31","_hash":"941e3f690d13015cb506b1a17e4a6ab862fe353878a92edb40cc35fe275e9d59"},"expire":1701130966069},"doi:10.1364/OPTICA.439170":{"value":{"indexed":{"date-parts":[[2022,11,14]],"date-time":"2022-11-14T18:29:16Z","timestamp":1668450556202},"reference-count":35,"publisher":"Optica Publishing Group","issue":"2","license":[{"start":{"date-parts":[[2022,2,2]],"date-time":"2022-02-02T00:00:00Z","timestamp":1643760000000},"content-version":"vor","delay-in-days":0,"URL":"https://doi.org/10.1364/OA_License_v1#VOR-OA"}],"funder":[{"name":"National Key Research and Development (R&D) Plan of China","award":["2018YFA0306501"]},{"DOI":"10.13039/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11425417","12104444","11975222","U1738140"]},{"name":"Shanghai Municipal Science and Technology Major Project","award":["2019SHZDZX01"]},{"name":"Anhui Initiative in Quantum Information Technologies"},{"name":"Chinese Academy of Sciences and the Shanghai Science and Technology Development Funds","award":["18JC1414700"]},{"name":"Key R&D Program of Guangdong province","award":["2018B030325001"]},{"DOI":"10.13039/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2021M693093"]}],"content-domain":{"domain":["opg.optica.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2022,2,20]]},"abstract":"<jats:p>The all-photonic quantum repeater (APQR) is a promising repeater scheme to realize long-distance quantum communication. For a practical APQR, an indispensable requirement is the robustness of the repeater graph state (RGS) against photon loss. We propose a new loss-tolerant scheme by applying the generalized Shor code to RGS, which can be experimentally demonstrated with current technology. Experimentally, we first prepare and verify the nine-qubit Shor code. Then, by applying the generalized Shor code to APQR and preparing a simplified encoded RGS with the structure of <jats:inline-formula>\n      <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n\t<mml:mrow class=\"MJX-TeXAtom-ORD\">\n\t  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n\t    <mml:mn>1</mml:mn>\n\t  </mml:mrow>\n\t</mml:mrow>\n\t<mml:mo>×<!-- × --></mml:mo>\n\t<mml:mrow class=\"MJX-TeXAtom-ORD\">\n\t  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n\t    <mml:mn>2</mml:mn>\n\t  </mml:mrow>\n\t</mml:mrow>\n      </mml:math>\n    </jats:inline-formula> based on the Shor code state, the effectiveness of our loss-tolerant scheme and the loss tolerance of the encoded RGS are respectively verified. Our results make an essential step toward a practical APQR and enrich the research of quantum error correction code.</jats:p>","DOI":"10.1364/optica.439170","type":"journal-article","created":{"date-parts":[[2021,12,21]],"date-time":"2021-12-21T17:02:33Z","timestamp":1640106153000},"page":"152","update-policy":"http://dx.doi.org/10.1364/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":"Loss-tolerant all-photonic quantum repeater with generalized Shor code","prefix":"10.1364","volume":"9","author":[{"ORCID":"http://orcid.org/0000-0001-5687-5576","authenticated-orcid":true,"given":"Rui","family":"Zhang","sequence":"first","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Li-Zheng","family":"Liu","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Zheng-Da","family":"Li","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Yue-Yang","family":"Fei","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Xu-Fei","family":"Yin","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Li","family":"Li","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Nai-Le","family":"Liu","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Yingqiu","family":"Mao","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Yu-Ao","family":"Chen","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Jian-Wei","family":"Pan","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]}],"member":"285","published-online":{"date-parts":[[2022,2,2]]},"reference":[],"container-title":"Optica","original-title":[],"language":"en","link":[{"URL":"https://opg.optica.org/viewmedia.cfm?URI=optica-9-2-152&seq=0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,9]],"date-time":"2022-08-09T18:28:48Z","timestamp":1660069728000},"score":1,"resource":{"primary":{"URL":"https://opg.optica.org/abstract.cfm?URI=optica-9-2-152"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,2]]},"references-count":35,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022]]},"published-print":{"date-parts":[[2022]]}},"URL":"http://dx.doi.org/10.1364/OPTICA.439170","relation":{},"ISSN":["2334-2536"],"subject":["Atomic and Molecular Physics, and Optics","Electronic, Optical and Magnetic Materials"],"container-title-short":"Optica","published":{"date-parts":[[2022,2,2]]},"assertion":[{"value":"Optica Publishing Group","name":"publisher","label":"This article is maintained by"},{"value":"https://doi.org/10.1364/OPTICA.439170","name":"articlelink","label":"Crossref DOI link to publisher maintained version"},{"value":"research-article","name":"content_type","label":"Article type"},{"value":"Screened by Similarity Check","name":"cross_check","label":"Similarity check"},{"value":"Yes","order":0,"name":"peer_reviewed","label":"Peer reviewed","group":{"name":"peer_review","label":"Peer review"}},{"value":"Single blind","order":1,"name":"review_process","label":"Review process","group":{"name":"peer_review","label":"Peer review"}},{"value":"29 July 2021","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"19 December 2021","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2 February 2022","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}},{"value":"© 2022 Optical Society of America under the terms of the OSA Open Access Publishing Agreement","name":"copyright","label":"Copyright"}],"id":"doi:10.1364/OPTICA.439170","_hash":"9169e646d5a63dfe5876af424ebed319ddb4499a54042932a90c3961204bd72e"},"expire":1701130966962},"doi:10.1073/pnas.2026250118":{"value":{"indexed":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T19:00:22Z","timestamp":1669316422209},"reference-count":41,"publisher":"Proceedings of the National Academy of Sciences","issue":"36","license":[{"start":{"date-parts":[[2021,9,3]],"date-time":"2021-09-03T00:00:00Z","timestamp":1630627200000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by-nc-nd/4.0/"}],"content-domain":{"domain":["www.pnas.org"],"crossmark-restriction":true},"published-print":{"date-parts":[[2021,9,7]]},"abstract":"<jats:title>Significance</jats:title>\n          <jats:p>Quantum teleportation and quantum error correction play crucial roles in fault-tolerant quantum computing. Here, we implemented error-correctable quantum teleportation to manipulate a logical qubit and observed the protection of quantum information. Our work presents a useful technology for scalable quantum computing and can serve as a quantum simulator for holographic quantum gravity.</jats:p>","DOI":"10.1073/pnas.2026250118","type":"journal-article","created":{"date-parts":[[2021,9,14]],"date-time":"2021-09-14T16:47:29Z","timestamp":1631638049000},"update-policy":"http://dx.doi.org/10.1073/pnas.cm10313","source":"Crossref","is-referenced-by-count":9,"title":"Quantum teleportation of physical qubits into logical code spaces","prefix":"10.1073","volume":"118","author":[{"given":"Yi-Han","family":"Luo","sequence":"first","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Chinese Academy of Sciences Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, Hefei, Anhui 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Among the possible candidate platforms for realizing quantum computing devices, the compatibility with mature nanofabrication technologies of silicon-based spin qubits offers promise to overcome the challenges in scaling up device sizes from the prototypes of today to large-scale computers<jats:sup>3–5</jats:sup>. Recent advances in silicon-based qubits have enabled the implementations of high-quality one-qubit and two-qubit systems<jats:sup>6–8</jats:sup>. However, the demonstration of QEC, which requires three or more coupled qubits<jats:sup>1</jats:sup>, and involves a three-qubit gate<jats:sup>9–11</jats:sup> or measurement-based feedback, remains an open challenge. Here we demonstrate a three-qubit phase-correcting code in silicon, in which an encoded three-qubit state is protected against any phase-flip error on one of the three qubits. The correction to this encoded state is performed by a three-qubit conditional rotation, which we implement by an efficient single-step resonantly driven iToffoli gate. As expected, the error correction mitigates the errors owing to one-qubit phase-flip, as well as the intrinsic dephasing mainly owing to quasi-static phase noise. These results show successful implementation of QEC and the potential of a silicon-based platform for large-scale quantum computing.</jats:p>","DOI":"10.1038/s41586-022-04986-6","type":"journal-article","created":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T16:04:14Z","timestamp":1661357054000},"page":"682-686","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":"Quantum error correction with silicon spin qubits","prefix":"10.1038","volume":"608","author":[{"ORCID":"http://orcid.org/0000-0003-1240-1103","authenticated-orcid":false,"given":"Kenta","family":"Takeda","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-9145-0303","authenticated-orcid":false,"given":"Akito","family":"Noiri","sequence":"additional","affiliation":[]},{"given":"Takashi","family":"Nakajima","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-2841-8129","authenticated-orcid":false,"given":"Takashi","family":"Kobayashi","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-7465-0135","authenticated-orcid":false,"given":"Seigo","family":"Tarucha","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,8,24]]},"reference":[],"container-title":"Nature","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41586-022-04986-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41586-022-04986-6","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41586-022-04986-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,12]],"date-time":"2022-11-12T19:09:14Z","timestamp":1668280154000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41586-022-04986-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,24]]},"references-count":45,"journal-issue":{"issue":"7924","published-print":{"date-parts":[[2022,8,25]]}},"alternative-id":["4986"],"URL":"http://dx.doi.org/10.1038/s41586-022-04986-6","relation":{},"ISSN":["0028-0836","1476-4687"],"subject":["Multidisciplinary"],"container-title-short":"Nature","published":{"date-parts":[[2022,8,24]]},"assertion":[{"value":"21 January 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 June 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 August 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"id":"doi:10.1038/s41586-022-04986-6","_hash":"97a1c7c63bfb638cf92682dbc825579431ed9098c6176451c0e0ed95fe21a190"},"expire":1701130972983},"doi:10.1038/s41467-022-29906-0":{"value":{"indexed":{"date-parts":[[2022,11,26]],"date-time":"2022-11-26T05:33:18Z","timestamp":1669440798295},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,4,28]],"date-time":"2022-04-28T00:00:00Z","timestamp":1651104000000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,4,28]],"date-time":"2022-04-28T00:00:00Z","timestamp":1651104000000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/100006754","name":"United States Department of Defense | United States Army | U.S. Army Research, Development and Engineering Command | Army Research Laboratory","doi-asserted-by":"publisher","award":["W911NF-17-S-0008","W911NF-17-S-0008","W911NF-17-S-0008","W911NF-17-S-0008","W911NF-17-S-0008"]},{"DOI":"10.13039/501100001742","name":"United States-Israel Binational Science Foundation","doi-asserted-by":"publisher","award":["735/18"]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"<jats:title>Abstract</jats:title><jats:p>The storage and processing of quantum information are susceptible to external noise, resulting in computational errors. A powerful method to suppress these effects is quantum error correction. Typically, quantum error correction is executed in discrete rounds, using entangling gates and projective measurement on ancillary qubits to complete each round of error correction. Here we use direct parity measurements to implement a continuous quantum bit-flip correction code in a resource-efficient manner, eliminating entangling gates, ancillary qubits, and their associated errors. An FPGA controller actively corrects errors as they are detected, achieving an average bit-flip detection efficiency of up to 91%. Furthermore, the protocol increases the relaxation time of the protected logical qubit by a factor of 2.7 over the relaxation times of the bare comprising qubits. Our results showcase resource-efficient stabilizer measurements in a multi-qubit architecture and demonstrate how continuous error correction codes can address challenges in realizing a fault-tolerant system.</jats:p>","DOI":"10.1038/s41467-022-29906-0","type":"journal-article","created":{"date-parts":[[2022,4,28]],"date-time":"2022-04-28T10:04:48Z","timestamp":1651140288000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":"Experimental demonstration of continuous quantum error correction","prefix":"10.1038","volume":"13","author":[{"ORCID":"http://orcid.org/0000-0001-8399-0975","authenticated-orcid":false,"given":"William P.","family":"Livingston","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-9736-4158","authenticated-orcid":false,"given":"Machiel S.","family":"Blok","sequence":"additional","affiliation":[]},{"given":"Emmanuel","family":"Flurin","sequence":"additional","affiliation":[]},{"given":"Justin","family":"Dressel","sequence":"additional","affiliation":[]},{"given":"Andrew N.","family":"Jordan","sequence":"additional","affiliation":[]},{"given":"Irfan","family":"Siddiqi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,28]]},"reference":[],"container-title":"Nature Communications","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41467-022-29906-0.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-022-29906-0","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-022-29906-0.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T17:00:23Z","timestamp":1669309223000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41467-022-29906-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,28]]},"references-count":45,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["29906"],"URL":"http://dx.doi.org/10.1038/s41467-022-29906-0","relation":{},"ISSN":["2041-1723"],"subject":["General Physics and Astronomy","General Biochemistry, Genetics and Molecular Biology","General Chemistry","Multidisciplinary"],"container-title-short":"Nat Commun","published":{"date-parts":[[2022,4,28]]},"assertion":[{"value":"31 January 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 April 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 April 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"2307","id":"doi:10.1038/s41467-022-29906-0","_hash":"0c6435c6396560802a82b003e827bd2511595c7e5cf423ce4fa600e28202eaa3"},"expire":1701130973994},"doi:10.1038/nature14270":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T22:50:19Z","timestamp":1670021419168},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"7541","license":[{"start":{"date-parts":[[2015,3,4]],"date-time":"2015-03-04T00:00:00Z","timestamp":1425427200000},"content-version":"tdm","delay-in-days":0,"URL":"https://www.springer.com/tdm"},{"start":{"date-parts":[[2015,3,4]],"date-time":"2015-03-04T00:00:00Z","timestamp":1425427200000},"content-version":"vor","delay-in-days":0,"URL":"https://www.springer.com/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2015,3,5]]},"DOI":"10.1038/nature14270","type":"journal-article","created":{"date-parts":[[2015,3,3]],"date-time":"2015-03-03T16:48:51Z","timestamp":1425401331000},"page":"66-69","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":606,"title":"State preservation by repetitive error detection in a superconducting quantum circuit","prefix":"10.1038","volume":"519","author":[{"given":"J.","family":"Kelly","sequence":"first","affiliation":[]},{"given":"R.","family":"Barends","sequence":"additional","affiliation":[]},{"given":"A. 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This includes a need to determine how well these devices support the techniques required for quantum error correction. In this paper we introduce the <jats:monospace>topological_codes</jats:monospace> module of Qiskit-Ignis, which is designed to provide the tools necessary to perform such tests. Specifically, we use the <jats:monospace>RepetitionCode</jats:monospace> and <jats:monospace>GraphDecoder</jats:monospace> classes to run tests based on the repetition code and process the results. As an example, data from a 43 qubit code running on IBM’s <jats:italic>Rochester</jats:italic> device is presented.</jats:p>","DOI":"10.1088/2058-9565/aba038","type":"journal-article","created":{"date-parts":[[2020,6,26]],"date-time":"2020-06-26T16:27:09Z","timestamp":1593188829000},"page":"044004","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":14,"title":"Benchmarking near-term devices with quantum error correction","prefix":"10.1088","volume":"5","author":[{"ORCID":"http://orcid.org/0000-0003-1943-5306","authenticated-orcid":false,"given":"James R","family":"Wootton","sequence":"first","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2020,7,31]]},"reference":[],"container-title":"Quantum Science and Technology","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038","content-type":"text/html","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,27]],"date-time":"2021-11-27T16:33:56Z","timestamp":1638030836000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,31]]},"references-count":29,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,7,31]]},"published-print":{"date-parts":[[2020,10,1]]}},"URL":"http://dx.doi.org/10.1088/2058-9565/aba038","relation":{},"ISSN":["2058-9565"],"subject":["Electrical and Electronic Engineering","Physics and Astronomy (miscellaneous)","Materials Science (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum Sci. 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Published by IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2020-01-28","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-06-25","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-07-31","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/2058-9565/aba038","_hash":"fc272c23a0d0870f08909343789cf9a7e4a6084a35fa50f339d121a474f25345"},"expire":1701130975985},"doi:10.1103/PhysRevA.97.052313":{"value":{"indexed":{"date-parts":[[2022,12,1]],"date-time":"2022-12-01T18:33:39Z","timestamp":1669919619321},"reference-count":21,"publisher":"American Physical Society (APS)","issue":"5","license":[{"start":{"date-parts":[[2018,5,10]],"date-time":"2018-05-10T00:00:00Z","timestamp":1525910400000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"funder":[{"DOI":"10.13039/501100001711","name":"Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung","doi-asserted-by":"publisher"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physreva.97.052313","type":"journal-article","created":{"date-parts":[[2018,5,10]],"date-time":"2018-05-10T10:03:46Z","timestamp":1525946626000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":33,"title":"Repetition code of 15 qubits","prefix":"10.1103","volume":"97","author":[{"given":"James R.","family":"Wootton","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Loss","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2018,5,10]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevA.97.052313","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.97.052313/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2018,5,10]],"date-time":"2018-05-10T10:03:50Z","timestamp":1525946630000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.97.052313"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,10]]},"references-count":21,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2018,5]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.052313","relation":{},"ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. 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When instantiated with 3 classical LDPC codes, this \"XYZ product\" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Denise Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes also depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved.Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes with constant rate might display a minimum distance as large as <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x0398;</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>N</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>2</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes that can be embedded in three dimensions with local interaction. Some of these codes seem to share properties with Haah's cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier.</jats:p>","DOI":"10.22331/q-2022-07-14-766","type":"journal-article","created":{"date-parts":[[2022,7,14]],"date-time":"2022-07-14T13:42:11Z","timestamp":1657806131000},"page":"766","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Quantum XYZ Product Codes","prefix":"10.22331","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0002-6707-1458","authenticated-orcid":false,"given":"Anthony","family":"Leverrier","sequence":"first","affiliation":[{"name":"Inria, France"}]},{"ORCID":"http://orcid.org/0000-0003-3823-6804","authenticated-orcid":false,"given":"Simon","family":"Apers","sequence":"additional","affiliation":[{"name":"CNRS, IRIF, Université Paris 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Inform. Theory","published":{"date-parts":[[2014,2]]},"id":"doi:10.1109/TIT.2013.2292061","_hash":"eb497befcd1f372b1a4167b5dcf4853e21f8b766f460fe79d8d6803b52d6da50"},"expire":1701131024695},"doi:10.22331/q-2019-02-04-120":{"value":{"indexed":{"date-parts":[[2022,7,28]],"date-time":"2022-07-28T14:29:59Z","timestamp":1659018599702},"reference-count":51,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2019,2,4]],"date-time":"2019-02-04T00:00:00Z","timestamp":1549238400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"<jats:p>A method for the implementation of a universal set of fault-tolerant logical gates is presented using homological product codes. In particular, it is shown that one can fault-tolerantly map between different encoded representations of a given logical state, enabling the application of different classes of transversal gates belonging to the underlying quantum codes. This allows for the circumvention of no-go results pertaining to universal sets of transversal gates and provides a general scheme for fault-tolerant computation while keeping the stabilizer generators of the code sparse.</jats:p>","DOI":"10.22331/q-2019-02-04-120","type":"journal-article","created":{"date-parts":[[2019,2,4]],"date-time":"2019-02-04T11:50:45Z","timestamp":1549281045000},"page":"120","source":"Crossref","is-referenced-by-count":8,"title":"Fault-tolerant gates via homological product codes","prefix":"10.22331","volume":"3","author":[{"ORCID":"http://orcid.org/0000-0002-8265-4774","authenticated-orcid":false,"given":"Tomas","family":"Jochym-O'Connor","sequence":"first","affiliation":[{"name":"Walter Burke Institute for Theoretical Physics, Institute for Quantum Information & Matter, California Institute of Technology, Pasadena, CA 91125, USA"}]}],"member":"9598","published-online":{"date-parts":[[2019,2,4]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2019-02-04-120/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2019,2,4]],"date-time":"2019-02-04T11:51:47Z","timestamp":1549281107000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2019-02-04-120/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,4]]},"references-count":51,"URL":"http://dx.doi.org/10.22331/q-2019-02-04-120","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2019,2,4]]},"article-number":"120","id":"doi:10.22331/q-2019-02-04-120","_hash":"f6048c598de9bcf066faac478d879468aa749da59b5966f0297490a3c36d1744"},"expire":1701131025753},"doi:10.22331/q-2021-03-10-406":{"value":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T13:19:00Z","timestamp":1648559940083},"reference-count":17,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2021,3,10]],"date-time":"2021-03-10T00:00:00Z","timestamp":1615334400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Homological product codes are a class of codes that can have improved distance while retaining relatively low stabilizer weight. We show how to build union-find decoders for these codes, using a union-find decoder for one of the codes in the product and a brute force decoder for the other code. We apply this construction to the specific case of the product of a surface code with a small code such as a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:math> code, which we call an augmented surface code. The distance of the augmented surface code is the product of the distance of the surface code with that of the small code, and the union-find decoder, with slight modifications, can decode errors up to half the distance. We present numerical simulations, showing that while the threshold of these augmented codes is lower than that of the surface code, the low noise performance is improved.</jats:p>","DOI":"10.22331/q-2021-03-10-406","type":"journal-article","created":{"date-parts":[[2021,3,10]],"date-time":"2021-03-10T13:12:34Z","timestamp":1615381954000},"page":"406","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Union-Find Decoders For Homological Product Codes","prefix":"10.22331","volume":"5","author":[{"given":"Nicolas","family":"Delfosse","sequence":"first","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"name":"Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA"},{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"}]}],"member":"9598","published-online":{"date-parts":[[2021,3,10]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2021-03-10-406/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2021,3,10]],"date-time":"2021-03-10T13:13:11Z","timestamp":1615381991000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2021-03-10-406/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,10]]},"references-count":17,"URL":"http://dx.doi.org/10.22331/q-2021-03-10-406","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2021,3,10]]},"article-number":"406","id":"doi:10.22331/q-2021-03-10-406","_hash":"4afecabbcf6e66825b5834af24aa90b5c51c60753d93295160813f53e75c59ed"},"expire":1701131026794},"doi:10.22331/q-2022-02-24-661":{"value":{"indexed":{"date-parts":[[2022,12,1]],"date-time":"2022-12-01T13:18:53Z","timestamp":1669900733329},"reference-count":38,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,2,24]],"date-time":"2022-02-24T00:00:00Z","timestamp":1645660800000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/anr","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["QCDA"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>We introduce the hemicubic codes, a family of quantum codes obtained by associating qubits with the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>p</mml:mi></mml:math>-faces of the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>-cube (for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi><mml:mo>&amp;#x003E;</mml:mo><mml:mi>p</mml:mi></mml:math>) and stabilizer constraints with faces of dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>&amp;#x00B1;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. The quantum code obtained by identifying antipodal faces of the resulting complex encodes one logical qubit into <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>n</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mi>p</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mrow><mml:mrow class=\"MJX-TeXAtom-OPEN\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mfrac linethickness=\"0\"><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:mfrac><mml:mrow class=\"MJX-TeXAtom-CLOSE\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math> physical qubits and displays local testability with a soundness of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x03A9;</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math> beating the current state-of-the-art of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>log</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> due to Hastings. We exploit this local testability to devise an efficient decoding algorithm that corrects arbitrary errors of size less than the minimum distance, up to polylog factors.We then extend this code family by considering the quotient of the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>-cube by arbitrary linear classical codes of length <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>. We establish the parameters of these generalized hemicubic codes. Interestingly, if the soundness of the hemicubic code could be shown to be constant, similarly to the ordinary <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>-cube, then the generalized hemicubic codes could yield quantum locally testable codes of length not exceeding an exponential or even polynomial function of the code dimension.</jats:p>","DOI":"10.22331/q-2022-02-24-661","type":"journal-article","created":{"date-parts":[[2022,2,24]],"date-time":"2022-02-24T19:54:29Z","timestamp":1645732469000},"page":"661","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":"Towards local testability for quantum coding","prefix":"10.22331","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0002-6707-1458","authenticated-orcid":false,"given":"Anthony","family":"Leverrier","sequence":"first","affiliation":[{"name":"Inria, France"}]},{"given":"Vivien","family":"Londe","sequence":"additional","affiliation":[{"name":"Microsoft, France"}]},{"ORCID":"http://orcid.org/0000-0002-6041-9554","authenticated-orcid":false,"given":"Gilles","family":"Zémor","sequence":"additional","affiliation":[{"name":"Institut de Mathématiques de Bordeaux, UMR 5251, France"}]}],"member":"9598","published-online":{"date-parts":[[2022,2,24]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-02-24-661/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,3,9]],"date-time":"2022-03-09T20:59:53Z","timestamp":1646859593000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-02-24-661/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,24]]},"references-count":38,"URL":"http://dx.doi.org/10.22331/q-2022-02-24-661","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,2,24]]},"article-number":"661","id":"doi:10.22331/q-2022-02-24-661","_hash":"21c9fc410af34064e140ff2180d8fa6221b0a16fbfc514db2259e4936c8558c2"},"expire":1701131027728},"doi:10.1103/PhysRevA.97.032346":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T22:50:17Z","timestamp":1670021417165},"reference-count":135,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2018,3,30]],"date-time":"2018-03-30T00:00:00Z","timestamp":1522368000000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"},{"start":{"date-parts":[[2019,3,30]],"date-time":"2019-03-30T00:00:00Z","timestamp":1553904000000},"content-version":"am","delay-in-days":365,"URL":"https://link.aps.org/licenses/aps-default-accepted-manuscript-license"}],"funder":[{"DOI":"10.13039/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["W911NF-14-1-0011","W911NF-14-1-0563","W911NF-16-1-0349","W911NF1410011"]},{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["EFMA-1640959","DMR-1609326"]},{"DOI":"10.13039/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-14-1-0052","FA9550-15-1-0015"]},{"DOI":"10.13039/100000879","name":"Alfred P. 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T.","family":"Brierley","sequence":"additional","affiliation":[]},{"given":"Philip","family":"Reinhold","sequence":"additional","affiliation":[]},{"given":"Christophe","family":"Vuillot","sequence":"additional","affiliation":[]},{"given":"Linshu","family":"Li","sequence":"additional","affiliation":[]},{"given":"Chao","family":"Shen","sequence":"additional","affiliation":[]},{"given":"S. M.","family":"Girvin","sequence":"additional","affiliation":[]},{"given":"Barbara M.","family":"Terhal","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2018,3,30]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/accepted/10.1103/PhysRevA.97.032346","content-type":"application/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"https://link.aps.org/article/10.1103/PhysRevA.97.032346","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.97.032346/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,10,30]],"date-time":"2020-10-30T16:07:03Z","timestamp":1604074023000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.97.032346"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,3,30]]},"references-count":135,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2018,3]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.032346","relation":{},"ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. 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Rev. A","published":{"date-parts":[[2014,12,10]]},"article-number":"062317","id":"doi:10.1103/PhysRevA.90.062317","_hash":"cad75a55f074cbb12dc36312461aac46cfad2d79c971b18f2c780f0ff269ede5"},"expire":1701131030835},"doi:10.1007/JHEP08(2019)152":{"value":{"indexed":{"date-parts":[[2022,4,7]],"date-time":"2022-04-07T14:04:42Z","timestamp":1649340282215},"reference-count":53,"publisher":"Springer Science and Business Media LLC","issue":"8","license":[{"start":{"date-parts":[[2019,8,1]],"date-time":"2019-08-01T00:00:00Z","timestamp":1564617600000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"},{"start":{"date-parts":[[2019,8,27]],"date-time":"2019-08-27T00:00:00Z","timestamp":1566864000000},"content-version":"vor","delay-in-days":26,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2019,8]]},"abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>The eigenstate thermalization hypothesis (ETH) is a powerful conjecture for understanding how statistical mechanics emerges in a large class of many-body quantum systems. It has also been interpreted in a CFT context, and, in particular, holographic CFTs are expected to satisfy ETH. Recently, it was observed that the ETH condition corresponds to a necessary and sufficient condition for an approximate quantum error correcting code (AQECC), implying the presence of AQECCs in systems satisfying ETH. In this paper, we explore the properties of ETH as an error correcting code and show that there exists an explicit universal recovery channel for the code. Based on the analysis, we discuss a generalization that all chaotic theories contain error correcting codes. We then specialize to AdS/CFT to demonstrate the possibility of total bulk reconstruction in black holes with a well-defined macroscopic geometry. When combined with the existing AdS/CFT error correction story, this shows that black holes are enormously robust against erasure errors.</jats:p>","DOI":"10.1007/jhep08(2019)152","type":"journal-article","created":{"date-parts":[[2019,8,28]],"date-time":"2019-08-28T14:05:27Z","timestamp":1567001127000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":"Eigenstate thermalization hypothesis and approximate quantum error correction","prefix":"10.1007","volume":"2019","author":[{"given":"Ning","family":"Bao","sequence":"first","affiliation":[]},{"given":"Newton","family":"Cheng","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,8,27]]},"reference":[],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","link":[{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP08(2019)152.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/article/10.1007/JHEP08(2019)152/fulltext.html","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP08(2019)152.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,3]],"date-time":"2021-09-03T17:59:42Z","timestamp":1630691982000},"score":1,"resource":{"primary":{"URL":"https://link.springer.com/10.1007/JHEP08(2019)152"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8]]},"references-count":53,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2019,8]]}},"alternative-id":["11156"],"URL":"http://dx.doi.org/10.1007/JHEP08(2019)152","relation":{},"ISSN":["1029-8479"],"subject":["Nuclear and High Energy Physics"],"container-title-short":"J. High Energ. Phys.","published":{"date-parts":[[2019,8]]},"assertion":[{"value":"25 June 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 August 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 August 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"152","id":"doi:10.1007/JHEP08(2019)152","_hash":"2e4136e0c2c5498ff6a58be5cbe6ea9202297305c01862b534911f6e40db4de7"},"expire":1701131031983},"doi:10.1103/PhysRevLett.123.110502":{"value":{"indexed":{"date-parts":[[2022,11,18]],"date-time":"2022-11-18T20:18:06Z","timestamp":1668802686404},"reference-count":35,"publisher":"American Physical Society (APS)","issue":"11","license":[{"start":{"date-parts":[[2019,9,11]],"date-time":"2019-09-11T00:00:00Z","timestamp":1568160000000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"funder":[{"DOI":"10.13039/100000936","name":"Gordon and Betty Moore Foundation","doi-asserted-by":"publisher","award":["GBMF-12500028"]},{"DOI":"10.13039/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["385612"]},{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["PHY-1125565","PHY-1125915"]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevlett.123.110502","type":"journal-article","created":{"date-parts":[[2019,9,11]],"date-time":"2019-09-11T14:36:37Z","timestamp":1568212597000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":23,"title":"Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains","prefix":"10.1103","volume":"123","author":[{"given":"Fernando G. S. L.","family":"Brandão","sequence":"first","affiliation":[]},{"given":"Elizabeth","family":"Crosson","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4488-2300","authenticated-orcid":true,"given":"M. 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Conventional schemes for error correction with the surface code place qubits on a planar grid and assume native CNOT gates between the data qubits with nearest-neighbor ancilla qubits.Here, we present surface code error-correction schemes using <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">only</mml:mtext></mml:mrow></mml:math> Pauli measurements on single qubits and on pairs of nearest-neighbor qubits. In particular, we provide several qubit layouts that offer favorable trade-offs between qubit overhead, circuit depth and connectivity degree. We also develop minimized measurement sequences for syndrome extraction, enabling reduced logical error rates and improved fault-tolerance thresholds.Our work applies to topologically protected qubits realized with Majorana zero modes and to similar systems in which multi-qubit Pauli measurements rather than CNOT gates are the native operations.</jats:p>","DOI":"10.22331/q-2020-10-28-352","type":"journal-article","created":{"date-parts":[[2020,10,28]],"date-time":"2020-10-28T11:06:16Z","timestamp":1603883176000},"page":"352","source":"Crossref","is-referenced-by-count":12,"title":"Optimization of the surface code design for Majorana-based qubits","prefix":"10.22331","volume":"4","author":[{"given":"Rui","family":"Chao","sequence":"first","affiliation":[{"name":"University of Southern California, Los Angeles, CA, USA"}]},{"given":"Michael E.","family":"Beverland","sequence":"additional","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA, USA"}]},{"given":"Nicolas","family":"Delfosse","sequence":"additional","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA, USA"}]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA, USA"}]}],"member":"9598","published-online":{"date-parts":[[2020,10,28]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2020-10-28-352/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2020,10,28]],"date-time":"2020-10-28T11:06:19Z","timestamp":1603883179000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2020-10-28-352/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,28]]},"references-count":20,"URL":"http://dx.doi.org/10.22331/q-2020-10-28-352","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2020,10,28]]},"article-number":"352","id":"doi:10.22331/q-2020-10-28-352","_hash":"b93bb89e6c6b7920278591eba87518774c6211b44614fbc0a8691cd6eb3f0dce"},"expire":1701131047318},"doi:10.22331/q-2022-09-21-813":{"value":{"indexed":{"date-parts":[[2022,9,22]],"date-time":"2022-09-22T05:45:36Z","timestamp":1663825536452},"reference-count":17,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T00:00:00Z","timestamp":1663718400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>We improve the planar honeycomb code by describing boundaries that need no additional physical connectivity, and by optimizing the shape of the qubit patch. We then benchmark the code using Monte Carlo sampling to estimate logical error rates and derive metrics including thresholds, lambdas, and teraquop qubit counts. We determine that the planar honeycomb code can create a logical qubit with one-in-a-trillion logical error rates using 7000 physical qubits at a 0.1% gate-level error rate (or 900 physical qubits given native two-qubit parity measurements). Our results cement the honeycomb code as a promising candidate for two-dimensional qubit architectures with sparse connectivity.</jats:p>","DOI":"10.22331/q-2022-09-21-813","type":"journal-article","created":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T12:09:46Z","timestamp":1663762186000},"page":"813","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Benchmarking the Planar Honeycomb Code","prefix":"10.22331","volume":"6","author":[{"given":"Craig","family":"Gidney","sequence":"first","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Matt","family":"McEwen","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"},{"name":"University of California, Santa Barbara, 93106, USA"}]}],"member":"9598","published-online":{"date-parts":[[2022,9,21]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-09-21-813/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T12:09:53Z","timestamp":1663762193000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-09-21-813/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,21]]},"references-count":17,"URL":"http://dx.doi.org/10.22331/q-2022-09-21-813","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,9,21]]},"article-number":"813","id":"doi:10.22331/q-2022-09-21-813","_hash":"7dbaa53da2454a63694155ce21078e26bff1866178dd006ddb892be692355450"},"expire":1701131048163},"doi:10.22331/q-2021-12-20-605":{"value":{"indexed":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T08:31:37Z","timestamp":1669797097092},"reference-count":44,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2021,12,20]],"date-time":"2021-12-20T00:00:00Z","timestamp":1639958400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Recently, Hastings &amp; Haah introduced a quantum memory defined on the honeycomb lattice. Remarkably, this honeycomb code assembles weight-six parity checks using only two-local measurements. The sparse connectivity and two-local measurements are desirable features for certain hardware, while the weight-six parity checks enable robust performance in the circuit model.In this work, we quantify the robustness of logical qubits preserved by the honeycomb code using a correlated minimum-weight perfect-matching decoder. Using Monte Carlo sampling, we estimate the honeycomb code's threshold in different error models, and project how efficiently it can reach the \"teraquop regime\" where trillions of quantum logical operations can be executed reliably. We perform the same estimates for the rotated surface code, and find a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.2</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>0.3</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> for the honeycomb code compared to a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.5</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>0.7</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> for the surface code in a controlled-not circuit model. In a circuit model with native two-body measurements, the honeycomb code achieves a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1.5</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi><mml:mo>&amp;#x003C;</mml:mo><mml:mi>p</mml:mi><mml:mo>&amp;#x003C;</mml:mo><mml:mn>2.0</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>p</mml:mi></mml:math> is the collective error rate of the two-body measurement gate - including both measurement and correlated data depolarization error processes. With such gates at a physical error rate of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mn>10</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>, we project that the honeycomb code can reach the teraquop regime with only <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>600</mml:mn></mml:math> physical qubits. </jats:p>","DOI":"10.22331/q-2021-12-20-605","type":"journal-article","created":{"date-parts":[[2022,1,30]],"date-time":"2022-01-30T10:03:32Z","timestamp":1643537012000},"page":"605","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":10,"title":"A Fault-Tolerant Honeycomb Memory","prefix":"10.22331","volume":"5","author":[{"given":"Craig","family":"Gidney","sequence":"first","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Austin","family":"Fowler","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Michael","family":"Broughton","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]}],"member":"9598","published-online":{"date-parts":[[2021,12,20]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2021-12-20-605/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,1,30]],"date-time":"2022-01-30T10:03:35Z","timestamp":1643537015000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2021-12-20-605/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,20]]},"references-count":44,"URL":"http://dx.doi.org/10.22331/q-2021-12-20-605","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and 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B","published":{"date-parts":[[2022,8,15]]},"article-number":"085122","id":"doi:10.1103/PhysRevB.106.085122","_hash":"b32e7efe9dac609e9c38cbf437aaac3746acf5e1802e13484e4c6a4e3cec0b0d"},"expire":1701131050126},"doi:10.22331/q-2022-04-21-693":{"value":{"indexed":{"date-parts":[[2022,11,26]],"date-time":"2022-11-26T05:28:41Z","timestamp":1669440521129},"reference-count":14,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,4,21]],"date-time":"2022-04-21T00:00:00Z","timestamp":1650499200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>We introduce a simple construction of boundary conditions for the honeycomb code \\cite{honeycomb} that uses only pairwise checks and allows parallelogram geometries at the cost of modifying the bulk measurement sequence. We discuss small instances of the code.</jats:p>","DOI":"10.22331/q-2022-04-21-693","type":"journal-article","created":{"date-parts":[[2022,4,21]],"date-time":"2022-04-21T08:03:42Z","timestamp":1650528222000},"page":"693","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":"Boundaries for the Honeycomb Code","prefix":"10.22331","volume":"6","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"},{"name":"Station Q, Microsoft Quantum, Santa Barbara, CA 93106-6105, USA"}]}],"member":"9598","published-online":{"date-parts":[[2022,4,21]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-04-21-693/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,4,21]],"date-time":"2022-04-21T08:03:51Z","timestamp":1650528231000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-04-21-693/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,21]]},"references-count":14,"URL":"http://dx.doi.org/10.22331/q-2022-04-21-693","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,4,21]]},"article-number":"693","id":"doi:10.22331/q-2022-04-21-693","_hash":"5197165b1ed7a1fe723bc3dda2c98ae516d9fa3fc4ed1d0cd60aab54aa6df835"},"expire":1701131050966},"doi:10.1142/S1230161208000043":{"value":{"indexed":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T19:18:43Z","timestamp":1669317523408},"reference-count":14,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2008,3]]},"abstract":"<jats:p> We give a short proof that the coherent information is an achievable rate for the transmission of quantum information through a noisy quantum channel. Our method is to produce random codes by performing a unitarily covariant projective measurement on a typical subspace of a tensor power state. We show that, provided the rank of each measurement operator is sufficiently small, the transmitted data will, with high probability, be decoupled from the channel environment. We also show that our construction leads to random codes whose average input is close to a product state and outline a modification yielding unitarily invariant ensembles of maximally entangled codes. </jats:p>","DOI":"10.1142/s1230161208000043","type":"journal-article","created":{"date-parts":[[2008,3,26]],"date-time":"2008-03-26T04:30:06Z","timestamp":1206505806000},"page":"7-19","source":"Crossref","is-referenced-by-count":87,"title":"A Decoupling Approach to the Quantum Capacity","prefix":"10.1142","volume":"15","author":[{"given":"Patrick","family":"Hayden","sequence":"first","affiliation":[{"name":"School of Computer Science, McGill University, Montreal, Quebec, Canada"}]},{"given":"Michał","family":"Horodecki","sequence":"additional","affiliation":[{"name":"Institute of Theoretical Physics and Astrophysics, University of Gdańsk, Poland"},{"name":"National Quantum Information Centre of Gdańsk, Gen. W. Andersa 27, 81 – 824 Sopot, Poland"}]},{"given":"Andreas","family":"Winter","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bristol, UK"}]},{"given":"Jon","family":"Yard","sequence":"additional","affiliation":[{"name":"CNLS (Center for Nonlinear Studies), CCS-3 (Computer, Computational and Statistical Sciences), Los Alamos National Laboratories, Los Alamos, NM, USA"},{"name":"Institute for Quantum Information, Caltech, Pasadena, CA, USA"}]}],"member":"219","published-online":{"date-parts":[[2012,1,25]]},"reference":[],"container-title":"Open Systems &amp; Information Dynamics","original-title":[],"language":"en","link":[{"URL":"https://www.worldscientific.com/doi/pdf/10.1142/S1230161208000043","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T21:56:35Z","timestamp":1565128595000},"score":1,"resource":{"primary":{"URL":"https://www.worldscientific.com/doi/abs/10.1142/S1230161208000043"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,3]]},"references-count":14,"journal-issue":{"issue":"01","published-online":{"date-parts":[[2012,1,25]]},"published-print":{"date-parts":[[2008,3]]}},"alternative-id":["10.1142/S1230161208000043"],"URL":"http://dx.doi.org/10.1142/S1230161208000043","relation":{},"ISSN":["1230-1612","1793-7191"],"subject":["Mathematical Physics","Statistics and Probability","Statistical and Nonlinear Physics"],"container-title-short":"Open Syst. 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The ultimate end-to-end rates of quantum communication networks are known to be achievable by an optimal entanglement distillation protocol followed by teleportation. In this work, we give a practical design for this achievability. Our ultimate design is an iterative approach, where each purification step operates on shared entangled states and detects loss errors at the highest rates allowed by physics. As a simpler design, we show that the first round of iterations can purify completely at high rates. We propose an experimental implementation using linear optics and photon-number measurements which is robust to inefficient operations and measurements, showcasing its near-term potential for real-world practical applications.</jats:p>","DOI":"10.1038/s41534-022-00641-0","type":"journal-article","created":{"date-parts":[[2022,11,8]],"date-time":"2022-11-08T03:02:32Z","timestamp":1667876552000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":"Achieving the ultimate end-to-end rates of lossy quantum communication networks","prefix":"10.1038","volume":"8","author":[{"ORCID":"http://orcid.org/0000-0003-3457-4451","authenticated-orcid":false,"given":"Matthew S.","family":"Winnel","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-9990-6341","authenticated-orcid":false,"given":"Joshua J.","family":"Guanzon","sequence":"additional","affiliation":[]},{"given":"Nedasadat","family":"Hosseinidehaj","sequence":"additional","affiliation":[]},{"given":"Timothy C.","family":"Ralph","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,11,8]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41534-022-00641-0.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-022-00641-0","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-022-00641-0.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,8]],"date-time":"2022-11-08T03:06:26Z","timestamp":1667876786000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-022-00641-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,8]]},"references-count":62,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["641"],"URL":"http://dx.doi.org/10.1038/s41534-022-00641-0","relation":{},"ISSN":["2056-6387"],"subject":["Computational Theory and Mathematics","Computer Networks and Communications","Statistical and Nonlinear Physics","Computer Science (miscellaneous)"],"container-title-short":"npj Quantum Inf","published":{"date-parts":[[2022,11,8]]},"assertion":[{"value":"25 March 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 October 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 November 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"129","id":"doi:10.1038/s41534-022-00641-0","_hash":"2005456e99b1fa2b5ea95b41e47a6596f58c05a86a2dd03690e8ab21ac06eefe"},"expire":1701131111115},"doi:10.1103/PhysRevLett.82.4556":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T22:56:07Z","timestamp":1670021767715},"reference-count":19,"publisher":"American Physical Society (APS)","issue":"22","license":[{"start":{"date-parts":[[1999,5,31]],"date-time":"1999-05-31T00:00:00Z","timestamp":928108800000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physrevlett.82.4556","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:16:54Z","timestamp":1027721814000},"page":"4556-4559","source":"Crossref","is-referenced-by-count":226,"title":"Concatenating Decoherence-Free Subspaces with Quantum Error Correcting Codes","prefix":"10.1103","volume":"82","author":[{"given":"D. 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In situations where the Heisenberg scaling is achievable, we provide a semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cramér-Rao (HCR) bound for pure states. We provide examples of significant advantages offered by our joint-QEC protocols, that sense all the parameters utilizing a single error-corrected subspace, over separate-QEC protocols where each parameter is effectively sensed in a separate subspace.</jats:p>","DOI":"10.22331/q-2020-07-02-288","type":"journal-article","created":{"date-parts":[[2020,7,2]],"date-time":"2020-07-02T17:47:57Z","timestamp":1593712077000},"page":"288","source":"Crossref","is-referenced-by-count":18,"title":"Optimal probes and error-correction schemes in multi-parameter quantum metrology","prefix":"10.22331","volume":"4","author":[{"ORCID":"http://orcid.org/0000-0001-9912-9186","authenticated-orcid":false,"given":"Wojciech","family":"Górecki","sequence":"first","affiliation":[{"name":"Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, 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The need for understanding the limits of covariant quantum error correction arises in various realms of physics including fault-tolerant quantum computation, condensed matter physics and quantum gravity. Here, we explore covariant quantum error correction with respect to continuous symmetries from the perspectives of quantum metrology and quantum resource theory, establishing solid connections between these formerly disparate fields. We prove new and powerful lower bounds on the infidelity of covariant quantum error correction, which not only extend the scope of previous no-go results but also provide a substantial improvement over existing bounds. Explicit lower bounds are derived for both erasure and depolarizing noises. 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In particular, several researchers have considered the individual code qubits being encoded with the continuous variable GottesmanKitaev-Preskill (GKP) code, and then imposed an outer discrete-variable code such as the surface code on these GKP qubits. Under such a concatenation scheme, the analog information from the inner GKP error correction improves the noise threshold of the outer code. However, the surface code has vanishing rate and demands a lot of resources with growing distance. In this work, we concatenate the GKP code with generic quantum low-density parity-check (QLDPC) codes and demonstrate a natural way to exploit the GKP analog information in iterative decoding algorithms. We first show the noise thresholds for two lifted product QLDPC code families, and then show the improvements of noise thresholds when the iterative decoder – a hardware-friendly min-sum algorithm (MSA) – utilizes the GKP analog information. We also show that, when the GKP analog information is combined with a sequential update schedule for MSA, the scheme surpasses the well-known CSS Hamming bound for these code families. Furthermore, we observe that the GKP analog information helps the iterative decoder in escaping harmful trapping sets in the Tanner graph of the QLDPC code, thereby eliminating or significantly lowering the error floor of the logical error rate curves. Finally, we discuss new fundamental and practical questions that arise from this work on channel capacity under GKP analog information, and on improving decoder design and analysis.</jats:p>","DOI":"10.22331/q-2022-07-20-767","type":"journal-article","created":{"date-parts":[[2022,7,20]],"date-time":"2022-07-20T13:39:18Z","timestamp":1658324358000},"page":"767","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":"Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound","prefix":"10.22331","volume":"6","author":[{"given":"Nithin","family":"Raveendran","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]},{"given":"Narayanan","family":"Rengaswamy","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]},{"given":"Filip","family":"Rozpędek","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Chicago, IL 60637, USA"}]},{"given":"Ankur","family":"Raina","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462066, India"}]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Chicago, IL 60637, USA"}]},{"given":"Bane","family":"Vasić","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]}],"member":"9598","published-online":{"date-parts":[[2022,7,20]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-07-20-767/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,7,20]],"date-time":"2022-07-20T13:39:45Z","timestamp":1658324385000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-07-20-767/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,20]]},"references-count":61,"URL":"http://dx.doi.org/10.22331/q-2022-07-20-767","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,7,20]]},"article-number":"767","id":"doi:10.22331/q-2022-07-20-767","_hash":"ca5bd48b8ce93634d7f330cfc52b3ed2807a4b0ea66d8b9ed78b62bc66d86df1"},"expire":1701131137987},"doi:10.22331/q-2022-02-10-648":{"value":{"indexed":{"date-parts":[[2022,5,20]],"date-time":"2022-05-20T14:49:34Z","timestamp":1653058174421},"reference-count":53,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,2,10]],"date-time":"2022-02-10T00:00:00Z","timestamp":1644451200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"BMBF","award":["RealistiQ"]},{"name":"BMBF","award":["PhoQuant"]},{"name":"BMBF","award":["QPIC-1"]},{"name":"BMBF","award":["QSolid"]},{"name":"DFG","award":["CRC 183"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>We examine general Gottesman-Kitaev-Preskill (GKP) codes for continuous-variable quantum error correction, including concatenated GKP codes, through the lens of lattice theory, in order to better understand the structure of this class of stabilizer codes. 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Publishing Services by Elsevier B.V. on behalf of KeAi Communications Co. Ltd.","name":"copyright","label":"Copyright"}],"id":"doi:10.1016/j.fmre.2020.12.006","_hash":"e370db8b702a3ebb97756bb142bb401a416b70e03783251f875aa5e2ecf2b8bb"},"expire":1701131164173},"doi:10.1088/2058-9565/abe989":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T22:53:31Z","timestamp":1670021611865},"reference-count":246,"publisher":"IOP Publishing","issue":"3","license":[{"start":{"date-parts":[[2021,4,1]],"date-time":"2021-04-01T00:00:00Z","timestamp":1617235200000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"},{"start":{"date-parts":[[2021,4,1]],"date-time":"2021-04-01T00:00:00Z","timestamp":1617235200000},"content-version":"tdm","delay-in-days":0,"URL":"https://iopscience.iop.org/info/page/text-and-data-mining"}],"funder":[{"DOI":"10.13039/501100001381","name":"Singapore National Research Foundation","doi-asserted-by":"crossref","award":["Fellowship (Class of 2020)"]}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2021,7,1]]},"abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>The unique features of quantum theory offer a powerful new paradigm for information processing. Translating these mathematical abstractions into useful algorithms and applications requires quantum systems with significant complexity and sufficiently low error rates. Such quantum systems must be made from robust hardware that can coherently store, process, and extract the encoded information, as well as possess effective quantum error correction (QEC) protocols to detect and correct errors. Circuit quantum electrodynamics (cQED) provides a promising hardware platform for implementing robust quantum devices. In particular, bosonic encodings in cQED that use multi-photon states of superconducting cavities to encode information have shown success in realizing hardware-efficient QEC. Here, we review recent developments in the theory and implementation of QEC with bosonic codes and report the progress made toward realizing fault-tolerant quantum information processing with cQED devices.</jats:p>","DOI":"10.1088/2058-9565/abe989","type":"journal-article","created":{"date-parts":[[2021,2,24]],"date-time":"2021-02-24T22:31:28Z","timestamp":1614205888000},"page":"033001","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":30,"title":"Quantum information processing with bosonic qubits in circuit QED","prefix":"10.1088","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0003-1445-2086","authenticated-orcid":false,"given":"Atharv","family":"Joshi","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-6318-8472","authenticated-orcid":false,"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-6360-9679","authenticated-orcid":false,"given":"Yvonne Y","family":"Gao","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2021,4,1]]},"reference":[],"container-title":"Quantum Science and Technology","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,19]],"date-time":"2022-01-19T03:08:42Z","timestamp":1642561722000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,1]]},"references-count":246,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,4,1]]},"published-print":{"date-parts":[[2021,7,1]]}},"URL":"http://dx.doi.org/10.1088/2058-9565/abe989","relation":{},"ISSN":["2058-9565"],"subject":["Electrical and Electronic Engineering","Physics and Astronomy (miscellaneous)","Materials Science (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum Sci. 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Published by IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2020-09-01","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2021-02-24","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2021-04-01","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/2058-9565/abe989","_hash":"f7bd44fb534df433e88a770c13e3ceb30164ff8146662519a785c8959990b073"},"expire":1701131165296},"doi:10.1103/PhysRevA.75.042316":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T22:57:08Z","timestamp":1670021828977},"reference-count":28,"publisher":"American Physical Society (APS)","issue":"4","license":[{"start":{"date-parts":[[2007,4,16]],"date-time":"2007-04-16T00:00:00Z","timestamp":1176681600000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.75.042316","type":"journal-article","created":{"date-parts":[[2007,4,16]],"date-time":"2007-04-16T21:02:25Z","timestamp":1176757345000},"source":"Crossref","is-referenced-by-count":36,"title":"Protecting an optical qubit against photon loss","prefix":"10.1103","volume":"75","author":[{"given":"Wojciech","family":"Wasilewski","sequence":"first","affiliation":[]},{"given":"Konrad","family":"Banaszek","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2007,4,16]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.75.042316","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.75.042316/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,17]],"date-time":"2017-06-17T16:24:55Z","timestamp":1497716695000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.75.042316"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,4,16]]},"references-count":28,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2007,4]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.75.042316","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. 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Despite its simplicity compared with the conventional measurement-based quantum error correction, it is still a far from practical technique because of significant hardware overhead. We propose an autonomous quantum error correction scheme for a rotational symmetric bosonic code in a four-photon Kerr parametric oscillator. Our scheme is the simplest possible error correction scheme that can surpass the break-even point—it requires only a single continuous microwave tone. We also introduce an unconditional reset scheme that requires one more continuous microwave tone in addition to that for the error correction. The key properties underlying this simplicity are protected quasienergy states of a four-photon Kerr parametric oscillator and the degeneracy in its quasienergy level structure. These properties eliminate the need for state-by-state correction in the Fock basis. 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Here we consider qubits encoded with constant space overhead (i.e. finite encoding rate) in the limit of arbitrarily large code distance<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>through the use of topological codes associated to triangulations of hyperbolic surfaces. We introduce explicit protocols to demonstrate how Dehn twists of the hyperbolic surface can be implemented on the code through constant depth unitary circuits, without increasing the space overhead. The circuit for a given Dehn twist consists of a permutation of physical qubits, followed by a constant depth local unitary circuit, where locality here is defined with respect to a hyperbolic metric that defines the code. Applying our results to the hyperbolic Fibonacci Turaev-Viro code implies the possibility of applying universal logical gate sets on encoded qubits through constant depth unitary circuits and with constant space overhead. Our circuits are inherently protected from errors as they map local operators to local operators while changing the size of their support by at most a constant factor; in the presence of noisy syndrome measurements, our results suggest the possibility of universal fault tolerant quantum computation with constant space overhead and time overhead of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>d</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. For quantum circuits that allow parallel gate operations, this yields the optimal scaling of space-time overhead known to date.</jats:p>","DOI":"10.22331/q-2019-08-26-180","type":"journal-article","created":{"date-parts":[[2019,8,26]],"date-time":"2019-08-26T14:31:27Z","timestamp":1566829887000},"page":"180","source":"Crossref","is-referenced-by-count":14,"title":"Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes","prefix":"10.22331","volume":"3","author":[{"given":"Ali","family":"Lavasani","sequence":"first","affiliation":[{"name":"Department of Physics, Condensed Matter Theory Center, University of Maryland, College Park, Maryland 20742, USA and Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA"}]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[{"name":"Department of Physics, Condensed Matter Theory Center, University of Maryland, College Park, Maryland 20742, USA and Joint 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A transistor, despite its roots in energy band structure, does not invoke quantum mechanically coherent transmission between terminals. The impressive progress in the past decade in mesoscopic physics, when combined with studies that have analysed a totally quantum mechanical computational process, suggest that we may be ready to move toward more quantum mechanical procedures for information processing. This paper is a warning signal; this possibility is beset by problems. The case will be made via two separate but complementary arguments. First, by summarizing this author's published comments on computation via totally quantum mechanical coherent Hamiltonians. The computation is likely to suffer from\n            <jats:italic>localization</jats:italic>\n            , i.e. from reflection of the computational trajectory, causing the computation to turn around. Additionally, small errors will accumulate and cause the computation to go off track. This is supplemented by analysis of specific proposals that suggest more detailed machinery than invoked in the general literature on quantum mechanical Hamiltonian computation.\n          </jats:p>","DOI":"10.1098/rsta.1995.0106","type":"journal-article","created":{"date-parts":[[2006,12,15]],"date-time":"2006-12-15T19:15:05Z","timestamp":1166210105000},"page":"367-376","source":"Crossref","is-referenced-by-count":101,"title":"Is quantum mechanics useful?","prefix":"10.1098","volume":"353","member":"175","published-online":{"date-parts":[[1997,1]]},"reference":[],"container-title":"Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences","original-title":[],"language":"en","link":[{"URL":"https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1995.0106","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1995.0106","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,20]],"date-time":"2021-02-20T19:21:17Z","timestamp":1613848877000},"score":1,"resource":{"primary":{"URL":"https://royalsocietypublishing.org/doi/10.1098/rsta.1995.0106"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,12,15]]},"references-count":45,"journal-issue":{"issue":"1703","published-print":{"date-parts":[[1995,12,15]]}},"alternative-id":["10.1098/rsta.1995.0106"],"URL":"http://dx.doi.org/10.1098/rsta.1995.0106","relation":{},"ISSN":["0962-8428","2054-0299"],"subject":["Pharmacology (medical)","Complementary and alternative medicine","Pharmaceutical Science"],"container-title-short":"Phil. Trans. R. Soc. Lond. A","published":{"date-parts":[[1995,12,15]]},"id":"doi:10.1098/rsta.1995.0106","_hash":"a2007c279c67eb3f5e8e2b9527d42ef5ee7bb7f90fb0234b3a7246ddc9ae5068"},"expire":1702831892960},"doi:10.1145/1568318.1568324":{"value":{"indexed":{"date-parts":[[2022,12,21]],"date-time":"2022-12-21T19:31:48Z","timestamp":1671651108260},"reference-count":38,"publisher":"Association for Computing Machinery (ACM)","issue":"6","funder":[{"DOI":"10.13039/501100004965","name":"Sixth Framework Programme","doi-asserted-by":"publisher","award":["15848"]},{"DOI":"10.13039/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["DAAD19-03-1-0082"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2009,9]]},"abstract":"<jats:p>\n            Our main result is a reduction from worst-case lattice problems such as GapSVP and SIVP to a certain learning problem. This learning problem is a natural extension of the “learning from parity with error” problem to higher moduli. It can also be viewed as the problem of decoding from a random linear code. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quantum. Hence, an efficient solution to the learning problem implies a\n            <jats:italic>quantum</jats:italic>\n            algorithm for GapSVP and SIVP. A main open question is whether this reduction can be made classical (i.e., nonquantum).\n          </jats:p>\n          <jats:p>\n            We also present a (classical) public-key cryptosystem whose security is based on the hardness of the learning problem. By the main result, its security is also based on the worst-case quantum hardness of GapSVP and SIVP. The new cryptosystem is much more efficient than previous lattice-based cryptosystems: the public key is of size Õ(\n            <jats:italic>n</jats:italic>\n            <jats:sup>2</jats:sup>\n            ) and encrypting a message increases its size by a factor of Õ(\n            <jats:italic>n</jats:italic>\n            ) (in previous cryptosystems these values are Õ(\n            <jats:italic>n</jats:italic>\n            <jats:sup>4</jats:sup>\n            ) and Õ(\n            <jats:italic>n</jats:italic>\n            <jats:sup>2</jats:sup>\n            ), respectively). In fact, under the assumption that all parties share a random bit string of length Õ(\n            <jats:italic>n</jats:italic>\n            <jats:sup>2</jats:sup>\n            ), the size of the public key can be reduced to Õ(\n            <jats:italic>n</jats:italic>\n            ).\n          </jats:p>","DOI":"10.1145/1568318.1568324","type":"journal-article","created":{"date-parts":[[2009,9,8]],"date-time":"2009-09-08T12:53:03Z","timestamp":1252414383000},"page":"1-40","source":"Crossref","is-referenced-by-count":844,"title":"On lattices, learning with errors, random linear codes, and cryptography","prefix":"10.1145","volume":"56","author":[{"given":"Oded","family":"Regev","sequence":"first","affiliation":[{"name":"Tel Aviv University, Tel Aviv, Israel"}]}],"member":"320","published-online":{"date-parts":[[2009,9,8]]},"reference":[],"container-title":"Journal of the ACM","original-title":[],"language":"en","link":[{"URL":"https://dl.acm.org/doi/pdf/10.1145/1568318.1568324","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,5]],"date-time":"2022-12-05T17:11:00Z","timestamp":1670260260000},"score":1,"resource":{"primary":{"URL":"https://dl.acm.org/doi/10.1145/1568318.1568324"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,9]]},"references-count":38,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2009,9]]}},"alternative-id":["10.1145/1568318.1568324"],"URL":"http://dx.doi.org/10.1145/1568318.1568324","relation":{},"ISSN":["0004-5411","1557-735X"],"subject":["Artificial Intelligence","Hardware and Architecture","Information Systems","Control and Systems Engineering","Software"],"container-title-short":"J. ACM","published":{"date-parts":[[2009,9]]},"id":"doi:10.1145/1568318.1568324","_hash":"63ad40b554ff88226cc2cd4b9579a30f40ccf5937a3fe759a360ae28351ab486"},"expire":1702831894229},"doi:10.1112/blms/19.2.113":{"value":{"indexed":{"date-parts":[[2022,11,16]],"date-time":"2022-11-16T05:54:49Z","timestamp":1668578089412},"reference-count":0,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[1987,3,1]],"date-time":"1987-03-01T00:00:00Z","timestamp":541555200000},"content-version":"tdm","delay-in-days":0,"URL":"http://doi.wiley.com/10.1002/tdm_license_1.1"},{"start":{"date-parts":[[2016,12,23]],"date-time":"2016-12-23T00:00:00Z","timestamp":1482451200000},"content-version":"vor","delay-in-days":10890,"URL":"http://onlinelibrary.wiley.com/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[1987,3]]},"DOI":"10.1112/blms/19.2.113","type":"journal-article","created":{"date-parts":[[2007,2,14]],"date-time":"2007-02-14T02:46:39Z","timestamp":1171421199000},"page":"113-134","source":"Crossref","is-referenced-by-count":143,"title":"From Groups to Groupoids: a Brief Survey","prefix":"10.1112","volume":"19","author":[{"given":"Ronald","family":"Brown","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2016,12,23]]},"container-title":"Bulletin of the London Mathematical Society","original-title":[],"language":"en","link":[{"URL":"https://api.wiley.com/onlinelibrary/tdm/v1/articles/10.1112%2Fblms%2F19.2.113","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://onlinelibrary.wiley.com/wol1/doi/10.1112/blms/19.2.113/fullpdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,20]],"date-time":"2021-07-20T11:22:08Z","timestamp":1626780128000},"score":1,"resource":{"primary":{"URL":"http://doi.wiley.com/10.1112/blms/19.2.113"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1987,3]]},"references-count":0,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1987,3]]}},"URL":"http://dx.doi.org/10.1112/blms/19.2.113","relation":{},"ISSN":["0024-6093"],"subject":["General Mathematics"],"container-title-short":"Bulletin of the London Mathematical Society","published":{"date-parts":[[1987,3]]},"reference":[],"id":"doi:10.1112/blms/19.2.113","_hash":"f37e1d70eb08e42ab422726cc78e280e11dc1fcb612548299c23b2d0d042ebf5"},"expire":1702831894998},"doi:10.1038/nphoton.2014.280":{"value":{"indexed":{"date-parts":[[2022,12,21]],"date-time":"2022-12-21T02:48:06Z","timestamp":1671590886702},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2014,11,17]],"date-time":"2014-11-17T00:00:00Z","timestamp":1416182400000},"content-version":"tdm","delay-in-days":0,"URL":"http://www.springer.com/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2015,1]]},"DOI":"10.1038/nphoton.2014.280","type":"journal-article","created":{"date-parts":[[2014,11,17]],"date-time":"2014-11-17T16:10:48Z","timestamp":1416240648000},"page":"48-53","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":83,"title":"Photon number resolution enables quantum receiver for realistic coherent optical communications","prefix":"10.1038","volume":"9","author":[{"given":"F. E.","family":"Becerra","sequence":"first","affiliation":[]},{"given":"J.","family":"Fan","sequence":"additional","affiliation":[]},{"given":"A.","family":"Migdall","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2014,11,17]]},"reference":[],"container-title":"Nature Photonics","original-title":[],"language":"en","link":[{"URL":"http://www.nature.com/articles/nphoton.2014.280.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/nphoton.2014.280","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/nphoton.2014.280.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T07:03:43Z","timestamp":1638428623000},"score":1,"resource":{"primary":{"URL":"http://www.nature.com/articles/nphoton.2014.280"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,11,17]]},"references-count":45,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2015,1]]}},"alternative-id":["BFnphoton2014280"],"URL":"http://dx.doi.org/10.1038/nphoton.2014.280","relation":{},"ISSN":["1749-4885","1749-4893"],"subject":["Atomic and Molecular Physics, and Optics","Electronic, Optical and Magnetic Materials"],"container-title-short":"Nature Photon","published":{"date-parts":[[2014,11,17]]},"assertion":[{"value":"26 May 2014","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 October 2014","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 November 2014","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing financial interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"id":"doi:10.1038/nphoton.2014.280","_hash":"2fbda0eeb355708ef57f0447af4ac91308df07d069683d5e2d8bef61ef7b1e85"},"expire":1702831896094},"doi:10.1038/s41534-022-00573-9":{"value":{"indexed":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T05:59:32Z","timestamp":1669355972248},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,5,27]],"date-time":"2022-05-27T00:00:00Z","timestamp":1653609600000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,5,27]],"date-time":"2022-05-27T00:00:00Z","timestamp":1653609600000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["ECCS 1927674"]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"<jats:title>Abstract</jats:title><jats:p>We present a systematic study of quantum receivers and modulation methods enabling resource efficient quantum-enhanced optical communication. We introduce quantum-inspired modulation schemes that theoretically yield a better resource efficiency than legacy protocols. Experimentally, we demonstrate below the shot-noise limit symbol error rates for <jats:italic>M</jats:italic> ≤ 16 legacy and quantum-inspired communication alphabets using software-configurable optical communication time-resolving quantum receiver testbed. Further, we experimentally verify that our quantum-inspired modulation schemes boost the accuracy of practical quantum measurements and significantly optimize the combined use of energy and bandwidth for communication alphabets that are longer than <jats:italic>M</jats:italic> = 4 symbols.</jats:p>","DOI":"10.1038/s41534-022-00573-9","type":"journal-article","created":{"date-parts":[[2022,5,27]],"date-time":"2022-05-27T10:03:49Z","timestamp":1653645829000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":"Energy and bandwidth efficiency optimization of quantum-enabled optical communication channels","prefix":"10.1038","volume":"8","author":[{"ORCID":"http://orcid.org/0000-0001-6022-6085","authenticated-orcid":false,"given":"M. V.","family":"Jabir","sequence":"first","affiliation":[]},{"given":"N. Fajar R.","family":"Annafianto","sequence":"additional","affiliation":[]},{"given":"I. A.","family":"Burenkov","sequence":"additional","affiliation":[]},{"given":"A.","family":"Battou","sequence":"additional","affiliation":[]},{"given":"S. V.","family":"Polyakov","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,5,27]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41534-022-00573-9.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-022-00573-9","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-022-00573-9.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T18:50:50Z","timestamp":1669315850000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-022-00573-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,27]]},"references-count":39,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["573"],"URL":"http://dx.doi.org/10.1038/s41534-022-00573-9","relation":{},"ISSN":["2056-6387"],"subject":["Computational Theory and Mathematics","Computer Networks and Communications","Statistical and Nonlinear Physics","Computer Science (miscellaneous)"],"container-title-short":"npj Quantum Inf","published":{"date-parts":[[2022,5,27]]},"assertion":[{"value":"15 September 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 April 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 May 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"63","id":"doi:10.1038/s41534-022-00573-9","_hash":"a5e53a90a575e7b1aa3b46e135e13b50b4f816b39c66d53cace69203b22a8608"},"expire":1702831896941},"doi:10.1364/OSAC.409200":{"value":{"indexed":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T19:20:10Z","timestamp":1669317610502},"reference-count":29,"publisher":"Optica Publishing Group","issue":"12","license":[{"start":{"date-parts":[[2020,11,19]],"date-time":"2020-11-19T00:00:00Z","timestamp":1605744000000},"content-version":"vor","delay-in-days":0,"URL":"https://doi.org/10.1364/OA_License_v1#VOR-OA"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["ECCS 1927674"]}],"content-domain":{"domain":["opg.optica.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2020,12,15]]},"abstract":"<jats:p>We implement the cyclic quantum receiver based on the theoretical proposal of Roy Bondurant and demonstrate experimentally below the shot-noise limit (SNL) discrimination of quadrature phase-shift keying signals (PSK). We also experimentally test the receiver generalized for longer communication alphabet lengths and coherent frequency shift keying (CFSK) encoding. Using off-the-shelf components, we obtain state discrimination error rates that are 3 dB and 4.6 dB below the SNLs of ideal classical receivers for quadrature PSK and CFSK encodings, respectively. The receiver unconditionally surpasses the SNL for M=8 PSK and CFSK. This receiver can be used for the simple and robust practical implementation of quantum-enhanced optical communication.</jats:p>","DOI":"10.1364/osac.409200","type":"journal-article","created":{"date-parts":[[2020,11,5]],"date-time":"2020-11-05T06:30:06Z","timestamp":1604557806000},"page":"3324","update-policy":"http://dx.doi.org/10.1364/crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":"Experimental demonstration of the near-quantum optimal receiver","prefix":"10.1364","volume":"3","author":[{"ORCID":"http://orcid.org/0000-0001-6022-6085","authenticated-orcid":true,"given":"M. V.","family":"Jabir","sequence":"first","affiliation":[]},{"given":"I. A.","family":"Burenkov","sequence":"additional","affiliation":[]},{"given":"N. Fajar R.","family":"Annafianto","sequence":"additional","affiliation":[]},{"given":"A.","family":"Battou","sequence":"additional","affiliation":[]},{"given":"S. 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Continuum","original-title":[],"language":"en","link":[{"URL":"https://opg.optica.org/viewmedia.cfm?URI=osac-3-12-3324","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://opg.optica.org/viewmedia.cfm?URI=osac-3-12-3324&seq=0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,7,22]],"date-time":"2022-07-22T19:48:02Z","timestamp":1658519282000},"score":1,"resource":{"primary":{"URL":"https://opg.optica.org/abstract.cfm?URI=osac-3-12-3324"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,19]]},"references-count":29,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2020]]},"published-print":{"date-parts":[[2020]]}},"URL":"http://dx.doi.org/10.1364/OSAC.409200","relation":{},"ISSN":["2578-7519"],"subject":["Electrical and Electronic Engineering","Atomic and Molecular Physics, and Optics","Electronic, Optical and 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We present detailed measurement sequences for magic state distillation protocols which can suppress arbitrary errors on any part of a protocol, assuming the independence of errors across qubits. Provided with input magic states, our protocol operates on a two-dimensional square grid by measurements of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi><mml:mi>Z</mml:mi></mml:math> on horizontal pairs of qubits, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi><mml:mi>X</mml:mi></mml:math> on vertical pairs, and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi></mml:math> on single qubits.</jats:p>","DOI":"10.22331/q-2021-01-20-383","type":"journal-article","created":{"date-parts":[[2021,1,20]],"date-time":"2021-01-20T17:24:30Z","timestamp":1611163470000},"page":"383","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Measurement sequences for magic state distillation","prefix":"10.22331","volume":"5","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Microsoft Quantum, Redmond, Washington, USA"}]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"name":"Microsoft Quantum, Santa Barbara, California, USA"},{"name":"Microsoft Quantum, Redmond, Washington, USA"}]}],"member":"9598","published-online":{"date-parts":[[2021,1,20]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2021-01-20-383/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2021,1,20]],"date-time":"2021-01-20T17:24:49Z","timestamp":1611163489000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2021-01-20-383/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,20]]},"references-count":20,"URL":"http://dx.doi.org/10.22331/q-2021-01-20-383","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2021,1,20]]},"article-number":"383","id":"doi:10.22331/q-2021-01-20-383","_hash":"9e886ba8a94589c03d799deacb490f8db5f258691aae229590d791052d1e0712"},"expire":1702831913975},"doi:10.26421/QIC22.11-12-3":{"value":{"indexed":{"date-parts":[[2022,9,28]],"date-time":"2022-09-28T05:40:12Z","timestamp":1664343612453},"reference-count":0,"publisher":"Rinton Press","issue":"11&12","content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"<jats:p>Quantum error correcting codes protect quantum computation from errors caused by decoherence and other noise. Here we study the problem of designing logical operations for quantum error correcting codes. We present an automated procedure that generates logical operations given known encoding and correcting procedures. Our technique is to use variational circuits for learning both the logical gates and the physical operations implementing them. This procedure can be implemented on near-term quantum computers via quantum process tomography. It enables automatic discovery of logical gates from analytically designed error correcting codes and can be extended to error correcting codes found by numerical optimization. We test the procedure by simulating small quantum codes of four to fifteen qubits showing that our procedure finds most logical gates known in the current literature. Additionally, it generates logical gates not found in the current literature for the [[5,1,2]] code, the [[6,3,2]] code, the [[8,3,2]] code, and the [[10,1,2]] code.</jats:p>","DOI":"10.26421/qic22.11-12-3","type":"journal-article","created":{"date-parts":[[2022,9,27]],"date-time":"2022-09-27T03:32:00Z","timestamp":1664249520000},"page":"947-964","source":"Crossref","is-referenced-by-count":0,"title":"Automated discovery of logical gates for quantum error correction (with Supplementary (153 pages))","prefix":"10.26421","volume":"22","author":[{"given":"Hongxiang","family":"Chen","sequence":"first","affiliation":[]},{"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[]},{"given":"Nikolas P.","family":"Breuckmann","sequence":"additional","affiliation":[]},{"given":"Edward","family":"Grant","sequence":"additional","affiliation":[]}],"member":"10955","published-online":{"date-parts":[[2022,8]]},"container-title":"Quantum Information and Computation","original-title":[],"deposited":{"date-parts":[[2022,9,27]],"date-time":"2022-09-27T03:32:05Z","timestamp":1664249525000},"score":1,"resource":{"primary":{"URL":"https://www.rintonpress.com/journals/doi/QIC22.11-12-3.html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8]]},"references-count":0,"journal-issue":{"issue":"11&12","published-online":{"date-parts":[[2022,8]]}},"URL":"http://dx.doi.org/10.26421/QIC22.11-12-3","relation":{},"ISSN":["1533-7146"],"subject":["Computational Theory and Mathematics","General Physics and Astronomy","Mathematical Physics","Nuclear and High Energy Physics","Statistical and Nonlinear Physics","Theoretical Computer Science"],"container-title-short":"QIC","published":{"date-parts":[[2022,8]]},"reference":[],"id":"doi:10.26421/QIC22.11-12-3","_hash":"c1a0a4c1bebb6ab68dfa6cf34961c5909430e9063835f2b54d1731f7b69b45fd"},"expire":1702831915046},"doi:10.1038/s41534-018-0085-z":{"value":{"indexed":{"date-parts":[[2022,12,21]],"date-time":"2022-12-21T05:48:38Z","timestamp":1671601718358},"reference-count":19,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2018,9,12]],"date-time":"2018-09-12T00:00:00Z","timestamp":1536710400000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2018,9,12]],"date-time":"2018-09-12T00:00:00Z","timestamp":1536710400000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1254119","CCF-1254119"]},{"DOI":"10.13039/100000183","name":"DOD | Army Research Office","doi-asserted-by":"publisher","award":["W911NF-12-1-0541","W911NF-12-1-0541"]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"<jats:title>Abstract</jats:title><jats:p>Reliable qubits are difficult to engineer, but standard fault-tolerance schemes use seven or more physical qubits to encode each logical qubit, with still more qubits required for error correction. The large overhead makes it hard to experiment with fault-tolerance schemes with multiple encoded qubits. Here, we study the 15-qubit Hamming code, which protects seven encoded qubits to distance three. We give fault-tolerant procedures for applying arbitrary Clifford operations on these encoded qubits, using only two extra qubits, 17 in total. In particular, individual encoded qubits within the code block can be targeted. Fault-tolerant universal computation is possible with four extra qubits, 19 in total. The procedures could enable testing more sophisticated protected circuits in small-scale quantum devices. Our main technique is to use gadgets to protect gates against correlated faults. We also take advantage of special code symmetries, and use pieceable fault tolerance.</jats:p>","DOI":"10.1038/s41534-018-0085-z","type":"journal-article","created":{"date-parts":[[2018,9,6]],"date-time":"2018-09-06T07:46:02Z","timestamp":1536219962000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":55,"title":"Fault-tolerant quantum computation with few qubits","prefix":"10.1038","volume":"4","author":[{"ORCID":"http://orcid.org/0000-0002-2570-2426","authenticated-orcid":false,"given":"Rui","family":"Chao","sequence":"first","affiliation":[]},{"given":"Ben W.","family":"Reichardt","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,9,12]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41534-018-0085-z.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-018-0085-z","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-018-0085-z.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T23:02:22Z","timestamp":1671577342000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-018-0085-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,12]]},"references-count":19,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,12]]}},"alternative-id":["85"],"URL":"http://dx.doi.org/10.1038/s41534-018-0085-z","relation":{},"ISSN":["2056-6387"],"subject":["Computational Theory and Mathematics","Computer Networks and Communications","Statistical and Nonlinear Physics","Computer Science (miscellaneous)"],"container-title-short":"npj Quantum Inf","published":{"date-parts":[[2018,9,12]]},"assertion":[{"value":"22 December 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 June 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 July 2018","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 September 2018","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"42","id":"doi:10.1038/s41534-018-0085-z","_hash":"d18c301ea384ae1d8742f39e0e24e389f24f829418a207eb6af5903bab464426"},"expire":1702831916066},"doi:10.1103/PhysRevA.54.4741":{"value":{"indexed":{"date-parts":[[2022,12,8]],"date-time":"2022-12-08T15:28:20Z","timestamp":1670513300426},"reference-count":9,"publisher":"American Physical Society (APS)","issue":"6","license":[{"start":{"date-parts":[[1996,12,1]],"date-time":"1996-12-01T00:00:00Z","timestamp":849398400000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.54.4741","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T02:27:51Z","timestamp":1027736871000},"page":"4741-4751","source":"Crossref","is-referenced-by-count":243,"title":"Simple quantum error-correcting codes","prefix":"10.1103","volume":"54","author":[{"given":"A. 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Rev. A","published":{"date-parts":[[1995,2,1]]},"id":"doi:10.1103/PhysRevA.51.992","_hash":"ccc68444bb3a8ec50bd24967f0e63cb70506583bcbcd8be6b2fb11c51aa411e4"},"expire":1702831921024},"doi:10.22331/q-2017-04-25-4":{"value":{"indexed":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T15:25:22Z","timestamp":1668093922603},"reference-count":56,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2017,4,25]],"date-time":"2017-04-25T00:00:00Z","timestamp":1493078400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>We study the fundamental limits on the reliable storage of quantum information in lattices of qubits by deriving tradeoff bounds for approximate quantum error correcting codes. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing various exact error-correction criteria. Our tradeoff bounds relate the number of physical qubits <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>, the number of encoded qubits <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math>, the code distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>, the accuracy parameter <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B4;</mml:mi></mml:math> that quantifies how well the erasure channel can be reversed, and the locality parameter <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x2113;</mml:mi></mml:math> that specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracy <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B4;</mml:mi></mml:math> that is exponentially small in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x2113;</mml:mi></mml:math>, which is the case for perturbations of local commuting projector codes, our bound reads <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi><mml:msup><mml:mi>d</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mfrac><mml:mn>2</mml:mn><mml:mrow><mml:mi>D</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mo>&amp;#x2264;</mml:mo><mml:mi>O</mml:mi><mml:mrow class=\"MJX-TeXAtom-OPEN\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mi>n</mml:mi><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>D</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mrow class=\"MJX-TeXAtom-CLOSE\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math> for codes on <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>D</mml:mi></mml:math>-dimensional lattices of Euclidean metric. We also find that the code distance of any local approximate code cannot exceed <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mrow class=\"MJX-TeXAtom-OPEN\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mi>&amp;#x2113;</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>D</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mrow class=\"MJX-TeXAtom-CLOSE\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math> if <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B4;</mml:mi><mml:mo>&amp;#x2264;</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>&amp;#x2113;</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. As a corollary of our formulation of correctability in terms of logical operator avoidance, we show that the code distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> and the size <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">&amp;#x007E;</mml:mo></mml:mover></mml:mrow></mml:math> of a minimal region that can support all approximate logical operators satisfies <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">&amp;#x007E;</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>D</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mo>&amp;#x2264;</mml:mo><mml:mi>O</mml:mi><mml:mrow class=\"MJX-TeXAtom-OPEN\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mi>n</mml:mi><mml:msup><mml:mi>&amp;#x2113;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mfrac><mml:mi>D</mml:mi><mml:mrow><mml:mi>D</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mrow class=\"MJX-TeXAtom-CLOSE\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math>, where the logical operators are accurate up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mrow class=\"MJX-TeXAtom-OPEN\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mi>&amp;#x03B4;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow class=\"MJX-TeXAtom-CLOSE\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math> in operator norm. Finally, we prove that for two-dimensional systems if logical operators can be approximated by operators supported on constant-width flexible strings, then the dimension of the code space must be bounded. This supports one of the assumptions of algebraic anyon theories, that there exist only finitely many anyon types.</jats:p>","DOI":"10.22331/q-2017-04-25-4","type":"journal-article","created":{"date-parts":[[2017,4,25]],"date-time":"2017-04-25T08:51:31Z","timestamp":1493110291000},"page":"4","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":20,"title":"Limits on the storage of quantum information in a volume of space","prefix":"10.22331","volume":"1","author":[{"given":"Steven T.","family":"Flammia","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Australia"},{"name":"Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, USA"}]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[{"name":"Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington, USA"},{"name":"Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, USA"}]},{"given":"Michael J.","family":"Kastoryano","sequence":"additional","affiliation":[{"name":"NBIA, Niels Bohr Institute, University of Copenhagen, Denmark"}]},{"given":"Isaac H.","family":"Kim","sequence":"additional","affiliation":[{"name":"IBM T. 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Theory","published":{"date-parts":[[1962,9]]},"id":"doi:10.1109/TIT.1962.1057777","_hash":"54de4d46cbdf30478ce3b68fc251ef505b82283b88a3107150e13dd4fc2f9583"},"expire":1703605015293},"doi:10.1007/978-3-642-29011-4_31":{"value":{"indexed":{"date-parts":[[2022,12,30]],"date-time":"2022-12-30T22:16:49Z","timestamp":1672438609517},"publisher-location":"Berlin, Heidelberg","reference-count":31,"publisher":"Springer Berlin Heidelberg","isbn-type":[{"value":"9783642290107","type":"print"},{"value":"9783642290114","type":"electronic"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2012]]},"DOI":"10.1007/978-3-642-29011-4_31","type":"book-chapter","created":{"date-parts":[[2012,4,5]],"date-time":"2012-04-05T07:05:53Z","timestamp":1333609553000},"page":"520-536","source":"Crossref","is-referenced-by-count":147,"title":"Decoding Random Binary Linear Codes in 2 n/20: How 1 + 1 = 0 Improves Information Set Decoding","prefix":"10.1007","author":[{"given":"Anja","family":"Becker","sequence":"first","affiliation":[]},{"given":"Antoine","family":"Joux","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"May","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Meurer","sequence":"additional","affiliation":[]}],"member":"297","reference":[],"container-title":"Advances in Cryptology – EUROCRYPT 2012","original-title":[],"link":[{"URL":"http://link.springer.com/content/pdf/10.1007/978-3-642-29011-4_31.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,5,4]],"date-time":"2021-05-04T11:18:41Z","timestamp":1620127121000},"score":1,"resource":{"primary":{"URL":"http://link.springer.com/10.1007/978-3-642-29011-4_31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012]]},"ISBN":["9783642290107","9783642290114"],"references-count":31,"URL":"http://dx.doi.org/10.1007/978-3-642-29011-4_31","relation":{},"ISSN":["0302-9743","1611-3349"],"published":{"date-parts":[[2012]]},"id":"doi:10.1007/978-3-642-29011-4_31","_hash":"fc1a6fe095d6193f314bf05e6ccbd42c02c58507f59d3103f5d229a31c595df2"},"expire":1703605016338},"doi:10.1007/JHEP09(2019)021":{"value":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T12:04:33Z","timestamp":1648555473999},"reference-count":79,"publisher":"Springer Science and Business Media LLC","issue":"9","license":[{"start":{"date-parts":[[2019,9,1]],"date-time":"2019-09-01T00:00:00Z","timestamp":1567296000000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"},{"start":{"date-parts":[[2019,9,3]],"date-time":"2019-09-03T00:00:00Z","timestamp":1567468800000},"content-version":"vor","delay-in-days":2,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2019,9]]},"abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>Motivated by the close relationship between quantum error-correction, topological order, the holographic AdS/CFT duality, and tensor networks, we initiate the study of approximate quantum error-detecting codes in matrix product states (MPS). We first show that using open-boundary MPS to define boundary to bulk encoding maps yields at most constant distance error-detecting codes. These are degenerate ground spaces of gapped local Hamiltonians. To get around this no-go result, we consider excited states, i.e., we use the excitation ansatz to construct encoding maps: these yield error-detecting codes with distance Ω(<jats:italic>n</jats:italic>\n            <jats:sup>1−ν</jats:sup> ) for any <jats:italic>ν</jats:italic> ∈ (0<jats:italic>,</jats:italic> 1) and Ω(log <jats:italic>n</jats:italic>) encoded qubits. This shows that gapped systems contain — within isolated energy bands — error-detecting codes spanned by momentum eigenstates. We also consider the gapless Heisenberg-XXX model, whose energy eigenstates can be described via Bethe ansatz tensor networks. We show that it contains — within its low-energy eigenspace — an error-detecting code with the same parameter scaling. All these codes detect arbitrary <jats:italic>d</jats:italic>-local (not necessarily geometrically local) errors even though they are not permutation-invariant. This suggests that a wide range of naturally occurring many-body systems possess intrinsic error-detecting features.</jats:p>","DOI":"10.1007/jhep09(2019)021","type":"journal-article","created":{"date-parts":[[2019,9,5]],"date-time":"2019-09-05T15:02:47Z","timestamp":1567695767000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":"Quantum error-detection at low energies","prefix":"10.1007","volume":"2019","author":[{"given":"Martina","family":"Gschwendtner","sequence":"first","affiliation":[]},{"given":"Robert","family":"König","sequence":"additional","affiliation":[]},{"given":"Burak","family":"Şahinoğlu","sequence":"additional","affiliation":[]},{"given":"Eugene","family":"Tang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,9,3]]},"reference":[],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","link":[{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP09(2019)021.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/article/10.1007/JHEP09(2019)021/fulltext.html","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP09(2019)021.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,3]],"date-time":"2021-09-03T18:13:22Z","timestamp":1630692802000},"score":1,"resource":{"primary":{"URL":"https://link.springer.com/10.1007/JHEP09(2019)021"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9]]},"references-count":79,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2019,9]]}},"alternative-id":["11204"],"URL":"http://dx.doi.org/10.1007/JHEP09(2019)021","relation":{},"ISSN":["1029-8479"],"subject":["Nuclear and High Energy Physics"],"container-title-short":"J. High Energ. Phys.","published":{"date-parts":[[2019,9]]},"assertion":[{"value":"14 June 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 August 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 September 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"21","id":"doi:10.1007/JHEP09(2019)021","_hash":"aedf3a33353963fc29b4c62502b6835ca4216df47ebc6fc530e03d7e0bbf540d"},"expire":1703605017393},"doi:10.1007/JHEP01(2022)170":{"value":{"indexed":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T15:28:08Z","timestamp":1668094088262},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,1,27]],"date-time":"2022-01-27T00:00:00Z","timestamp":1643241600000},"content-version":"vor","delay-in-days":26,"URL":"https://creativecommons.org/licenses/by/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2022,1]]},"abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n                     </jats:title><jats:p>There are two parts to this work: first, we study the error correction properties of the real-space renormalization group (RG). The long-distance operators are the (approximately) correctable operators encoded in the physical algebra of short-distance operators. This is closely related to modeling the holographic map as a quantum error correction code. As opposed to holography, the real-space RG of a many-body quantum system does not have the complementary recovery property. We discuss the role of large <jats:italic>N</jats:italic> and a large gap in the spectrum of operators in the emergence of complementary recovery.</jats:p><jats:p>Second, we study the operator algebra exact quantum error correction for any von Neumann algebra. We show that similar to the finite dimensional case, for any error map in between von Neumann algebras the Petz dual of the error map is a recovery map if the inclusion of the correctable subalgebra of operators has finite index.</jats:p>","DOI":"10.1007/jhep01(2022)170","type":"journal-article","created":{"date-parts":[[2022,1,28]],"date-time":"2022-01-28T03:03:27Z","timestamp":1643339007000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":"Real-space RG, error correction and Petz map","prefix":"10.1007","volume":"2022","author":[{"given":"Keiichiro","family":"Furuya","sequence":"first","affiliation":[]},{"given":"Nima","family":"Lashkari","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-4143-5276","authenticated-orcid":false,"given":"Shoy","family":"Ouseph","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,1,27]]},"reference":[],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","link":[{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP01(2022)170.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/article/10.1007/JHEP01(2022)170/fulltext.html","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP01(2022)170.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,25]],"date-time":"2022-04-25T21:26:33Z","timestamp":1650921993000},"score":1,"resource":{"primary":{"URL":"https://link.springer.com/10.1007/JHEP01(2022)170"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1]]},"references-count":48,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,1]]}},"alternative-id":["17662"],"URL":"http://dx.doi.org/10.1007/JHEP01(2022)170","relation":{},"ISSN":["1029-8479"],"subject":["Nuclear and High Energy Physics"],"container-title-short":"J. High Energ. 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Rev. Lett.","published":{"date-parts":[[2007,3,26]]},"article-number":"130501","id":"doi:10.1103/PhysRevLett.98.130501","_hash":"3bebec79db69e3879d3689444dd6cdc5349cd6dab5b517bf6a6b12027ca77cd7"},"expire":1703605019790},"doi:10.22331/q-2022-09-29-821":{"value":{"indexed":{"date-parts":[[2022,9,30]],"date-time":"2022-09-30T05:07:25Z","timestamp":1664514445154},"reference-count":35,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T00:00:00Z","timestamp":1664409600000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"ARO","award":["W911NF-18-1-0020"]},{"name":"ARO","award":["W911NF-18-1-0212"]},{"name":"ARO MURI","award":["W911NF-16-1-0349"]},{"name":"ARO MURI","award":["W911NF-21-1-0325"]},{"name":"AFOSR MURI","award":["FA9550-19-1-0399"]},{"name":"AFOSR MURI","award":["FA9550-21-1-0209"]},{"DOI":"10.13039/100006602","name":"AFRL","doi-asserted-by":"crossref","award":["FA8649-21-P-0781"]},{"name":"NSF","award":["OMA1936118"]},{"name":"NSF","award":["EEC-1941583"]},{"name":"NSF","award":["OMA-2137642"]},{"name":"NTT Research, and the Packard Foundation","award":["2020-71479"]},{"name":"ERC","award":["134847"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Bosonic qubits encoded in continuous-variable systems provide a promising alternative to two-level qubits for quantum computation and communication. So far, photon loss has been the dominant source of errors in bosonic qubits, but the significant reduction of photon loss in recent bosonic qubit experiments suggests that dephasing errors should also be considered. However, a detailed understanding of the combined photon loss and dephasing channel is lacking. Here, we show that, unlike its constituent parts, the combined loss-dephasing channel is non-degradable, pointing towards a richer structure of this channel. We provide bounds for the capacity of the loss-dephasing channel and use numerical optimization to find optimal single-mode codes for a wide range of error rates.</jats:p>","DOI":"10.22331/q-2022-09-29-821","type":"journal-article","created":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T12:26:49Z","timestamp":1664454409000},"page":"821","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Quantum capacity and codes for the bosonic loss-dephasing channel","prefix":"10.22331","volume":"6","author":[{"given":"Peter","family":"Leviant","sequence":"first","affiliation":[{"name":"Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel"}]},{"given":"Qian","family":"Xu","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA"}]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA"}]},{"given":"Serge","family":"Rosenblum","sequence":"additional","affiliation":[{"name":"Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel"}]}],"member":"9598","published-online":{"date-parts":[[2022,9,29]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-09-29-821/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T12:27:53Z","timestamp":1664454473000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-09-29-821/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,29]]},"references-count":35,"URL":"http://dx.doi.org/10.22331/q-2022-09-29-821","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,9,29]]},"article-number":"821","id":"doi:10.22331/q-2022-09-29-821","_hash":"96bec0481798cafaafa0b8232d56dfde88a97b3fa589719107cd558c8b584d64"},"expire":1703605021750},"doi:10.1103/PhysRevD.106.105007":{"value":{"indexed":{"date-parts":[[2022,11,11]],"date-time":"2022-11-11T05:52:54Z","timestamp":1668145974433},"reference-count":40,"publisher":"American Physical Society (APS)","issue":"10","license":[{"start":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T00:00:00Z","timestamp":1668038400000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["PHY-1911298"]},{"DOI":"10.13039/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE- SC0007884"]},{"name":"QuantiSED Fermilab consortium"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevd.106.105007","type":"journal-article","created":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T15:03:08Z","timestamp":1668092588000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":"Renormalization group and approximate error correction","prefix":"10.1103","volume":"106","author":[{"ORCID":"http://orcid.org/0000-0002-1289-6831","authenticated-orcid":true,"given":"Keiichiro","family":"Furuya","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-3446-5933","authenticated-orcid":true,"given":"Nima","family":"Lashkari","sequence":"additional","affiliation":[]},{"given":"Mudassir","family":"Moosa","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2022,11,10]]},"reference":[],"container-title":"Physical Review D","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevD.106.105007","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevD.106.105007/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T15:03:15Z","timestamp":1668092595000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevD.106.105007"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,10]]},"references-count":40,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2022,11]]}},"URL":"http://dx.doi.org/10.1103/PhysRevD.106.105007","relation":{},"ISSN":["2470-0010","2470-0029"],"container-title-short":"Phys. 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We propose a new encryption with a public key matrix by considering the adding of a random distortion matrix over     F  q m      of full column rank n. We show that IND-CPA security is achievable for our encryption under assumption of the Decisional Rank Syndrome Decoding problem. Furthermore, we also prove some bounds for the number of matrices of a fixed rank with entries over a finite field. Our proposal allows the choice of the error terms with rank up to     r 2    , where r is the error-correcting capability of a code. Our encryption based on Gabidulin codes has public key size of     13 . 68     KB, which is 82 times smaller than the public key size of McEliece Cryptosystem based on Goppa codes. For similar post-quantum security level of     2 140     bits, our encryption scheme has a smaller public key size than the key size suggested by LOI17 Encryption.</jats:p>","DOI":"10.3390/cryptography2040032","type":"journal-article","created":{"date-parts":[[2018,10,16]],"date-time":"2018-10-16T06:52:53Z","timestamp":1539672773000},"page":"32","source":"Crossref","is-referenced-by-count":5,"title":"A New Technique in Rank Metric Code-Based Encryption","prefix":"10.3390","volume":"2","author":[{"given":"Terry","family":"Lau","sequence":"first","affiliation":[]},{"given":"Chik","family":"Tan","sequence":"additional","affiliation":[]}],"member":"1968","published-online":{"date-parts":[[2018,10,15]]},"container-title":"Cryptography","original-title":[],"language":"en","link":[{"URL":"http://www.mdpi.com/2410-387X/2/4/32/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2018,10,16]],"date-time":"2018-10-16T06:59:22Z","timestamp":1539673162000},"score":1,"resource":{"primary":{"URL":"http://www.mdpi.com/2410-387X/2/4/32"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,15]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,12]]}},"alternative-id":["cryptography2040032"],"URL":"http://dx.doi.org/10.3390/cryptography2040032","relation":{},"ISSN":["2410-387X"],"subject":["Applied Mathematics","Computational Theory and Mathematics","Computer Networks and Communications","Computer Science Applications","Software"],"container-title-short":"Cryptography","published":{"date-parts":[[2018,10,15]]},"reference":[],"id":"doi:10.3390/cryptography2040032","_hash":"416a9eb7210550bd1f1160406fb074e984d6f72937194e25a1653b156cc34815"},"expire":1705266811927},"doi:10.1088/1367-2630/16/6/063038":{"value":{"indexed":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T16:36:24Z","timestamp":1671554184791},"reference-count":68,"publisher":"IOP Publishing","issue":"6","license":[{"start":{"date-parts":[[2014,6,17]],"date-time":"2014-06-17T00:00:00Z","timestamp":1402963200000},"content-version":"tdm","delay-in-days":0,"URL":"http://iopscience.iop.org/info/page/text-and-data-mining"},{"start":{"date-parts":[[2014,6,17]],"date-time":"2014-06-17T00:00:00Z","timestamp":1402963200000},"content-version":"vor","delay-in-days":0,"URL":"http://creativecommons.org/licenses/by/3.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1088/1367-2630/16/6/063038","type":"journal-article","created":{"date-parts":[[2014,6,17]],"date-time":"2014-06-17T16:55:28Z","timestamp":1403024128000},"page":"063038","source":"Crossref","is-referenced-by-count":43,"title":"Fast decoders for qudit topological codes","prefix":"10.1088","volume":"16","author":[{"given":"Hussain","family":"Anwar","sequence":"first","affiliation":[]},{"given":"Benjamin J","family":"Brown","sequence":"additional","affiliation":[]},{"given":"Earl T","family":"Campbell","sequence":"additional","affiliation":[]},{"given":"Dan E","family":"Browne","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2014,6,17]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"link":[{"URL":"http://stacks.iop.org/1367-2630/16/i=6/a=063038/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://stacks.iop.org/1367-2630/16/i=6/a=063038?key=crossref.cfc8882fe938407cb31511bc503a0006","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http://stacks.iop.org/1367-2630/16/i=6/a=063038/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"http://stacks.iop.org/1367-2630/16/i=6/a=063038?key=crossref.cfc8882fe938407cb31511bc503a0006","content-type":"text/html","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,8]],"date-time":"2022-04-08T15:46:40Z","timestamp":1649432800000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/16/6/063038"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,6,17]]},"references-count":68,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2014,6,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/16/6/063038","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. 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Quantum measurements to discriminate multiple states at the single-photon level are essential for optimizing information transfer in low-power optical communications and quantum communications, and can enhance the capabilities of many quantum information protocols. Here, we theoretically investigate and experimentally demonstrate the discrimination of multiple coherent states of light with sensitivities surpassing the quantum noise limit (QNL) at the single-photon level under realistic conditions of loss and noise based on strategies implementing globally-optimized adaptive measurements with single photon counting and displacement operations. These discrimination strategies can provide realistic advantages to enhance information transfer at low powers, and are compatible with photon number resolving detection, which provides robustness at high powers, thus allowing for surpassing the QNL at arbitrary input power levels under realistic conditions.</jats:p>","DOI":"10.1038/s41534-017-0042-2","type":"journal-article","created":{"date-parts":[[2017,10,9]],"date-time":"2017-10-09T16:04:29Z","timestamp":1507565069000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":"Multi-state discrimination below the quantum noise limit at the single-photon level","prefix":"10.1038","volume":"3","author":[{"given":"A. R.","family":"Ferdinand","sequence":"first","affiliation":[]},{"given":"M. T.","family":"DiMario","sequence":"additional","affiliation":[]},{"given":"F. 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A recent work by Fawzi, Grospellier and Leverrier (FOCS 2018) building on a result by Gottesman (QIC 2013) has shown that the space overhead can be asymptotically reduced to a constant independent of the circuit provided we only consider circuits with a length bounded by a polynomial in the width. In this work, using a minimal model for quantum fault tolerance, we establish a general lower bound on the space overhead required to achieve fault tolerance. \r\nFor any non-unitary qubit channel 𝒩 and any quantum fault tolerance schemes against i.i.d. noise modeled by 𝒩, we prove a lower bound of max{Q(𝒩)^{-1}n,α_𝒩 log T} on the number of physical qubits, for circuits of length T and width n. Here, Q(𝒩) denotes the quantum capacity of 𝒩 and α_𝒩 > 0 is a constant only depending on the channel 𝒩. In our model, we allow for qubits to be replaced by fresh ones during the execution of the circuit and in the case of unital noise, we allow classical computation to be free and perfect. This improves upon results that assumed classical computations to be also affected by noise, and that sometimes did not allow for fresh qubits to be added. Along the way, we prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude damping noise resolving a conjecture by Ben-Or, Gottesman and Hassidim (2013).","container-title":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","DOI":"10.4230/LIPICS.ITCS.2022.68","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","title":"A Lower Bound on the Space Overhead of Fault-Tolerant Quantum Computation","URL":"https://drops.dagstuhl.de/opus/volltexte/2022/15664/","copyright":"Creative Commons Attribution 4.0 International license","reference":[],"_hash":"8d2072ca718b40d6ed2fb41d2639e0d4f5ab8c9bdacc01ade7c83a8eb6094767"},"expire":1708162723395},"doi:10.22331/q-2022-10-06-828":{"value":{"indexed":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T11:10:18Z","timestamp":1675249818437},"reference-count":202,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,10,6]],"date-time":"2022-10-06T00:00:00Z","timestamp":1665014400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/501100001870","name":"the Foundation for Polish Science","doi-asserted-by":"crossref","award":["MAB/2018/5"]},{"name":"General Research Fund","award":["GRF/16305121"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Quantum error correction is believed to be a necessity for large-scale fault-tolerant quantum computation. In the past two decades, various constructions of quantum error-correcting codes (QECCs) have been developed, leading to many good code families. However, the majority of these codes are not suitable for near-term quantum devices. Here we present VarQEC, a noise-resilient variational quantum algorithm to search for quantum codes with a hardware-efficient encoding circuit. The cost functions are inspired by the most general and fundamental requirements of a QECC, the Knill-Laflamme conditions. Given the target noise channel (or the target code parameters) and the hardware connectivity graph, we optimize a shallow variational quantum circuit to prepare the basis states of an eligible code. In principle, VarQEC can find quantum codes for any error model, whether additive or non-additive, degenerate or non-degenerate, pure or impure. We have verified its effectiveness by (re)discovering some symmetric and asymmetric codes, e.g., <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>n</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math> from 7 to 14. We also found new <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>6</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>7</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> codes that are not equivalent to any stabilizer code, and extensive numerical evidence with VarQEC suggests that a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>7</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> code does not exist. Furthermore, we found many new channel-adaptive codes for error models involving nearest-neighbor correlated errors. Our work sheds new light on the understanding of QECC in general, which may also help to enhance near-term device performance with channel-adaptive error-correcting codes.</jats:p>","DOI":"10.22331/q-2022-10-06-828","type":"journal-article","created":{"date-parts":[[2022,10,6]],"date-time":"2022-10-06T13:21:02Z","timestamp":1665062462000},"page":"828","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":"Quantum variational learning for quantum error-correcting codes","prefix":"10.22331","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0001-5589-7503","authenticated-orcid":false,"given":"Chenfeng","family":"Cao","sequence":"first","affiliation":[{"name":"Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China"}]},{"ORCID":"http://orcid.org/0000-0002-2093-7496","authenticated-orcid":false,"given":"Chao","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China"}]},{"ORCID":"http://orcid.org/0000-0002-9349-1325","authenticated-orcid":false,"given":"Zipeng","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China"}]},{"ORCID":"http://orcid.org/0000-0002-3720-5195","authenticated-orcid":false,"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University of Gdansk, 80-309 Gdansk, Poland"}]},{"ORCID":"http://orcid.org/0000-0003-3989-4948","authenticated-orcid":false,"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[{"name":"Department of 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The Epstein zeta function of <jats:italic>f</jats:italic> is denned to be</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0017089500007047_eqnU1\" /></jats:disp-formula></jats:p><jats:p>Rankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s &gt; 0,</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0017089500007047_eqn1\" /></jats:disp-formula></jats:p><jats:p>We prove a corresponding result for theta functions. 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Leveraging such protection requires testing its robustness against uncontrolled environmental interactions. Using 47 superconducting qubits, we implement the one-dimensional kicked Ising model, which exhibits nonlocal Majorana edge modes (MEMs) with\n            <jats:inline-formula>\n              <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n                <mml:mrow>\n                  <mml:msub>\n                    <mml:mi>ℤ</mml:mi>\n                    <mml:mn>2</mml:mn>\n                  </mml:msub>\n                </mml:mrow>\n              </mml:math>\n            </jats:inline-formula>\n            parity symmetry. We find that any multiqubit Pauli operator overlapping with the MEMs exhibits a uniform late-time decay rate comparable to single-qubit relaxation rates, irrespective of its size or composition. This characteristic allows us to accurately reconstruct the exponentially localized spatial profiles of the MEMs. Furthermore, the MEMs are found to be resilient against certain symmetry-breaking noise owing to a prethermalization mechanism. Our work elucidates the complex interplay between noise and symmetry-protected edge modes in a solid-state environment.\n          </jats:p>","DOI":"10.1126/science.abq5769","type":"journal-article","created":{"date-parts":[[2022,11,17]],"date-time":"2022-11-17T18:58:33Z","timestamp":1668711513000},"page":"785-790","source":"Crossref","is-referenced-by-count":3,"title":"Noise-resilient edge modes on a chain of superconducting qubits","prefix":"10.1126","volume":"378","author":[{"ORCID":"http://orcid.org/0000-0003-0507-0211","authenticated-orcid":true,"given":"X.","family":"Mi","sequence":"first","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"http://orcid.org/0000-0002-9003-5914","authenticated-orcid":true,"given":"M.","family":"Sonner","sequence":"additional","affiliation":[{"name":"Department of Theoretical Physics, University of Geneva, Geneva, Switzerland."}]},{"given":"M. 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Unfortunately, their exchange that can reveal their exotic statistics needs manipulations that are still beyond our experimental capabilities. Here we take an alternative approach. Through the Jordan–Wigner transformation, the Kitaev's chain supporting two Majorana zero modes is mapped to the spin-1/2 chain. We experimentally simulated the spin system and its evolution with a photonic quantum simulator. This allows us to probe the geometric phase, which corresponds to the exchange of two Majorana zero modes positioned at the ends of a three-site chain. Finally, we demonstrate the immunity of quantum information encoded in the Majorana zero modes against local errors through the simulator. Our photonic simulator opens the way for the efficient realization and manipulation of Majorana zero modes in complex architectures.</jats:p>","DOI":"10.1038/ncomms13194","type":"journal-article","created":{"date-parts":[[2016,10,25]],"date-time":"2016-10-25T09:37:45Z","timestamp":1477388265000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":"Simulating the exchange of Majorana zero modes with a photonic system","prefix":"10.1038","volume":"7","author":[{"given":"Jin-Shi","family":"Xu","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-3018-7219","authenticated-orcid":false,"given":"Kai","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Yong-Jian","family":"Han","sequence":"additional","affiliation":[]},{"given":"Chuan-Feng","family":"Li","sequence":"additional","affiliation":[]},{"given":"Jiannis K.","family":"Pachos","sequence":"additional","affiliation":[]},{"given":"Guang-Can","family":"Guo","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2016,10,25]]},"reference":[],"container-title":"Nature 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Call a configuration sharp if there are <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m\">\n  <mml:semantics>\n    <mml:mi>m</mml:mi>\n    <mml:annotation encoding=\"application/x-tex\">m</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> distances between distinct points in it and it is a spherical <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis 2 m minus 1 right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo stretchy=\"false\">(</mml:mo>\n      <mml:mn>2</mml:mn>\n      <mml:mi>m</mml:mi>\n      <mml:mo>−<!-- − --></mml:mo>\n      <mml:mn>1</mml:mn>\n      <mml:mo stretchy=\"false\">)</mml:mo>\n    </mml:mrow>\n    <mml:annotation encoding=\"application/x-tex\">(2m-1)</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula>-design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E 8\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>E</mml:mi>\n      <mml:mn>8</mml:mn>\n    </mml:msub>\n    <mml:annotation encoding=\"application/x-tex\">E_8</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> and Leech lattices. We also prove the same result for the vertices of the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"600\">\n  <mml:semantics>\n    <mml:mn>600</mml:mn>\n    <mml:annotation encoding=\"application/x-tex\">600</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula>-cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact two-point homogeneous spaces, and we conclude with an extension to Euclidean space.</p>","DOI":"10.1090/s0894-0347-06-00546-7","type":"journal-article","created":{"date-parts":[[2006,10,14]],"date-time":"2006-10-14T01:46:17Z","timestamp":1160790377000},"page":"99-148","source":"Crossref","is-referenced-by-count":193,"title":"Universally optimal distribution of points on spheres","prefix":"10.1090","volume":"20","author":[{"given":"Henry","family":"Cohn","sequence":"first","affiliation":[]},{"given":"Abhinav","family":"Kumar","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,9,5]]},"reference":[],"container-title":"Journal of the American Mathematical Society","original-title":[],"language":"en","link":[{"URL":"http://www.ams.org/jams/2007-20-01/S0894-0347-06-00546-7/S0894-0347-06-00546-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://www.ams.org/jams/2007-20-01/S0894-0347-06-00546-7/S0894-0347-06-00546-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,29]],"date-time":"2021-07-29T21:43:35Z","timestamp":1627595015000},"score":1,"resource":{"primary":{"URL":"https://www.ams.org/jams/2007-20-01/S0894-0347-06-00546-7/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,9,5]]},"references-count":62,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["S0894-0347-06-00546-7"],"URL":"http://dx.doi.org/10.1090/S0894-0347-06-00546-7","relation":{},"ISSN":["0894-0347","1088-6834"],"subject":["Applied Mathematics","General Mathematics"],"container-title-short":"J. 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Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.","name":"license_information","label":"License information"},{"value":"2018-11-13","name":"date_received","label":"Date received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2019-01-24","name":"date_accepted","label":"Date accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2019-03-28","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/ab0199","_hash":"78dd9a0055ffd1b60a52c554f811ecc329ca1a08bb6826747c436dcd84f340f9"},"expire":1712098433985},"doi:10.1103/PhysRevA.107.042407":{"value":{"indexed":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T05:13:14Z","timestamp":1680844394312},"reference-count":54,"publisher":"American Physical Society (APS)","issue":"4","license":[{"start":{"date-parts":[[2023,4,6]],"date-time":"2023-04-06T00:00:00Z","timestamp":1680739200000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"funder":[{"name":"Plan France 2030","award":["ANR-22-PETQ-0006"]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physreva.107.042407","type":"journal-article","created":{"date-parts":[[2023,4,6]],"date-time":"2023-04-06T14:03:55Z","timestamp":1680789835000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":"Two-photon driven Kerr quantum oscillator with multiple spectral degeneracies","prefix":"10.1103","volume":"107","author":[{"ORCID":"http://orcid.org/0000-0003-2003-7030","authenticated-orcid":true,"given":"Diego","family":"Ruiz","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-6136-6688","authenticated-orcid":true,"given":"Ronan","family":"Gautier","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-6507-9344","authenticated-orcid":true,"given":"Jérémie","family":"Guillaud","sequence":"additional","affiliation":[]},{"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2023,4,6]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevA.107.042407","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.107.042407/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,6]],"date-time":"2023-04-06T14:04:30Z","timestamp":1680789870000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.107.042407"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,6]]},"references-count":54,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,4]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.107.042407","relation":{},"ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. 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Sloan Foundation","doi-asserted-by":"publisher","award":["BR2013-049"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2018,7,20]]},"abstract":"<jats:title>Fault-tolerant quantum coding</jats:title>\n          <jats:p>\n            Noise and imperfections in a quantum system can result in the presence and propagation of errors through the system. A reliable quantum processor will need to be able to correct for these errors and error syndromes. Rosenblum\n            <jats:italic>et al.</jats:italic>\n            used higher quantum states of a superconducting-based quantum circuit to demonstrate a method for the fault-tolerant measurement of an error-correctable logical qubit. Such fault-tolerant measurements will allow more frequent interrogations of the state of the logical qubit, ultimately leading to the implementation of more quantum operations and more complex entangled quantum circuits.\n          </jats:p>\n          <jats:p>\n            <jats:italic>Science</jats:italic>\n            , this issue p.\n            <jats:related-article xmlns:xlink=\"http://www.w3.org/1999/xlink\" ext-link-type=\"doi\" issue=\"6399\" page=\"266\" related-article-type=\"in-this-issue\" vol=\"361\" xlink:href=\"10.1126/science.aat3996\">266</jats:related-article>\n          </jats:p>","DOI":"10.1126/science.aat3996","type":"journal-article","created":{"date-parts":[[2018,7,19]],"date-time":"2018-07-19T18:05:33Z","timestamp":1532023533000},"page":"266-270","source":"Crossref","is-referenced-by-count":89,"title":"Fault-tolerant detection of a quantum error","prefix":"10.1126","volume":"361","author":[{"given":"S.","family":"Rosenblum","sequence":"first","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]},{"ORCID":"http://orcid.org/0000-0002-8141-1842","authenticated-orcid":true,"given":"P.","family":"Reinhold","sequence":"additional","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]},{"given":"M.","family":"Mirrahimi","sequence":"additional","affiliation":[{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."},{"name":"QUANTIC team, INRIA de Paris, 2 Rue Simone Iff, 75012 Paris, France."}]},{"ORCID":"http://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]},{"ORCID":"http://orcid.org/0000-0002-0272-5481","authenticated-orcid":true,"given":"L.","family":"Frunzio","sequence":"additional","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]},{"ORCID":"http://orcid.org/0000-0002-2469-5068","authenticated-orcid":true,"given":"R. J.","family":"Schoelkopf","sequence":"additional","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]}],"member":"221","reference":[],"container-title":"Science","original-title":[],"language":"en","link":[{"URL":"https://syndication.highwire.org/content/doi/10.1126/science.aat3996","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"https://syndication.highwire.org/content/doi/10.1126/science.aat3996","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,13]],"date-time":"2022-01-13T20:57:29Z","timestamp":1642107449000},"score":1,"resource":{"primary":{"URL":"https://www.science.org/doi/10.1126/science.aat3996"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,20]]},"references-count":23,"journal-issue":{"issue":"6399","published-print":{"date-parts":[[2018,7,20]]}},"alternative-id":["10.1126/science.aat3996"],"URL":"http://dx.doi.org/10.1126/science.aat3996","relation":{},"ISSN":["0036-8075","1095-9203"],"subject":["Multidisciplinary"],"container-title-short":"Science","published":{"date-parts":[[2018,7,20]]},"id":"doi:10.1126/science.aat3996","_hash":"5f11a35a2b98324879304d709db301d9a23e5e9fe357582f5bf9bd71a8cf7244"},"expire":1712098435953},"doi:10.1038/nature13436":{"value":{"indexed":{"date-parts":[[2023,3,28]],"date-time":"2023-03-28T13:51:26Z","timestamp":1680011486778},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"7510","license":[{"start":{"date-parts":[[2014,7,1]],"date-time":"2014-07-01T00:00:00Z","timestamp":1404172800000},"content-version":"tdm","delay-in-days":0,"URL":"http://www.springer.com/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2014,7]]},"DOI":"10.1038/nature13436","type":"journal-article","created":{"date-parts":[[2014,7,11]],"date-time":"2014-07-11T14:48:46Z","timestamp":1405090126000},"page":"444-448","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":175,"title":"Tracking photon jumps with repeated quantum non-demolition parity measurements","prefix":"10.1038","volume":"511","author":[{"given":"L.","family":"Sun","sequence":"first","affiliation":[]},{"given":"A.","family":"Petrenko","sequence":"additional","affiliation":[]},{"given":"Z.","family":"Leghtas","sequence":"additional","affiliation":[]},{"given":"B.","family":"Vlastakis","sequence":"additional","affiliation":[]},{"given":"G.","family":"Kirchmair","sequence":"additional","affiliation":[]},{"given":"K. 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Typical measurement paradigms for state discrimination involve a minimum probability of error or unambiguous discrimination with a minimum probability of inconclusive results. Alternatively, an optimal inconclusive measurement, a non-projective measurement, achieves minimal error for a given inconclusive probability. This more general measurement encompasses the standard measurement paradigms for state discrimination and provides a much more powerful tool for quantum information and communication. Here, we experimentally demonstrate the optimal inconclusive measurement for the discrimination of binary coherent states using linear optics and single-photon detection. Our demonstration uses coherent displacement operations based on interference, single-photon detection, and fast feedback to prepare the optimal feedback policy for the optimal non-projective quantum measurement with high fidelity. This generalized measurement allows us to transition among standard measurement paradigms in an optimal way from minimum error to unambiguous measurements for binary coherent states. As a particular case, we use this general measurement to implement the optimal minimum error measurement for phase-coherent states, which is the optimal modulation for communications under the average power constraint. Moreover, we propose a hybrid measurement that leverages the binary optimal inconclusive measurement in conjunction with sequential, unambiguous state elimination to realize higher dimensional inconclusive measurements of coherent states.</jats:p>","DOI":"10.1038/s41534-022-00595-3","type":"journal-article","created":{"date-parts":[[2022,7,18]],"date-time":"2022-07-18T11:03:02Z","timestamp":1658142182000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":"Demonstration of optimal non-projective measurement of binary coherent states with photon counting","prefix":"10.1038","volume":"8","author":[{"given":"M. T.","family":"DiMario","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-2928-310X","authenticated-orcid":false,"given":"F. 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Most of the new packings are nonlattice packings. These new packings increase the previously greatest known numbers of spheres which one sphere may touch, and, except in dimensions 9, 12, 14, 15, they include denser packings than any previously known. The density Δ of the packings in <jats:italic>E<jats:sup>n</jats:sup></jats:italic> for <jats:italic>n</jats:italic> = <jats:italic>2<jats:sup>m</jats:sup></jats:italic> satisfies</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0008414X00053724_eqn1\" /></jats:disp-formula></jats:p><jats:p>In this paper we make systematic use of error-correcting codes to obtain sphere packings in <jats:italic>E<jats:sup>n</jats:sup>,</jats:italic> including several of the densest packings known and several new packings.</jats:p>","DOI":"10.4153/cjm-1971-081-3","type":"journal-article","created":{"date-parts":[[2010,12,7]],"date-time":"2010-12-07T23:24:44Z","timestamp":1291764284000},"page":"718-745","source":"Crossref","is-referenced-by-count":150,"title":"Sphere Packings and Error-Correcting Codes","prefix":"10.4153","volume":"23","author":[{"given":"John","family":"Leech","sequence":"first","affiliation":[]},{"given":"N. 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A famous necessary,\nbut not sufficient, condition for the existence of a topological\nboundary condition is that the chiral central charge\n<jats:inline-formula><jats:alternatives><jats:tex-math>c_-</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msub><mml:mi>c</mml:mi><mml:mo>−</mml:mo></mml:msub></mml:math></jats:alternatives></jats:inline-formula>\nhas to vanish. In this paper, we consider conditions associated with\n``higher\" central charges, which have been introduced recently in\nthe math literature. In terms of these new obstructions, we identify\nnecessary and sufficient conditions for the existence of a topological\nboundary in the case of bosonic, Abelian TQFTs, providing an alternative\nto the identification of a Lagrangian subgroup. Our proof relies on\ngeneral aspects of gauging generalized global symmetries. For\nnon-Abelian TQFTs, we give a geometric way of studying topological\nboundary conditions, and explain certain necessary conditions given\nagain in terms of the higher central charges. Along the way, we find a\ncurious duality in the partition functions of Abelian TQFTs, which begs\nfor an explanation via the 3d-3d correspondence.</jats:p>","DOI":"10.21468/scipostphys.13.3.067","type":"journal-article","created":{"date-parts":[[2022,9,26]],"date-time":"2022-09-26T07:33:00Z","timestamp":1664177580000},"update-policy":"http://dx.doi.org/10.21468/scipost.crossmarkpolicy","source":"Crossref","is-referenced-by-count":7,"title":"Higher central charges and topological boundaries in 2+1-dimensional  TQFTs","prefix":"10.21468","volume":"13","author":[{"given":"Justin","family":"Kaidi","sequence":"first","affiliation":[{"name":"Stony Brook University"}]},{"given":"Zohar","family":"Komargodski","sequence":"additional","affiliation":[{"name":"Stony Brook University"}]},{"given":"Kantaro","family":"Ohmori","sequence":"additional","affiliation":[{"name":"University of Tokyo"},{"name":"Stony Brook University"}]},{"given":"Sahand","family":"Seifnashri","sequence":"additional","affiliation":[{"name":"Stony Brook University"}]},{"given":"Shu-Heng","family":"Shao","sequence":"additional","affiliation":[{"name":"C.N. 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It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a <jats:italic>K</jats:italic>-matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.</jats:p>","DOI":"10.1088/1751-8121/abc6c0","type":"journal-article","created":{"date-parts":[[2020,11,2]],"date-time":"2020-11-02T22:15:47Z","timestamp":1604355347000},"page":"505203","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":"In and around abelian anyon models\n                  <sup>*</sup>","prefix":"10.1088","volume":"53","author":[{"given":"Liang","family":"Wang","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-5253-6400","authenticated-orcid":false,"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2020,11,24]]},"reference":[],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0","content-type":"text/html","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,5]],"date-time":"2022-01-05T10:35:20Z","timestamp":1641378920000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,18]]},"references-count":32,"journal-issue":{"issue":"50","published-online":{"date-parts":[[2020,11,24]]},"published-print":{"date-parts":[[2020,11,18]]}},"URL":"http://dx.doi.org/10.1088/1751-8121/abc6c0","relation":{},"ISSN":["1751-8113","1751-8121"],"subject":["General Physics and Astronomy","Mathematical Physics","Modeling and Simulation","Statistics and Probability","Statistical and Nonlinear Physics"],"container-title-short":"J. 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The notion is a\ncondition on the ground state subspace, rather than on the Hamiltonian,\nand demands that given a collection of ball-like regions, any linear\ntransformation on the ground space be realized by an operator that\navoids the ball-like regions. We derive a bound on the ground state\ndegeneracy <jats:inline-formula><jats:alternatives><jats:tex-math>\\mathcal D</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mstyle mathvariant=\"script\"><mml:mi>𝒟</mml:mi></mml:mstyle></mml:math></jats:alternatives></jats:inline-formula>\nfor systems with homogeneous topological order on an arbitrary closed\nRiemannian manifold of dimension <jats:inline-formula><jats:alternatives><jats:tex-math>d</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>d</mml:mi></mml:math></jats:alternatives></jats:inline-formula>,\nwhich reads [ D c (L/a)^{d-2}.] Here, <jats:inline-formula><jats:alternatives><jats:tex-math>L</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>L</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\nis the diameter of the system, <jats:inline-formula><jats:alternatives><jats:tex-math>a</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>a</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\nis the lattice spacing, and <jats:inline-formula><jats:alternatives><jats:tex-math>c</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>c</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\nis a constant that only depends on the isometry class of the manifold,\nand <jats:inline-formula><jats:alternatives><jats:tex-math>\\mu</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>μ</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\nis a constant that only depends on the density of degrees of freedom. If\n<jats:inline-formula><jats:alternatives><jats:tex-math>d=2</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>,\nthe constant <jats:inline-formula><jats:alternatives><jats:tex-math>c</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>c</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\nis the (demi)genus of the space manifold. This bound is saturated up to\nconstants by known examples.examples.</jats:p>","DOI":"10.21468/scipostphys.10.1.011","type":"journal-article","created":{"date-parts":[[2021,1,20]],"date-time":"2021-01-20T08:06:32Z","timestamp":1611129992000},"update-policy":"http://dx.doi.org/10.21468/scipost.crossmarkpolicy","source":"Crossref","is-referenced-by-count":6,"title":"A degeneracy bound for homogeneous topological order","prefix":"10.21468","volume":"10","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Microsoft"}]}],"member":"8907","published-online":{"date-parts":[[2021,1,20]]},"reference":[],"container-title":"SciPost Physics","original-title":[],"link":[{"URL":"https://scipost.org/10.21468/SciPostPhys.10.1.011/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://scipost.org/10.21468/SciPostPhys.10.1.011/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,1,20]],"date-time":"2021-01-20T08:06:34Z","timestamp":1611129994000},"score":1,"resource":{"primary":{"URL":"https://scipost.org/10.21468/SciPostPhys.10.1.011"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,20]]},"references-count":28,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021]]}},"URL":"http://dx.doi.org/10.21468/SciPostPhys.10.1.011","relation":{"has-review":[{"id-type":"doi","id":"10.21468/SciPost.Report.2200","asserted-by":"object"},{"id-type":"doi","id":"10.21468/SciPost.Report.2127","asserted-by":"object"}]},"ISSN":["2542-4653"],"subject":["General Physics and Astronomy"],"container-title-short":"SciPost Phys.","published":{"date-parts":[[2021,1,20]]},"article-number":"011","id":"doi:10.21468/SciPostPhys.10.1.011","_hash":"f05f4e6399713056c1fb3fb92af95651b64787ab4edcff7f7ea37b36425f6541"},"expire":1712098559768},"doi:10.1103/PhysRevB.107.085134":{"value":{"indexed":{"date-parts":[[2023,2,22]],"date-time":"2023-02-22T05:27:38Z","timestamp":1677043658020},"reference-count":36,"publisher":"American Physical Society (APS)","issue":"8","license":[{"start":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T00:00:00Z","timestamp":1676937600000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"funder":[{"DOI":"10.13039/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevb.107.085134","type":"journal-article","created":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T16:39:53Z","timestamp":1676997593000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":"Disentangling modular Walker-Wang models via fermionic invertible boundaries","prefix":"10.1103","volume":"107","author":[{"ORCID":"http://orcid.org/0000-0002-4881-2003","authenticated-orcid":true,"given":"Andreas","family":"Bauer","sequence":"first","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2023,2,21]]},"reference":[],"container-title":"Physical Review B","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevB.107.085134","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevB.107.085134/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T16:41:01Z","timestamp":1676997661000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevB.107.085134"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,21]]},"references-count":36,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2023,2]]}},"URL":"http://dx.doi.org/10.1103/PhysRevB.107.085134","relation":{},"ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. 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Finally, the opportunities offered for quantum information processing by employing the cavity as a quantum bus are presented.</jats:p>","DOI":"10.1093/acprof:oso/9780199681181.003.0003","type":"book-chapter","created":{"date-parts":[[2014,9,18]],"date-time":"2014-09-18T07:05:22Z","timestamp":1411023922000},"page":"113-256","source":"Crossref","is-referenced-by-count":22,"title":"Circuit QED: superconducting qubits coupled to microwave photons","prefix":"10.1093","author":[{"given":"S. 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Here we propose a cellular automaton decoder for a variation of the color code where the bases of the physical qubits are locally rotated, which we call the XYZ color code. The local transformation means our decoder demonstrates key properties of a two-dimensional fractal code if the noise acting on the system is infinitely biased towards dephasing, namely, no string-like logical operators. As such, in the high-bias limit, our local decoder reproduces the behavior of a partially self-correcting memory. At low error rates, our simulations show that the memory time diverges polynomially with system size without intervention from a global decoder, up to some critical system size that grows as the error rate is lowered. Furthermore, although we find that we cannot reproduce partially self-correcting behavior at finite bias, our numerics demonstrate improved memory times at realistic noise biases. Our results therefore motivate the design of tailored cellular automaton decoders that help to reduce the bandwidth demands of global decoding for realistic noise models.</jats:p>","DOI":"10.22331/q-2023-03-09-940","type":"journal-article","created":{"date-parts":[[2023,3,9]],"date-time":"2023-03-09T12:21:56Z","timestamp":1678364516000},"page":"940","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"A cellular automaton decoder for a noise-bias tailored color code","prefix":"10.22331","volume":"7","author":[{"given":"Jonathan F. San","family":"Miguel","sequence":"first","affiliation":[{"name":"Department of Physics, Stanford University, Stanford, CA 94305"}]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[{"name":"Department of Physics, Stanford University, Stanford, CA 94305"},{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia"}]},{"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, 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(APS)","issue":"2","license":[{"start":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T00:00:00Z","timestamp":1680825600000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/prxquantum.4.020303","type":"journal-article","created":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T15:02:15Z","timestamp":1680879735000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":"Logical Blocks for Fault-Tolerant Topological Quantum Computation","prefix":"10.1103","volume":"4","author":[{"ORCID":"http://orcid.org/0000-0001-6728-5002","authenticated-orcid":true,"given":"Héctor","family":"Bombín","sequence":"first","affiliation":[]},{"given":"Chris","family":"Dawson","sequence":"additional","affiliation":[]},{"given":"Ryan 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The most interesting simulations will require a fault-tolerant quantum computer, and building such a device remains a long-term goal. However, the capabilities of existing noisy quantum processors have steadily improved, sparking an interest in running simulations that, while not necessarily classically intractable, may serve as device benchmarks and help elucidate the challenges to achieving practical applications on near-term devices. Systems of <jats:italic>non</jats:italic>-interacting fermions are ideally suited to serve these purposes. While they display rich physics and generate highly entangled states when simulated on a quantum processor, their classical tractability enables experimental results to be verified even at large system sizes that would typically defy classical simulation. In this work, we use a noisy superconducting quantum processor to prepare Majorana zero modes (MZMs) as eigenstates of the Kitaev chain Hamiltonian, a model of non-interacting fermions. Our work builds on previous experiments with non-interacting fermionic systems. Previous work demonstrated error mitigation techniques applicable to the special case of Slater determinants. Here, we show how to extend these techniques to the case of general fermionic Gaussian states, and demonstrate them by preparing MZMs on systems of up to seven qubits.</jats:p>","DOI":"10.1088/2058-9565/acb796","type":"journal-article","created":{"date-parts":[[2023,1,31]],"date-time":"2023-01-31T22:30:26Z","timestamp":1675204226000},"page":"025010","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":"Simulating Majorana zero modes on a noisy quantum processor","prefix":"10.1088","volume":"8","author":[{"ORCID":"http://orcid.org/0000-0001-6459-6374","authenticated-orcid":true,"given":"Kevin J","family":"Sung","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4077-9744","authenticated-orcid":true,"given":"Marko J","family":"Rančić","sequence":"additional","affiliation":[]},{"given":"Olivia T","family":"Lanes","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-6731-6814","authenticated-orcid":true,"given":"Nicholas T","family":"Bronn","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2023,2,9]]},"reference":[],"container-title":"Quantum Science and Technology","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,9]],"date-time":"2023-02-09T13:27:13Z","timestamp":1675949233000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,9]]},"references-count":41,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2,9]]},"published-print":{"date-parts":[[2023,4,1]]}},"URL":"http://dx.doi.org/10.1088/2058-9565/acb796","relation":{},"ISSN":["2058-9565"],"subject":["Electrical and Electronic Engineering","Physics and Astronomy (miscellaneous)","Materials Science (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum Sci. Technol.","published":{"date-parts":[[2023,2,9]]},"assertion":[{"value":"Simulating Majorana zero modes on a noisy quantum processor","name":"article_title","label":"Article Title"},{"value":"Quantum Science and Technology","name":"journal_title","label":"Journal Title"},{"value":"paper","name":"article_type","label":"Article Type"},{"value":"© 2023 IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2022-08-01","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2023-01-31","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2023-02-09","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/2058-9565/acb796","_hash":"49e07071ad2302deba5a24ecebfc6e2ce3f517179d21b5266a165a3db760a087"},"expire":1712590663879},"doi:10.22331/q-2023-03-09-942":{"value":{"indexed":{"date-parts":[[2023,3,10]],"date-time":"2023-03-10T05:51:13Z","timestamp":1678427473305},"reference-count":99,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,3,9]],"date-time":"2023-03-09T00:00:00Z","timestamp":1678320000000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"European Research Council","award":["804247"]},{"name":"European Comission, Horizon 2020","award":["820495"]},{"DOI":"10.13039/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"crossref","award":["449905436"]},{"name":"US A.R.O.","award":["W911NF-21-1-0007"]},{"name":"US A.R.O.","award":["W911NF-16-1-0070"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Active quantum error correction is a central ingredient to achieve robust quantum processors. In this paper we investigate the potential of quantum machine learning for quantum error correction in a quantum memory. Specifically, we demonstrate how quantum neural networks, in the form of quantum autoencoders, can be trained to learn optimal strategies for active detection and correction of errors, including spatially correlated computational errors as well as qubit losses. We highlight that the denoising capabilities of quantum autoencoders are not limited to the protection of specific states but extend to the entire logical codespace. We also show that quantum neural networks can be used to discover new logical encodings that are optimally adapted to the underlying noise. Moreover, we find that, even in the presence of moderate noise in the quantum autoencoders themselves, they may still be successfully used to perform beneficial quantum error correction and thereby extend the lifetime of a logical qubit.</jats:p>","DOI":"10.22331/q-2023-03-09-942","type":"journal-article","created":{"date-parts":[[2023,3,9]],"date-time":"2023-03-09T14:44:56Z","timestamp":1678373096000},"page":"942","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Quantum Error Correction with Quantum Autoencoders","prefix":"10.22331","volume":"7","author":[{"given":"David F.","family":"Locher","sequence":"first","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany"},{"name":"Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52425 Jülich, Germany"}]},{"given":"Lorenzo","family":"Cardarelli","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany"},{"name":"Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52425 Jülich, Germany"}]},{"given":"Markus","family":"Müller","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany"},{"name":"Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52425 Jülich, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2023,3,9]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2023-03-09-942/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,3,9]],"date-time":"2023-03-09T14:45:12Z","timestamp":1678373112000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2023-03-09-942/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,9]]},"references-count":99,"URL":"http://dx.doi.org/10.22331/q-2023-03-09-942","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2023,3,9]]},"article-number":"942","id":"doi:10.22331/q-2023-03-09-942","_hash":"ff4a9445f3b108803f21adbfb99b2a430f90d474edbe28482e3813a5899130f9"},"expire":1712590664962},"doi:10.1146/annurev-conmatphys-031720-030658":{"value":{"indexed":{"date-parts":[[2023,3,27]],"date-time":"2023-03-27T15:24:05Z","timestamp":1679930645482},"reference-count":311,"publisher":"Annual Reviews","issue":"1","license":[{"start":{"date-parts":[[2023,3,10]],"date-time":"2023-03-10T00:00:00Z","timestamp":1678406400000},"content-version":"unspecified","delay-in-days":0,"URL":"http://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2023,3,10]]},"abstract":"<jats:p> Quantum circuits—built from local unitary gates and local measurements—are a new playground for quantum many-body physics and a tractable setting to explore universal collective phenomena far from equilibrium. These models have shed light on longstanding questions about thermalization and chaos, and on the underlying universal dynamics of quantum information and entanglement. In addition, such models generate new sets of questions and give rise to phenomena with no traditional analog, such as dynamical phase transitions in quantum systems that are monitored by an external observer. Quantum circuit dynamics is also topical in view of experimental progress in building digital quantum simulators that allow control of precisely these ingredients. Randomness in the circuit elements allows a high level of theoretical control, with a key theme being mappings between real-time quantum dynamics and effective classical lattice models or dynamical processes. Many of the universal phenomena that can be identified in this tractable setting apply to much wider classes of more structured many-body dynamics. </jats:p>","DOI":"10.1146/annurev-conmatphys-031720-030658","type":"journal-article","created":{"date-parts":[[2022,12,12]],"date-time":"2022-12-12T20:39:21Z","timestamp":1670877561000},"page":"335-379","source":"Crossref","is-referenced-by-count":4,"title":"Random Quantum Circuits","prefix":"10.1146","volume":"14","author":[{"given":"Matthew P.A.","family":"Fisher","sequence":"first","affiliation":[{"name":"Department of Physics, University of California, Santa Barbara, California, USA;,"}]},{"given":"Vedika","family":"Khemani","sequence":"additional","affiliation":[{"name":"Department of Physics, Stanford University, Stanford, California, USA;"}]},{"given":"Adam","family":"Nahum","sequence":"additional","affiliation":[{"name":"Laboratoire de Physique, École Normale Supérieure, CNRS, Université PSL, Sorbonne Université, Université de Paris, Paris, France;"}]},{"given":"Sagar","family":"Vijay","sequence":"additional","affiliation":[{"name":"Department of Physics, University of California, Santa Barbara, California, USA;,"}]}],"member":"22","reference":[],"container-title":"Annual Review of Condensed Matter Physics","original-title":[],"language":"en","link":[{"URL":"https://www.annualreviews.org/doi/pdf/10.1146/annurev-conmatphys-031720-030658","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,10]],"date-time":"2023-03-10T23:37:12Z","timestamp":1678491432000},"score":1,"resource":{"primary":{"URL":"https://www.annualreviews.org/doi/10.1146/annurev-conmatphys-031720-030658"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,10]]},"references-count":311,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,3,10]]}},"alternative-id":["10.1146/annurev-conmatphys-031720-030658"],"URL":"http://dx.doi.org/10.1146/annurev-conmatphys-031720-030658","relation":{},"ISSN":["1947-5454","1947-5462"],"subject":["Condensed Matter Physics","General Materials Science"],"container-title-short":"Annu. 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In most QEC codes<jats:sup>2–8</jats:sup>, a logical qubit is encoded in some discrete variables, for example photon numbers, so that the encoded quantum information can be unambiguously extracted after processing. Over the past decade, repetitive QEC has been demonstrated with various discrete-variable-encoded scenarios<jats:sup>9–17</jats:sup>. However, extending the lifetimes of thus-encoded logical qubits beyond the best available physical qubit still remains elusive, which represents a break-even point for judging the practical usefulness of QEC. Here we demonstrate a QEC procedure in a circuit quantum electrodynamics architecture<jats:sup>18</jats:sup>, where the logical qubit is binomially encoded in photon-number states of a microwave cavity<jats:sup>8</jats:sup>, dispersively coupled to an auxiliary superconducting qubit. By applying a pulse featuring a tailored frequency comb to the auxiliary qubit, we can repetitively extract the error syndrome with high fidelity and perform error correction with feedback control accordingly, thereby exceeding the break-even point by about 16% lifetime enhancement. 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Optimizing the resource and time overheads needed to implement QEC is one of the most pressing challenges. Here, we introduce a new topological quantum error-correcting code, the three-dimensional subsystem toric code (3D STC). The 3D STC can be realized with geometrically-local parity checks of weight at most three on the cubic lattice with open boundary conditions. We prove that one round of parity-check measurements suffices to perform reliable QEC with the 3D STC even in the presence of measurement errors. We also propose an efficient single-shot QEC decoding strategy for the 3D STC and numerically estimate the resulting storage threshold against independent bit-flip, phase-flip and measurement errors to be<jats:italic>p</jats:italic><jats:sub>STC</jats:sub> ≈ 1.045%. Such a high threshold together with local parity-check measurements make the 3D STC particularly appealing for realizing fault-tolerant quantum computing.</jats:p>","DOI":"10.1038/s41467-022-33923-4","type":"journal-article","created":{"date-parts":[[2022,10,21]],"date-time":"2022-10-21T14:08:47Z","timestamp":1666361327000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":"Single-shot quantum error correction with the three-dimensional subsystem toric code","prefix":"10.1038","volume":"13","author":[{"ORCID":"http://orcid.org/0000-0001-8213-8190","authenticated-orcid":false,"given":"Aleksander","family":"Kubica","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-6711-5924","authenticated-orcid":false,"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,10,21]]},"reference":[],"container-title":"Nature 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Published by Elsevier Inc.","name":"copyright","label":"Copyright"}],"article-number":"169286","id":"doi:10.1016/j.aop.2023.169286","_hash":"96192b1ddfeb64b839f7308081c632c75fd6b63585a655f159b86ca7f39e2225"},"expire":1715612576796},"doi:10.1038/s41467-023-37725-0":{"value":{"indexed":{"date-parts":[[2023,5,11]],"date-time":"2023-05-11T17:32:26Z","timestamp":1683826346465},"reference-count":53,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T00:00:00Z","timestamp":1682035200000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T00:00:00Z","timestamp":1682035200000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"<jats:title>Abstract</jats:title><jats:p>Electrons are indivisible elementary particles, yet paradoxically a collection of them can act as a fraction of a single electron, exhibiting exotic and useful properties. One such collective excitation, known as a topological Majorana mode, is naturally stable against perturbations, such as unwanted local noise, and can thereby robustly store quantum information. As such, Majorana modes serve as the basic primitive of topological quantum computing, providing resilience to errors. However, their demonstration on quantum hardware has remained elusive. Here, we demonstrate a verifiable identification and braiding of topological Majorana modes using a superconducting quantum processor as a quantum simulator. By simulating fermions on a one-dimensional lattice subject to a periodic drive, we confirm the existence of Majorana modes localized at the edges, and distinguish them from other trivial modes. To simulate a basic logical operation of topological quantum computing known as braiding, we propose a non-adiabatic technique, whose implementation reveals correct braiding statistics in our experiments. This work could further be used to study topological models of matter using circuit-based simulations, and shows that long-sought quantum phenomena can be realized by anyone in cloud-run quantum simulations, whereby accelerating fundamental discoveries in quantum science and technology.</jats:p>","DOI":"10.1038/s41467-023-37725-0","type":"journal-article","created":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T08:02:54Z","timestamp":1682064174000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":"Observing and braiding topological Majorana modes on programmable quantum 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They break the fermion-boson dichotomy and obey non-Abelian braiding statistics: their interchanges yield unitary operations, rather than merely a phase factor, in a space spanned by topologically degenerate wavefunctions. They are the building blocks of topological quantum computing. However, experimental observation of non-Abelian anyons and their characterizing braiding statistics is notoriously challenging and has remained elusive hitherto, in spite of various theoretical proposals. Here, we report an experimental quantum digital simulation of projective non-Abelian anyons and their braiding statistics with up to 68 programmable superconducting qubits arranged on a two-dimensional lattice. By implementing the ground states of the toric-code model with twists through quantum circuits, we demonstrate that twists exchange electric and magnetic charges and behave as a particular type of non-Abelian anyons—the Ising anyons. 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Such a degeneracy is known as topological degeneracy and can be usually seen under the periodic boundary condition regardless of the choice of the system sizes L1 and L2 in each direction. In this work, we introduce a family of extensions of the Kitaev toric code to N level spins (N ≥ 2). The model realizes topologically ordered phases or symmetry-protected topological phases depending on the parameters in the model. The most remarkable feature of topologically ordered phases is that the ground state may be unique, depending on L1 and L2, despite that the translation symmetry of the model remains unbroken. Nonetheless, the topological entanglement entropy takes the nontrivial value. We argue that this behavior originates from the nontrivial action of translations permuting anyon species.</jats:p>","DOI":"10.1063/5.0134010","type":"journal-article","created":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T10:19:06Z","timestamp":1684318746000},"update-policy":"http://dx.doi.org/10.1063/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":"Ground state degeneracy on torus in a family of ZN toric code","prefix":"10.1063","volume":"64","author":[{"ORCID":"http://orcid.org/0000-0002-8112-021X","authenticated-orcid":false,"given":"Haruki","family":"Watanabe","sequence":"first","affiliation":[{"name":"Department of Applied Physics, The University of Tokyo 1 , Tokyo, Japan"}]},{"given":"Meng","family":"Cheng","sequence":"additional","affiliation":[{"name":"Department of Physics, Yale University 2 , New Haven, Connecticut 06520, USA"}]},{"ORCID":"http://orcid.org/0000-0002-3381-6642","authenticated-orcid":false,"given":"Yohei","family":"Fuji","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, The University of Tokyo 1 , Tokyo, Japan"}]}],"member":"317","published-online":{"date-parts":[[2023,5,17]]},"reference":[],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","link":[{"URL":"https://pubs.aip.org/aip/jmp/article-pdf/doi/10.1063/5.0134010/17614211/051901_1_5.0134010.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://pubs.aip.org/aip/jmp/article-pdf/doi/10.1063/5.0134010/17614211/051901_1_5.0134010.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T10:19:16Z","timestamp":1684318756000},"score":1,"resource":{"primary":{"URL":"https://pubs.aip.org/jmp/article/64/5/051901/2891377/Ground-state-degeneracy-on-torus-in-a-family-of-ZN"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,1]]},"references-count":54,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,5,1]]}},"URL":"http://dx.doi.org/10.1063/5.0134010","relation":{},"ISSN":["0022-2488","1089-7658"],"subject":["Mathematical Physics","Statistical and Nonlinear Physics"],"published-other":{"date-parts":[[2023,5,1]]},"published":{"date-parts":[[2023,5,1]]},"id":"doi:10.1063/5.0134010","_hash":"1fdf11bd915d05d4602acb584d34e5fa06289c8f81e3bb4563e0be2857d4a1ac"},"expire":1718831603903},"doi:10.1038/s41467-023-38247-5":{"value":{"indexed":{"date-parts":[[2023,5,19]],"date-time":"2023-05-19T05:13:29Z","timestamp":1684473209864},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T00:00:00Z","timestamp":1684368000000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T00:00:00Z","timestamp":1684368000000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/100011039","name":"ODNI | Intelligence Advanced Research Projects Activity","doi-asserted-by":"publisher","award":["W911NF-16-1-0114","youngseok.kim1@ibm.com","W911NF-16-1-0114","W911NF-16-1-0114","W911NF-16-1-0114","W911NF-16-1-0114","W911NF-16-1-0114","W911NF-16-1-0114","W911NF-16-1-0114"]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum error correction offers a promising path for performing high fidelity quantum computations. Although fully fault-tolerant executions of algorithms remain unrealized, recent improvements in control electronics and quantum hardware enable increasingly advanced demonstrations of the necessary operations for error correction. Here, we perform quantum error correction on superconducting qubits connected in a heavy-hexagon lattice. We encode a logical qubit with distance three and perform several rounds of fault-tolerant syndrome measurements that allow for the correction of any single fault in the circuitry. Using real-time feedback, we reset syndrome and flag qubits conditionally after each syndrome extraction cycle. We report decoder dependent logical error, with average logical error per syndrome measurement in Z(X)-basis of ~0.040 (~0.088) and ~0.037 (~0.087) for matching and maximum likelihood decoders, respectively, on leakage post-selected data.</jats:p>","DOI":"10.1038/s41467-023-38247-5","type":"journal-article","created":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T18:31:30Z","timestamp":1684434690000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":"Demonstrating multi-round subsystem quantum error correction using matching and maximum likelihood decoders","prefix":"10.1038","volume":"14","author":[{"ORCID":"http://orcid.org/0000-0002-0772-2853","authenticated-orcid":false,"given":"Neereja","family":"Sundaresan","sequence":"first","affiliation":[]},{"given":"Theodore J.","family":"Yoder","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-8486-9162","authenticated-orcid":false,"given":"Youngseok","family":"Kim","sequence":"additional","affiliation":[]},{"given":"Muyuan","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-7217-7137","authenticated-orcid":false,"given":"Edward H.","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Grace","family":"Harper","sequence":"additional","affiliation":[]},{"given":"Ted","family":"Thorbeck","sequence":"additional","affiliation":[]},{"given":"Andrew W.","family":"Cross","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-7800-0399","authenticated-orcid":false,"given":"Antonio D.","family":"Córcoles","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-8871-4638","authenticated-orcid":false,"given":"Maika","family":"Takita","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,5,18]]},"reference":[],"container-title":"Nature Communications","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41467-023-38247-5.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-023-38247-5","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-023-38247-5.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T18:31:40Z","timestamp":1684434700000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41467-023-38247-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,18]]},"references-count":46,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,12]]}},"alternative-id":["38247"],"URL":"http://dx.doi.org/10.1038/s41467-023-38247-5","relation":{},"ISSN":["2041-1723"],"subject":["General Physics and Astronomy","General Biochemistry, Genetics and Molecular Biology","General Chemistry","Multidisciplinary"],"container-title-short":"Nat Commun","published":{"date-parts":[[2023,5,18]]},"assertion":[{"value":"15 July 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 April 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 May 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"2852","id":"doi:10.1038/s41467-023-38247-5","_hash":"4e93b21633c5eaec23e9a0fcf05249ed1d5eb4daf7b1920b58098588cec04b8c"},"expire":1718831605074},"doi:10.22331/q-2023-06-07-1037":{"value":{"indexed":{"date-parts":[[2023,6,8]],"date-time":"2023-06-08T04:30:07Z","timestamp":1686198607392},"reference-count":46,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,6,7]],"date-time":"2023-06-07T00:00:00Z","timestamp":1686096000000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"EU Quantum Technology Flagship","award":["AQTION/820495"]},{"name":"BMBF","award":["MUNIQC-ATOMS"]},{"name":"European Research Council","award":["ERC/682726"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Belief propagation (BP) is well-known as a low complexity decoding algorithm with a strong performance for important classes of quantum error correcting codes, e.g. notably for the quantum low-density parity check (LDPC) code class of random expander codes. However, it is also well-known that the performance of BP breaks down when facing topological codes such as the surface code, where naive BP fails entirely to reach a below-threshold regime, i.e. the regime where error correction becomes useful. Previous works have shown, that this can be remedied by resorting to post-processing decoders outside the framework of BP. In this work, we present a generalized belief propagation method with an outer re-initialization loop that successfully decodes surface codes, i.e. opposed to naive BP it recovers the sub-threshold regime known from decoders tailored to the surface code and from statistical-mechanical mappings. We report a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">17%</mml:mtext></mml:mrow></mml:math> under independent bit-and phase-flip data noise (to be compared to the ideal threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">20.6%</mml:mtext></mml:mrow></mml:math>) and a threshold value of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">14%</mml:mtext></mml:mrow></mml:math> under depolarizing data noise (compared to the ideal threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">18.9%</mml:mtext></mml:mrow></mml:math>), which are on par with thresholds achieved by non-BP post-processing methods.</jats:p>","DOI":"10.22331/q-2023-06-07-1037","type":"journal-article","created":{"date-parts":[[2023,6,7]],"date-time":"2023-06-07T10:53:02Z","timestamp":1686135182000},"page":"1037","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Generalized Belief Propagation Algorithms for Decoding of Surface Codes","prefix":"10.22331","volume":"7","author":[{"given":"Josias","family":"Old","sequence":"first","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, Aachen, Germany"},{"name":"Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich, Jülich, Germany"}]},{"given":"Manuel","family":"Rispler","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, Aachen, Germany"},{"name":"Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich, Jülich, Germany"},{"name":"QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands"}]}],"member":"9598","published-online":{"date-parts":[[2023,6,7]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2023-06-07-1037/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,6,7]],"date-time":"2023-06-07T10:53:16Z","timestamp":1686135196000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2023-06-07-1037/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,7]]},"references-count":46,"URL":"http://dx.doi.org/10.22331/q-2023-06-07-1037","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2023,6,7]]},"article-number":"1037","id":"doi:10.22331/q-2023-06-07-1037","_hash":"888e955c0ce0e381d34fe03109132bc44c43bf500d7ee3074b63f68063f941e6"},"expire":1718831605998},"doi:10.1088/1367-2630/13/4/043016":{"value":{"indexed":{"date-parts":[[2023,6,19]],"date-time":"2023-06-19T08:30:42Z","timestamp":1687163442615},"reference-count":0,"publisher":"IOP Publishing","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1088/1367-2630/13/4/043016","type":"journal-article","created":{"date-parts":[[2011,4,15]],"date-time":"2011-04-15T03:34:39Z","timestamp":1302838479000},"page":"043016","source":"Crossref","is-referenced-by-count":139,"title":"Interacting quantum observables: categorical algebra and diagrammatics","prefix":"10.1088","volume":"13","author":[{"given":"Bob","family":"Coecke","sequence":"first","affiliation":[]},{"given":"Ross","family":"Duncan","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2011,4,14]]},"container-title":"New Journal of Physics","original-title":[],"deposited":{"date-parts":[[2020,4,11]],"date-time":"2020-04-11T15:15:12Z","timestamp":1586618112000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/13/4/043016"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,4,14]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2011,4,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/13/4/043016","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. Phys.","published":{"date-parts":[[2011,4,14]]},"reference":[],"id":"doi:10.1088/1367-2630/13/4/043016","_hash":"5152c192586b62cb6171ddc27668f1174492e7f6a3437dae777abee234e74cb5"},"expire":1718831606965},"doi:10.22331/q-2019-04-30-135":{"value":{"indexed":{"date-parts":[[2023,6,18]],"date-time":"2023-06-18T10:50:45Z","timestamp":1687085445826},"reference-count":32,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T00:00:00Z","timestamp":1556582400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"<jats:p>We present magic state factory constructions for producing<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>states and<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>states. For the<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>factory we apply the surface code lattice surgery construction techniques described in \\cite{fowler2018} to the fault-tolerant Toffoli \\cite{jones2013, eastin2013distilling}. The resulting factory has a footprint of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>12</mml:mn><mml:mi>d</mml:mi><mml:mo>×</mml:mo><mml:mn>6</mml:mn><mml:mi>d</mml:mi></mml:math>(where<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>is the code distance) and produces one<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>every<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>5.5</mml:mn><mml:mi>d</mml:mi></mml:math>surface code cycles. Our<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>state factory uses the<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>factory's output and a catalyst<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>state to exactly transform one<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>state into two<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>states. It has a footprint<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>25</mml:mn><mml:mi mathvariant=\"normal\">%</mml:mi></mml:math>smaller than the factory in \\cite{fowler2018} but outputs<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>states twice as quickly. We show how to generalize the catalyzed transformation to arbitrary phase angles, and note that the case<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>22.5</mml:mn><mml:mo>∘</mml:mo></mml:msup></mml:math>produces a particularly efficient circuit for producing<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msqrt><mml:mi>T</mml:mi></mml:msqrt><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>states. Compared to using the<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>12</mml:mn><mml:mi>d</mml:mi><mml:mo>×</mml:mo><mml:mn>8</mml:mn><mml:mi>d</mml:mi><mml:mo>×</mml:mo><mml:mn>6.5</mml:mn><mml:mi>d</mml:mi></mml:math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>factory of \\cite{fowler2018}, our<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>factory can quintuple the speed of algorithms that are dominated by the cost of applying Toffoli gates, including Shor's algorithm \\cite{shor1994} and the chemistry algorithm of Babbush et al. \\cite{babbush2018}. Assuming a physical gate error rate of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mn>10</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>, our CCZ factory can produce<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>∼</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>10</mml:mn></mml:mrow></mml:msup></mml:math>states on average before an error occurs. This is sufficient for classically intractable instantiations of the chemistry algorithm, but for more demanding algorithms such as Shor's algorithm the mean number of states until failure can be increased to<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>∼</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>12</mml:mn></mml:mrow></mml:msup></mml:math>by increasing the factory footprint<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>∼</mml:mo><mml:mn>20</mml:mn><mml:mi mathvariant=\"normal\">%</mml:mi></mml:math>.</jats:p>","DOI":"10.22331/q-2019-04-30-135","type":"journal-article","created":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T03:12:22Z","timestamp":1556593942000},"page":"135","source":"Crossref","is-referenced-by-count":46,"title":"Efficient magic state factories with a catalyzed<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>to<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>transformation","prefix":"10.22331","volume":"3","author":[{"given":"Craig","family":"Gidney","sequence":"first","affiliation":[{"name":"Google Inc., Santa Barbara, California 93117, USA"}]},{"given":"Austin G.","family":"Fowler","sequence":"additional","affiliation":[{"name":"Google Inc., Santa Barbara, California 93117, USA"}]}],"member":"9598","published-online":{"date-parts":[[2019,4,30]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2019-04-30-135/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2020,12,9]],"date-time":"2020-12-09T03:13:27Z","timestamp":1607483607000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2019-04-30-135/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,30]]},"references-count":32,"URL":"http://dx.doi.org/10.22331/q-2019-04-30-135","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2019,4,30]]},"article-number":"135","id":"doi:10.22331/q-2019-04-30-135","_hash":"6d4e746ea6d62d0218955310e6ec6696b5cfec48535d9cef7b22ffac8ceaec7d"},"expire":1718831608000},"doi:10.22331/q-2020-01-09-218":{"value":{"indexed":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T15:35:26Z","timestamp":1680881726443},"reference-count":47,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2020,1,9]],"date-time":"2020-01-09T00:00:00Z","timestamp":1578528000000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"<jats:p>A leading choice of error correction for scalable quantum computing is the surface code with lattice surgery. The basic lattice surgery operations, the merging and splitting of logical qubits, act non-unitarily on the logical states and are not easily captured by standard circuit notation. This raises the question of how best to design, verify, and optimise protocols that use lattice surgery, in particular in architectures with complex resource management issues. In this paper we demonstrate that the operations of the ZX calculus --- a form of quantum diagrammatic reasoning based on bialgebras --- match exactly the operations of lattice surgery. Red and green ``spider'' nodes match rough and smooth merges and splits, and follow the axioms of a dagger special associative Frobenius algebra. Some lattice surgery operations require non-trivial correction operations, which are captured natively in the use of the ZX calculus in the form of ensembles of diagrams. 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Rev. A","published":{"date-parts":[[2002,5,3]]},"article-number":"052316","id":"doi:10.1103/PhysRevA.65.052316","_hash":"78a4826461d12bff41f0e39b2abb20a8b820ee1525fcebcb235d1eaa91a92035"},"expire":1718831616967},"doi:10.21468/SciPostPhysLectNotes.72":{"value":{"indexed":{"date-parts":[[2023,6,17]],"date-time":"2023-06-17T04:31:58Z","timestamp":1686976318323},"reference-count":79,"publisher":"Stichting SciPost","license":[{"start":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T00:00:00Z","timestamp":1686873600000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"content-domain":{"domain":["scipost.org"],"crossmark-restriction":false},"abstract":"<jats:p>These are the lecture notes from the 2019 Les Houches Summer School on “Quantum Information Machines”. After a brief introduction to quantum error correction and bosonic codes, we focus on the case of cat qubits stabilized by a nonlinear multi-photon driven dissipation process. We argue that such a system can be seen as a self-correcting qubit where bit-flip errors are robustly and exponentially suppressed. Next, we provide some experimental directions to engineer such a multi-photon driven dissipation process with superconducting circuits. Finally, we analyze various logical gates that can be implemented without re-introducing bit-flip errors. This set of bias-preserving gates pave the way towards a hardware-efficient and fault-tolerant quantum processor.</jats:p>","DOI":"10.21468/scipostphyslectnotes.72","type":"journal-article","created":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T09:24:40Z","timestamp":1686907480000},"update-policy":"http://dx.doi.org/10.21468/scipost.crossmarkpolicy","source":"Crossref","is-referenced-by-count":0,"title":"Quantum computation with cat qubits","prefix":"10.21468","author":[{"given":"Jérémie","family":"Guillaud","sequence":"first","affiliation":[{"name":"Alice and Bob"}]},{"given":"Joachim","family":"Cohen","sequence":"additional","affiliation":[{"name":"Université de Sherbrooke"}]},{"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[{"name":"Physics Laboratory of the École Normale Supérieure"}]}],"member":"8907","published-online":{"date-parts":[[2023,6,16]]},"reference":[],"container-title":"SciPost Physics Lecture Notes","original-title":[],"link":[{"URL":"https://scipost.org/10.21468/SciPostPhysLectNotes.72/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://scipost.org/10.21468/SciPostPhysLectNotes.72/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T09:24:43Z","timestamp":1686907483000},"score":1,"resource":{"primary":{"URL":"https://scipost.org/10.21468/SciPostPhysLectNotes.72"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,16]]},"references-count":79,"URL":"http://dx.doi.org/10.21468/SciPostPhysLectNotes.72","relation":{},"ISSN":["2590-1990"],"subject":["Condensed Matter Physics","Nuclear and High Energy Physics","Atomic and Molecular Physics, and Optics","Statistical and Nonlinear Physics"],"container-title-short":"SciPost Phys. Lect. Notes","published":{"date-parts":[[2023,6,16]]},"article-number":"72","id":"doi:10.21468/SciPostPhysLectNotes.72","_hash":"a4da1b77cfcb34ee2c896010e9640dc4b58ec48493df27cfe3ab55c716d31255"},"expire":1718831617957},"doi:10.22331/q-2023-06-05-1032":{"value":{"indexed":{"date-parts":[[2023,6,6]],"date-time":"2023-06-06T04:39:56Z","timestamp":1686026396136},"reference-count":48,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T00:00:00Z","timestamp":1685923200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"crossref","award":["ANR-22-PETQ-0006"]},{"DOI":"10.13039/501100000780","name":"European Union","doi-asserted-by":"crossref","award":["H2020 CiViQ"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Quantum computers often manipulate physical qubits encoded on two-level quantum systems. Bosonic qubit codes depart from this idea by encoding information in a well-chosen subspace of an infinite-dimensional Fock space. This larger physical space provides a natural protection against experimental imperfections and allows bosonic codes to circumvent no-go results that apply to states constrained by a 2-dimensional Hilbert space. A bosonic qubit is usually defined in a single bosonic mode but it makes sense to look for multimode versions that could exhibit better performance.In this work, building on the observation that the cat code lives in the span of coherent states indexed by a finite subgroup of the complex numbers, we consider a two-mode generalisation living in the span of 24 coherent states indexed by the binary tetrahedral group <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>T</mml:mi></mml:math> of the quaternions. The resulting <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>T</mml:mi></mml:math>-qutrit naturally inherits the algebraic properties of the group <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>T</mml:mi></mml:math> and appears to be quite robust in the low-loss regime. We initiate its study and identify stabilisers as well as some logical operators for this bosonic code.</jats:p>","DOI":"10.22331/q-2023-06-05-1032","type":"journal-article","created":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T13:34:27Z","timestamp":1685972067000},"page":"1032","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"The <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>T</mml:mi></mml:math>-qutrit, a two-mode bosonic qutrit","prefix":"10.22331","volume":"7","author":[{"given":"Aurélie","family":"Denys","sequence":"first","affiliation":[{"name":"Inria Paris, France"}]},{"ORCID":"http://orcid.org/0000-0002-6707-1458","authenticated-orcid":false,"given":"Anthony","family":"Leverrier","sequence":"additional","affiliation":[{"name":"Inria Paris, France"}]}],"member":"9598","published-online":{"date-parts":[[2023,6,5]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2023-06-05-1032/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T13:34:34Z","timestamp":1685972074000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2023-06-05-1032/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,5]]},"references-count":48,"URL":"http://dx.doi.org/10.22331/q-2023-06-05-1032","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2023,6,5]]},"article-number":"1032","id":"doi:10.22331/q-2023-06-05-1032","_hash":"0f6247e3c14e9281652b61c4aadfb9562e70296f32cf88ca17c69955e167efc9"},"expire":1718831618909},"doi:10.22331/q-2023-05-31-1025":{"value":{"indexed":{"date-parts":[[2023,6,1]],"date-time":"2023-06-01T10:41:18Z","timestamp":1685616078944},"reference-count":59,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,5,31]],"date-time":"2023-05-31T00:00:00Z","timestamp":1685491200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"EPSRC Quantum Communications Hub","award":["EP/T001011/1"]},{"name":"NSF Center for Quantum Networks","award":["EEC-1941583"]},{"name":"PNRR MUR Project","award":["PE0000023-NQSTI"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Coherent states of the quantum electromagnetic field, the quantum description of ideal laser light, are prime candidates as information carriers for optical communications. A large body of literature exists on their quantum-limited estimation and discrimination. However, very little is known about the practical realizations of receivers for unambiguous state discrimination (USD) of coherent states. Here we fill this gap and outline a theory of USD with receivers that are allowed to employ: passive multimode linear optics, phase-space displacements, auxiliary vacuum modes, and on-off photon detection. Our results indicate that, in some regimes, these currently-available optical components are typically sufficient to achieve near-optimal unambiguous discrimination of multiple, multimode coherent states.</jats:p>","DOI":"10.22331/q-2023-05-31-1025","type":"journal-article","created":{"date-parts":[[2023,5,31]],"date-time":"2023-05-31T15:09:03Z","timestamp":1685545743000},"page":"1025","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Linear optics and photodetection achieve near-optimal unambiguous coherent state discrimination","prefix":"10.22331","volume":"7","author":[{"ORCID":"http://orcid.org/0000-0002-6167-8224","authenticated-orcid":false,"given":"Jasminder S.","family":"Sidhu","sequence":"first","affiliation":[{"name":"SUPA Department of Physics, The University of Strathclyde, Glasgow, G4 0NG, UK"}]},{"ORCID":"http://orcid.org/0000-0002-3528-7473","authenticated-orcid":false,"given":"Michael S.","family":"Bullock","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, The University of Arizona, Tucson, Arizona 85721, USA"}]},{"ORCID":"http://orcid.org/0000-0002-2581-4380","authenticated-orcid":false,"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, The University of Arizona, Tucson, Arizona 85721, USA"},{"name":"College of Optical Sciences, The University of Arizona, Tucson, Arizona 85721, USA"}]},{"ORCID":"http://orcid.org/0000-0002-5227-4009","authenticated-orcid":false,"given":"Cosmo","family":"Lupo","sequence":"additional","affiliation":[{"name":"Dipartimento Interateneo di Fisica, Politecnico & Università di Bari, 70126 Bari, Italy"},{"name":"INFN, Sezione di Bari, 70126 Bari, Italy"}]}],"member":"9598","published-online":{"date-parts":[[2023,5,31]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2023-05-31-1025/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,5,31]],"date-time":"2023-05-31T15:09:18Z","timestamp":1685545758000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2023-05-31-1025/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,31]]},"references-count":59,"URL":"http://dx.doi.org/10.22331/q-2023-05-31-1025","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2023,5,31]]},"article-number":"1025","id":"doi:10.22331/q-2023-05-31-1025","_hash":"4e82dae06e0390386ba0aee2fe5463a966ea23a4814ec79875ae1b3bdf5c8f53"},"expire":1718831619908},"doi:10.1103/PhysRevLett.92.027902":{"value":{"indexed":{"date-parts":[[2023,6,25]],"date-time":"2023-06-25T02:18:15Z","timestamp":1687659495766},"reference-count":19,"publisher":"American Physical Society (APS)","issue":"2","license":[{"start":{"date-parts":[[2004,1,15]],"date-time":"2004-01-15T00:00:00Z","timestamp":1074124800000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physrevlett.92.027902","type":"journal-article","created":{"date-parts":[[2004,1,16]],"date-time":"2004-01-16T03:55:00Z","timestamp":1074225300000},"source":"Crossref","is-referenced-by-count":247,"title":"Classical Capacity of the Lossy Bosonic Channel: The Exact Solution","prefix":"10.1103","volume":"92","author":[{"given":"V.","family":"Giovannetti","sequence":"first","affiliation":[]},{"given":"S.","family":"Guha","sequence":"additional","affiliation":[]},{"given":"S.","family":"Lloyd","sequence":"additional","affiliation":[]},{"given":"L.","family":"Maccone","sequence":"additional","affiliation":[]},{"given":"J. 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Reconstructions exist for any microstate, but no reconstruction works for all microstates. We refine this dichotomy, demonstrating that the same boundary operator can often be used for large subspaces of black hole microstates, corresponding to a constant fraction <jats:italic>α</jats:italic> of the black hole entropy. In the Schrödinger picture, the boundary subregion encodes the <jats:italic>α</jats:italic>-bits (a concept from quantum information) of a bulk region containing the black hole and bounded by extremal surfaces. These results have important consequences for the structure of AdS/CFT and for quantum information. Firstly, they imply that the bulk reconstruction is necessarily only approximate and allow us to place non-perturbative lower bounds on the error when doing so. Second, they provide a simple and tractable limit in which the entanglement wedge is state dependent, but in a highly controlled way. Although the state dependence of operators comes from ordinary quantum error correction, there are clear connections to the Papadodimas-Raju proposal for understanding operators behind black hole horizons. In tensor network toy models of AdS/CFT, we see how state dependence arises from the bulk operator being ‘pushed’ through the black hole itself. Finally, we show that black holes provide the first ‘explicit’ examples of capacity-achieving <jats:italic>α</jats:italic>-bit codes. Unintuitively, Hawking radiation always reveals the <jats:italic>α</jats:italic>-bits of a black hole as soon as possible. In an appendix, we apply a result from the quantum information literature to prove that entanglement wedge reconstruction can be made exact to all orders in 1<jats:italic>/N</jats:italic>.</jats:p>","DOI":"10.1007/jhep12(2019)007","type":"journal-article","created":{"date-parts":[[2019,12,5]],"date-time":"2019-12-05T17:16:52Z","timestamp":1575566212000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":54,"title":"Learning the Alpha-bits of black holes","prefix":"10.1007","volume":"2019","author":[{"given":"Patrick","family":"Hayden","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-8627-5237","authenticated-orcid":false,"given":"Geoffrey","family":"Penington","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,12,2]]},"reference":[],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","link":[{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP12(2019)007.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/article/10.1007/JHEP12(2019)007/fulltext.html","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP12(2019)007.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,3]],"date-time":"2021-09-03T19:54:45Z","timestamp":1630698885000},"score":1,"resource":{"primary":{"URL":"https://link.springer.com/10.1007/JHEP12(2019)007"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12]]},"references-count":58,"journal-issue":{"issue":"12","published-print":{"date-parts":[[2019,12]]}},"alternative-id":["11902"],"URL":"http://dx.doi.org/10.1007/JHEP12(2019)007","relation":{},"ISSN":["1029-8479"],"subject":["Nuclear and High Energy Physics"],"container-title-short":"J. 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Developing the subject 'from the ground up' it covers classical results as well as major advances of the past decade. Beginning with an extensive overview of classical information theory suitable for the non-expert, the author then turns his attention to quantum mechanics for quantum information theory, and the important protocols of teleportation, super-dense coding and entanglement distribution. He develops all of the tools necessary for understanding important results in quantum information theory, including capacity theorems for classical, entanglement-assisted, private and quantum communication. The book also covers important recent developments such as superadditivity of private, coherent and Holevo information, and the superactivation of quantum capacity. This book will be warmly welcomed by the upcoming generation of quantum information theorists and the already established community of classical information theorists.</jats:p>","DOI":"10.1017/cbo9781139525343","type":"monograph","created":{"date-parts":[[2013,5,5]],"date-time":"2013-05-05T04:17:06Z","timestamp":1367727426000},"source":"Crossref","is-referenced-by-count":604,"title":"Quantum Information Theory","prefix":"10.1017","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2013,5,5]]},"container-title":[],"original-title":[],"deposited":{"date-parts":[[2022,6,13]],"date-time":"2022-06-13T18:21:44Z","timestamp":1655144504000},"score":1,"resource":{"primary":{"URL":"https://www.cambridge.org/core/product/identifier/9781139525343/type/book"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,4,18]]},"ISBN":["9781107034259","9781139525343"],"references-count":0,"URL":"http://dx.doi.org/10.1017/CBO9781139525343","relation":{},"published":{"date-parts":[[2013,4,18]]},"reference":[],"id":"doi:10.1017/CBO9781139525343","_hash":"803666f83242193b9f3f627c62c7b7294df2c8ce32ed7f9f688da8fbe6c8f689"},"expire":1721576483929},"doi:10.22331/q-2023-07-06-1048":{"value":{"indexed":{"date-parts":[[2023,7,7]],"date-time":"2023-07-07T04:25:15Z","timestamp":1688703915460},"reference-count":65,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,7,6]],"date-time":"2023-07-06T00:00:00Z","timestamp":1688601600000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"National Science and Technology Council, Taiwan","award":["NSTC 112-2119-M-002-014"]},{"name":"National Science and Technology Council, Taiwan","award":["NSTC 111- 2119-M-002-006-MY3"]},{"name":"National Science and Technology Council, Taiwan","award":["NSTC 110-2627-M002-002"]},{"name":"National Science and Technology Council, Taiwan","award":["NSTC 111-2627-M-002-001"]},{"name":"Center for Advanced Computing and Imaging in Biomedicine","award":["NTU-112L900702"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>In this work, we compute the number of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mi>d</mml:mi></mml:msub></mml:math> stabilizer codes made up of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>-dimensional qudits, for arbitrary positive integers <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>. In a seminal work by Gross \\cite{Gross2006} the number of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mi>d</mml:mi></mml:msub></mml:math> stabilizer codes was computed for the case when <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> is a prime (or the power of a prime, i.e., <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mi>m</mml:mi></mml:msup></mml:math>, but when the qudits are Galois-qudits). The proof in \\cite{Gross2006} is inapplicable to the non-prime case. For our proof, we introduce a group structure to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mi>d</mml:mi></mml:msub></mml:math> codes, and use this in conjunction with the Chinese remainder theorem to count the number of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mi>d</mml:mi></mml:msub></mml:math> codes. Our work overlaps with \\cite{Gross2006} when <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> is a prime and in this case our results match exactly, but the results differ for the more generic case. Despite that, the overall order of magnitude of the number of stabilizer codes scales agnostic of whether the dimension is prime or non-prime. This is surprising since the method employed to count the number of stabilizer states (or more generally stabilizer codes) depends on whether <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> is prime or not. The cardinality of stabilizer states, which was so far known only for the prime-dimensional case (and the Galois qudit prime-power dimensional case) plays an important role as a quantifier in many topics in quantum computing. Salient among these are the resource theory of magic, design theory, de Finetti theorem for stabilizer states, the study and optimisation of the classical simulability of Clifford circuits, the study of quantum contextuality of small-dimensional systems and the study of Wigner-functions. Our work makes available this quantifier for the generic case, and thus is an important step needed to place results for quantum computing with non-prime dimensional quantum systems on the same pedestal as prime-dimensional systems.</jats:p>","DOI":"10.22331/q-2023-07-06-1048","type":"journal-article","created":{"date-parts":[[2023,7,6]],"date-time":"2023-07-06T14:16:26Z","timestamp":1688652986000},"page":"1048","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Counting stabiliser codes for arbitrary dimension","prefix":"10.22331","volume":"7","author":[{"ORCID":"http://orcid.org/0000-0001-5986-248X","authenticated-orcid":false,"given":"Tanmay","family":"Singal","sequence":"first","affiliation":[{"name":"Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5/7, 87-100 Toruń, 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In this work we begin by quantifying these requirements. We then introduce the construction of local neural network (NN) decoders using three-dimensional convolutions. These local decoders are adapted to circuit-level noise and can be applied to surface code volumes of arbitrary size. Their application removes errors arising from a certain number of faults, which serves to substantially reduce the syndrome density. Remaining errors can then be corrected by a global decoder, such as Blossom or union find, with their implementation significantly accelerated due to the reduced syndrome density. However, in the circuit-level setting, the corrections applied by the local decoder introduce many vertical pairs of highlighted vertices. To obtain a low syndrome density in the presence of vertical pairs, we consider a strategy of performing a syndrome collapse which removes many vertical pairs and reduces the size of the decoding graph used by the global decoder. We also consider a strategy of performing a vertical cleanup, which consists of removing all local vertical pairs prior to implementing the global decoder. By applying our local NN decoder and the vertical cleanup strategy to a <jats:italic>d</jats:italic> = 17 surface code volume, we show a <jats:inline-formula>\n                     <jats:tex-math><?CDATA $10^6 \\times$?></jats:tex-math>\n                     <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n                        <mml:msup>\n                           <mml:mn>10</mml:mn>\n                           <mml:mn>6</mml:mn>\n                        </mml:msup>\n                        <mml:mo>×</mml:mo>\n                     </mml:math>\n                     <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"qstace64dieqn1.gif\" xlink:type=\"simple\" />\n                  </jats:inline-formula> speedup of the minimum-weight perfect matching decoder. Lastly, we estimate the cost of implementing our local decoders on field programmable gate arrays.</jats:p>","DOI":"10.1088/2058-9565/ace64d","type":"journal-article","created":{"date-parts":[[2023,7,11]],"date-time":"2023-07-11T22:26:47Z","timestamp":1689114407000},"page":"045011","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":"Techniques for combining fast local decoders with global decoders under circuit-level noise","prefix":"10.1088","volume":"8","author":[{"ORCID":"http://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"given":"Luis","family":"Goncalves","sequence":"additional","affiliation":[]},{"given":"Prasahnt","family":"Sivarajah","sequence":"additional","affiliation":[]},{"given":"Eric","family":"Peterson","sequence":"additional","affiliation":[]},{"given":"Sebastian","family":"Grimberg","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2023,7,27]]},"reference":[],"container-title":"Quantum Science and Technology","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,27]],"date-time":"2023-07-27T08:17:12Z","timestamp":1690445832000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,27]]},"references-count":72,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,7,27]]},"published-print":{"date-parts":[[2023,10,1]]}},"URL":"http://dx.doi.org/10.1088/2058-9565/ace64d","relation":{},"ISSN":["2058-9565"],"subject":["Electrical and Electronic Engineering","Physics and Astronomy (miscellaneous)","Materials Science (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum Sci. 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zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,9,26]],"date-time":"2023-09-26T00:00:00Z","timestamp":1695686400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"Simons Collaboration on Ultra-Quantum Matter","award":["651438"]},{"name":"Institute for Quantum Information and Matter, an NSF Physics Frontiers Center","award":["PHY-1733907"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Recently, a class of fractal surface codes (FSCs), has been constructed on fractal lattices with Hausdorff dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mi>&amp;#x03F5;</mml:mi></mml:math>, which admits a fault-tolerant non-Clifford CCZ gate \\cite{zhu2021topological}. We investigate the performance of such FSCs as fault-tolerant quantum memories. We prove that there exist decoding strategies with non-zero thresholds for bit-flip and phase-flip errors in the FSCs with Hausdorff dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mi>&amp;#x03F5;</mml:mi></mml:math>. For the bit-flip errors, we adapt the sweep decoder, developed for string-like syndromes in the regular 3D surface code, to the FSCs by designing suitable modifications on the boundaries of the holes in the fractal lattice. Our adaptation of the sweep decoder for the FSCs maintains its self-correcting and single-shot nature. For the phase-flip errors, we employ the minimum-weight-perfect-matching (MWPM) decoder for the point-like syndromes. We report a sustainable fault-tolerant threshold (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x223C;</mml:mo><mml:mn>1.7</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>) under phenomenological noise for the sweep decoder and the code capacity threshold (lower bounded by <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2.95</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>) for the MWPM decoder for a particular FSC with Hausdorff dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>D</mml:mi><mml:mi>H</mml:mi></mml:msub><mml:mo>&amp;#x2248;</mml:mo><mml:mn>2.966</mml:mn></mml:math>. The latter can be mapped to a lower bound of the critical point of a  confinement-Higgs transition on the fractal lattice, which is tunable via the Hausdorff dimension.</jats:p>","DOI":"10.22331/q-2023-09-26-1122","type":"journal-article","created":{"date-parts":[[2023,9,26]],"date-time":"2023-09-26T14:03:22Z","timestamp":1695737002000},"page":"1122","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Quantum error correction with fractal topological codes","prefix":"10.22331","volume":"7","author":[{"given":"Arpit","family":"Dua","sequence":"first","affiliation":[{"name":"Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125 USA"}]},{"given":"Tomas","family":"Jochym-O&apos;Connor","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM T.J. 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zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,8,29]],"date-time":"2023-08-29T00:00:00Z","timestamp":1693267200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"JST Moonshot R&D Grant","award":["JPMJMS2061"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"monospace\">f</mml:mi><mml:mi mathvariant=\"monospace\">l</mml:mi><mml:mi mathvariant=\"monospace\">i</mml:mi><mml:mi mathvariant=\"monospace\">p</mml:mi></mml:mrow></mml:math> is an extremely simple and maximally local classical decoder which has been used to great effect in certain classes of classical codes. When applied to quantum codes there exist constant-weight errors (such as half of a stabiliser) which are uncorrectable for this decoder, so previous studies have considered modified versions of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"monospace\">f</mml:mi><mml:mi mathvariant=\"monospace\">l</mml:mi><mml:mi mathvariant=\"monospace\">i</mml:mi><mml:mi mathvariant=\"monospace\">p</mml:mi></mml:mrow></mml:math>, sometimes in conjunction with other decoders. We argue that this may not always be necessary, and present numerical evidence for the existence of a threshold for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"monospace\">f</mml:mi><mml:mi mathvariant=\"monospace\">l</mml:mi><mml:mi mathvariant=\"monospace\">i</mml:mi><mml:mi mathvariant=\"monospace\">p</mml:mi></mml:mrow></mml:math> when applied to the looplike syndromes of a three-dimensional toric code on a cubic lattice. This result can be attributed to the fact that the lowest-weight uncorrectable errors for this decoder are closer (in terms of Hamming distance) to correctable errors than to other uncorrectable errors, and so they are likely to become correctable in future code cycles after transformation by additional noise. Introducing randomness into the decoder can allow it to correct these \"uncorrectable\" errors with finite probability, and for a decoding strategy that uses a combination of belief propagation and probabilistic <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"monospace\">f</mml:mi><mml:mi mathvariant=\"monospace\">l</mml:mi><mml:mi mathvariant=\"monospace\">i</mml:mi><mml:mi mathvariant=\"monospace\">p</mml:mi></mml:mrow></mml:math> we observe a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x223C;</mml:mo><mml:mn>5.5</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> under phenomenological noise. This is comparable to the best known threshold for this code (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x223C;</mml:mo><mml:mn>7.1</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>) which was achieved using belief propagation and ordered statistics decoding [Higgott and Breuckmann, 2022], a strategy with a runtime of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math> as opposed to the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math> when parallelised) runtime of our local decoder. We expect that this strategy could be generalised to work well in other low-density parity check codes, and hope that these results will prompt investigation of other previously overlooked decoders.</jats:p>","DOI":"10.22331/q-2023-08-29-1093","type":"journal-article","created":{"date-parts":[[2023,8,29]],"date-time":"2023-08-29T11:18:41Z","timestamp":1693307921000},"page":"1093","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":"Local Probabilistic Decoding of a Quantum Code","prefix":"10.22331","volume":"7","author":[{"given":"T. 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zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,8,8]],"date-time":"2023-08-08T00:00:00Z","timestamp":1691452800000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"Office of the Director of National Intelligence - Intelligence Advanced Research Projects Activity","award":["W911NF-16-1-0082"]},{"DOI":"10.13039/100000183","name":"Army Research Office","doi-asserted-by":"crossref","award":["W911NF-21-1-0005"]},{"name":"National Science Foundation","award":["OMA-2120757"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>The Shor fault-tolerant error correction (FTEC) scheme uses transversal gates and ancilla qubits prepared in the cat state in syndrome extraction circuits to prevent propagation of errors caused by gate faults. For a stabilizer code of distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> that can correct up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x230A;</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>d</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x230B;</mml:mo></mml:math> errors, the traditional Shor scheme handles ancilla preparation and measurement faults by performing syndrome measurements until the syndromes are repeated <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math> times in a row; in the worst-case scenario, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:math> rounds of measurements are required. In this work, we improve the Shor FTEC scheme using an adaptive syndrome measurement technique. The syndrome for error correction is determined based on information from the differences of syndromes obtained from consecutive rounds. Our protocols that satisfy the strong and the weak FTEC conditions require no more than <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:math> rounds and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:mo>&amp;#x2212;</mml:mo><mml:mn>2</mml:mn></mml:math> rounds, respectively, and are applicable to any stabilizer code. Our simulations of FTEC protocols with the adaptive schemes on hexagonal color codes of small distances verify that our protocols preserve the code distance, can increase the pseudothreshold, and can decrease the average number of rounds compared to the traditional Shor scheme. We also find that for the code of distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>, our FTEC protocols with the adaptive schemes require no more than <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> rounds on average.</jats:p>","DOI":"10.22331/q-2023-08-08-1075","type":"journal-article","created":{"date-parts":[[2023,8,8]],"date-time":"2023-08-08T13:47:26Z","timestamp":1691502446000},"page":"1075","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":"Adaptive syndrome measurements for Shor-style error correction","prefix":"10.22331","volume":"7","author":[{"ORCID":"http://orcid.org/0000-0002-2865-0705","authenticated-orcid":false,"given":"Theerapat","family":"Tansuwannont","sequence":"first","affiliation":[{"name":"Duke Quantum Center, Duke University, Durham, NC 27701, USA"},{"name":"Department of Electrical and Computer 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In this regard, oscillator-to-oscillator codes not only provide an additional opportunity in bosonic encoding, but also extend the applicability of error correction to continuous-variable states ubiquitous in quantum sensing and communication. In this work, we derive the optimal oscillator-to-oscillator codes among the general family of Gottesman-Kitaev-Preskill (GKP)-stablizer codes for homogeneous noise. We prove that an arbitrary GKP-stabilizer code can be reduced to a generalized GKP two-mode-squeezing (TMS) code. The optimal encoding to minimize the geometric mean error can be constructed from GKP-TMS codes with an optimized GKP lattice and TMS gains. For single-mode data and ancilla, this optimal code design problem can be efficiently solved, and we further provide numerical evidence that a hexagonal GKP lattice is optimal and strictly better than the previously adopted square lattice. For the multimode case, general GKP lattice optimization is challenging. In the two-mode data and ancilla case, we identify the D4 lattice—a 4-dimensional dense-packing lattice—to be superior to a product of lower dimensional lattices. As a by-product, the code reduction allows us to prove a universal no-threshold-theorem for arbitrary oscillators-to-oscillators codes based on Gaussian encoding, even when the ancilla are not GKP states.</jats:p>","DOI":"10.22331/q-2023-08-16-1082","type":"journal-article","created":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T10:07:50Z","timestamp":1692180470000},"page":"1082","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Optimal encoding of oscillators into more oscillators","prefix":"10.22331","volume":"7","author":[{"given":"Jing","family":"Wu","sequence":"first","affiliation":[{"name":"James C. 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We study the limit where the number of fermions erased is large but small compared to the total number of fermions. We compute the <jats:italic>price</jats:italic> of the quantum error correcting code, defined as the number of physical qubits needed to reconstruct whether a given operator has been acted upon the thermal state or not. By thinking about reconstruction via quantum teleportation, we argue for a bound that relates the price to the ordinary operator size in systems that display so-called <jats:italic>detailed size winding</jats:italic> [1]. We then find that in SYK the price roughly saturates this bound. Computing the price requires computing modular flowed correlators with respect to the density matrix associated to a subset of fermions. We offer an interpretation of these correlators as probing a quantum extremal surface in the AdS dual of SYK. In the large <jats:italic>N</jats:italic> limit, the operator algebras associated to subsets of fermions in SYK satisfy half-sided modular inclusion, which is indicative of an emergent Type III1 von Neumann algebra. We discuss the relationship between the emergent algebra of half-sided modular inclusions and bulk symmetry generators.</jats:p>","DOI":"10.1007/jhep06(2022)039","type":"journal-article","created":{"date-parts":[[2022,6,9]],"date-time":"2022-06-09T11:21:23Z","timestamp":1654773683000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":"Quantum error correction in SYK and bulk emergence","prefix":"10.1007","volume":"2022","author":[{"given":"Venkatesa","family":"Chandrasekaran","sequence":"first","affiliation":[]},{"given":"Adam","family":"Levine","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,6,8]]},"reference":[],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","link":[{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP06(2022)039.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/article/10.1007/JHEP06(2022)039/fulltext.html","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP06(2022)039.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,26]],"date-time":"2023-03-26T23:00:21Z","timestamp":1679871621000},"score":1,"resource":{"primary":{"URL":"https://link.springer.com/10.1007/JHEP06(2022)039"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,8]]},"references-count":45,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["18519"],"URL":"http://dx.doi.org/10.1007/JHEP06(2022)039","relation":{},"ISSN":["1029-8479"],"subject":["Nuclear and High Energy Physics"],"container-title-short":"J. 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Technol.","published":{"date-parts":[[2019,2,5]]},"assertion":[{"value":"Quantum Science and Technology","name":"journal_title","label":"Journal title"},{"value":"paper","name":"article_type","label":"Article type"},{"value":"A theory of single-shot error correction for adversarial noise","name":"article_title","label":"Article title"},{"value":"© 2019 IOP Publishing Ltd","name":"copyright_information","label":"Copyright information"},{"value":"2018-08-16","name":"date_received","label":"Date received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2019-01-07","name":"date_accepted","label":"Date accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2019-02-05","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/2058-9565/aafc8f","_hash":"f50f36cdcbce73f671b6a76717eef9c712960e6975f75ddfb8ba631600681e3a"},"expire":1729266846834},"doi:10.21468/SciPostPhys.11.5.094":{"value":{"indexed":{"date-parts":[[2023,10,10]],"date-time":"2023-10-10T14:45:53Z","timestamp":1696949153427},"reference-count":49,"publisher":"Stichting SciPost","issue":"5","license":[{"start":{"date-parts":[[2021,11,19]],"date-time":"2021-11-19T00:00:00Z","timestamp":1637280000000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"content-domain":{"domain":["scipost.org"],"crossmark-restriction":false},"abstract":"<jats:p>In recent years quantum error correction (QEC) has become an\nimportant part of AdS/CFT. Unfortunately, there are no field-theoretic\narguments about why QEC holds in known holographic systems. The purpose\nof this paper is to fill this gap by studying the error correcting\nproperties of the fermionic sector of various large\n<jats:inline-formula><jats:alternatives><jats:tex-math>N</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>N</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\ntheories. Specifically we examine <jats:inline-formula><jats:alternatives><jats:tex-math>SU(N)</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mrow><mml:mo stretchy=\"true\" form=\"prefix\">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"true\" form=\"postfix\">)</mml:mo></mml:mrow></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>\nmatrix quantum mechanics and 3-rank tensor\n<jats:inline-formula><jats:alternatives><jats:tex-math>O(N)^3</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mi>O</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"true\" form=\"prefix\">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"true\" form=\"postfix\">)</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>\ntheories. Both of these theories contain large gauge groups. We argue\nthat gauge singlet states indeed form a quantum error correcting code.\nOur considerations are based purely on large\n<jats:inline-formula><jats:alternatives><jats:tex-math>N</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>N</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\nanalysis and do not appeal to a particular form of Hamiltonian or\nholography.</jats:p>","DOI":"10.21468/scipostphys.11.5.094","type":"journal-article","created":{"date-parts":[[2021,11,19]],"date-time":"2021-11-19T20:36:58Z","timestamp":1637354218000},"update-policy":"http://dx.doi.org/10.21468/scipost.crossmarkpolicy","source":"Crossref","is-referenced-by-count":3,"title":"Quantum error correction and large $N$","prefix":"10.21468","volume":"11","author":[{"given":"Alexey","family":"Milekhin","sequence":"first","affiliation":[{"name":"Princeton University"}]}],"member":"8907","published-online":{"date-parts":[[2021,11,19]]},"reference":[],"container-title":"SciPost Physics","original-title":[],"link":[{"URL":"https://scipost.org/10.21468/SciPostPhys.11.5.094/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://scipost.org/10.21468/SciPostPhys.11.5.094/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,19]],"date-time":"2021-11-19T20:37:02Z","timestamp":1637354222000},"score":1,"resource":{"primary":{"URL":"https://scipost.org/10.21468/SciPostPhys.11.5.094"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,19]]},"references-count":49,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021]]}},"URL":"http://dx.doi.org/10.21468/SciPostPhys.11.5.094","relation":{},"ISSN":["2542-4653"],"subject":["General Physics and Astronomy"],"container-title-short":"SciPost Phys.","published":{"date-parts":[[2021,11,19]]},"article-number":"094","id":"doi:10.21468/SciPostPhys.11.5.094","_hash":"1015dca57561e2911ce7e0f62d54ac58064c3814c7808a5e6a56819518fdb2ae"},"expire":1729266847936},"preset:PreskillNotes":{"value":{"_ready_formatted":{"flm":"J. Preskill.  \\emph{Lecture notes on Quantum Computation.} (1997–2020) \\href{http://theory.caltech.edu/~preskill/ph219/}{URL}"},"id":"preset:PreskillNotes","_hash":"df675189e33ce588fbee6398966862b13985d48d8839950be89e1d4b040d0c0a"},"expire":1698321650466},"manual:{Andries E. Brouwer, Bounds on linear codes, in: Vera S. Pless and W. Cary Huffman (Eds.), Handbook of Coding Theory, pp. 295-461, Elsevier, 1998.}":{"value":{"_ready_formatted":{"flm":"{Andries E. Brouwer, Bounds on linear codes, in: Vera S. Pless and W. Cary Huffman (Eds.), Handbook of Coding Theory, pp. 295-461, Elsevier, 1998.}"},"id":"manual:{Andries E. Brouwer, Bounds on linear codes, in: Vera S. Pless and W. Cary Huffman (Eds.), Handbook of Coding Theory, pp. 295-461, Elsevier, 1998.}","_hash":"8332901ba1eb44a21c81a6ef0d32667602c6495818784cc44c9b4846ed49cdad"},"expire":1698321590503}}
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Phys. Soc. Jpn.","published":{"date-parts":[[1986,10,15]]},"id":"doi:10.1143/JPSJ.55.3305","_hash":"d626718640ba60f749370d063bb489138065a17dc280ab492538dd365e0673e6"},"expire":1701130346055},"doi:10.7907/AHMQ-EG82":{"value":{"type":"thesis","id":"doi:10.7907/AHMQ-EG82","categories":["fault tolerance","Physics","error correction","fault-tolerant","computing"],"language":"en","author":[{"family":"Harrington","given":"James William"}],"issued":{"date-parts":[[2004]]},"abstract":"Quantum information theory is concerned with identifying how quantum mechanical resources, such as entangled quantum states, can be utilized for a number of information processing tasks, including data storage, computation, communication, and cryptography.  Efficient quantum algorithms and protocols have been developed for performing some tasks (e.g., factoring large numbers, securely communicating over a public channel, and simulating quantum mechanical systems) that appear to be very difficult with just classical resources.  In addition to identifying the separation between classical and quantum computational power, much of the theoretical focus in this field over the last decade has been concerned with finding novel ways of encoding quantum information that are robust against errors, which is an important step toward building practical quantum information processing devices.\n\nIn this thesis I present some results on the quantum error-correcting properties of oscillator codes (also described as symplectic lattice codes) and toric codes.  Any harmonic oscillator system, such as a mode of light, can be encoded with quantum information via symplectic lattice codes that are robust against shifts in the system's continuous quantum variables.  I show the existence of lattice codes whose achievable rates match the one-shot coherent information over the Gaussian quantum channel.  Also, I construct a family of symplectic self-dual lattices and search for optimal encodings of quantum information distributed between several oscillators.\n\nToric codes provide encodings of quantum information into two-dimensional spin lattices that are robust against local clusters of errors and which require only local quantum operations for error correction.  Numerical simulations of this system under realistic error models provide a calculation of the accuracy threshold for quantum memory using toric codes, which can be related to phase transitions in particular condensed matter models.  I also present a local classical processing scheme for correcting errors on toric codes, which demonstrates that quantum information can be maintained in two dimensions by purely local quantum and classical resources.","DOI":"10.7907/AHMQ-EG82","publisher":"California Institute of Technology","title":"Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Codes","URL":"https://resolver.caltech.edu/CaltechETD:etd-05122004-113132","copyright":"No commercial reproduction, distribution, display or performance rights in this work are provided.","version":"Final","reference":[],"_hash":"347fe3088c237ff38c67e94a1ba231070cc7858231e4c4d55cf02eaf6558b350"},"expire":1701130346859},"doi:10.6028/jres.069B.013":{"value":{"indexed":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T05:36:14Z","timestamp":1669354574244},"reference-count":0,"publisher":"National Institute of Standards and Technology (NIST)","issue":"1 and 2","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[1965,1]]},"DOI":"10.6028/jres.069b.013","type":"journal-article","created":{"date-parts":[[2012,8,17]],"date-time":"2012-08-17T15:49:20Z","timestamp":1345218560000},"page":"125","source":"Crossref","is-referenced-by-count":969,"title":"Maximum matching and a polyhedron with 0,1-vertices","prefix":"10.6028","volume":"69B","author":[{"given":"Jack","family":"Edmonds","sequence":"first","affiliation":[]}],"member":"4068","container-title":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","original-title":[],"language":"en","deposited":{"date-parts":[[2018,3,6]],"date-time":"2018-03-06T09:23:28Z","timestamp":1520328208000},"score":1,"resource":{"primary":{"URL":"https://nvlpubs.nist.gov/nistpubs/jres/69B/jresv69Bn1-2p125_A1b.pdf"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1965,1]]},"references-count":0,"journal-issue":{"issue":"1 and 2","published-print":{"date-parts":[[1965,1]]}},"URL":"http://dx.doi.org/10.6028/jres.069B.013","relation":{},"ISSN":["0022-4340"],"subject":["Community and Home Care"],"container-title-short":"J. RES. NATL. BUR. STAN. SECT. B. MATH. MATH. 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An edge is said to <jats:italic>join</jats:italic> its end-points.</jats:p><jats:p>A <jats:italic>matching</jats:italic> in <jats:italic>G</jats:italic> is a subset of its edges such that no two meet the same vertex. We describe an efficient algorithm for finding in a given graph a matching of maximum cardinality. This problem was posed and partly solved by C. 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Phys.","published":{"date-parts":[[1990,4]]},"id":"doi:10.1007/BF02096988","_hash":"bfa4336af1a3322d0451838d3e07f9ece23f222630418354c8e7f783e5862384"},"expire":1701130356034},"doi:10.7907/059V-MG69":{"value":{"type":"thesis","id":"doi:10.7907/059V-MG69","categories":["topological quantum codes","quantum many-body physics","quantum information","Physics","quantum computation","quantum error correction"],"language":"en","author":[{"family":"Kubica","given":"Aleksander Marek"}],"contributor":[{"literal":"Institute for Quantum Information and Matter"},{"literal":"IQIM"}],"issued":{"date-parts":[[2018]]},"abstract":"This thesis is devoted to studying a class of quantum error-correcting codes — topological quantum codes. We explore the question of how one can achieve fault- tolerant quantum computation with topological codes. We treat quantum error-correcting codes not only as a compelling ingredient needed to build a quantum computer, but also as a useful theoretical tool in other areas of physics. In particular, we explore what insights topological codes can provide into challenging questions, such as the classification of quantum phases of matter.\n\nIn this thesis, we focus on a family of topological codes — color codes, which are particularly intriguing due to the rich physics they display and their computational power. We start by introducing color codes and explaining their basic properties. Then, we show how to perform fault-tolerant universal quantum computation with three-dimensional color codes by transverse gates and code switching. We later compare the resource overhead of the code-switching approach with that of a state distillation scheme. We discuss how to perform error correction with the toric and color codes, as well as introduce local decoders for those two families of codes. By exploiting a connection between error correction and statistical mechanics we estimate the storage threshold error rates for bit-flip and phase-flip noise in the three-dimensional color code. We finish by showing that the color and toric code families in d dimensions are equivalent in a sense of local unitary transformations and explore implications of this equivalence.","DOI":"10.7907/059V-MG69","publisher":"California Institute of Technology","title":"The ABCs of the Color Code: A Study of Topological Quantum Codes as Toy Models for Fault-Tolerant Quantum Computation and Quantum Phases Of Matter","URL":"https://resolver.caltech.edu/CaltechTHESIS:05282018-173928314","copyright":"No commercial reproduction, distribution, display or performance rights in this work are provided.","version":"Final","reference":[],"_hash":"2b8fe7163dfb727bb3cf03d9c769010671607be7a2aa6a038ef7b6cd04ace1c1"},"expire":1701130356844},"doi:10.1088/1367-2630/13/4/043005":{"value":{"indexed":{"date-parts":[[2022,10,24]],"date-time":"2022-10-24T13:28:53Z","timestamp":1666618133823},"reference-count":29,"publisher":"IOP Publishing","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1088/1367-2630/13/4/043005","type":"journal-article","created":{"date-parts":[[2011,4,8]],"date-time":"2011-04-08T03:39:51Z","timestamp":1302233991000},"page":"043005","source":"Crossref","is-referenced-by-count":44,"title":"Clifford gates by code deformation","prefix":"10.1088","volume":"13","author":[{"given":"H","family":"Bombin","sequence":"first","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2011,4,7]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"deposited":{"date-parts":[[2020,4,11]],"date-time":"2020-04-11T15:15:00Z","timestamp":1586618100000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/13/4/043005"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,4,7]]},"references-count":29,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2011,4,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/13/4/043005","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. 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Its two-dimensional version, the triangular color code, may soon be realized with currently available superconducting hardware despite constrained qubit connectivity. To guide this experimental effort, we study the storage threshold of the triangular color code against circuit-level depolarizing noise. First, we adapt the Restriction Decoder to the setting of the triangular color code and to phenomenological noise. Then, we propose a fault-tolerant implementation of the stabilizer measurement circuits, which incorporates flag qubits. We show how information from flag qubits can be used in an efficient and scalable way with the Restriction Decoder to maintain the effective distance of the code. We numerically estimate the threshold of the triangular color code to be 0.2%, which is competitive with the thresholds of other topological quantum codes. We also prove that 1-flag stabilizer measurement circuits are sufficient to preserve the full code distance, which may be used to find simpler syndrome extraction circuits of the color code.</jats:p>","DOI":"10.1088/1367-2630/ab68fd","type":"journal-article","created":{"date-parts":[[2020,1,8]],"date-time":"2020-01-08T22:16:52Z","timestamp":1578521812000},"page":"023019","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":34,"title":"Triangular color codes on trivalent graphs with flag qubits","prefix":"10.1088","volume":"22","author":[{"ORCID":"http://orcid.org/0000-0003-3239-5783","authenticated-orcid":false,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"given":"Aleksander","family":"Kubica","sequence":"additional","affiliation":[]},{"given":"Theodore J","family":"Yoder","sequence":"additional","affiliation":[]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2020,2,7]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd","content-type":"text/html","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,24]],"date-time":"2021-11-24T21:51:16Z","timestamp":1637790676000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab68fd"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,1]]},"references-count":59,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2,7]]},"published-print":{"date-parts":[[2020,2,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/ab68fd","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. 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Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft","name":"copyright_information","label":"Copyright Information"},{"value":"2019-12-02","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-01-08","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-02-07","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/ab68fd","_hash":"24e4d1e3334fad341d085e8f1c07dd94f7f66753821de5d15c8dcac777444176"},"expire":1701130359855},"doi:10.1070/PU1982v025n04ABEH004537":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T21:08:43Z","timestamp":1670015323353},"reference-count":0,"publisher":"IOP Publishing","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[1982,4,30]]},"DOI":"10.1070/pu1982v025n04abeh004537","type":"journal-article","created":{"date-parts":[[2005,11,22]],"date-time":"2005-11-22T09:22:37Z","timestamp":1132651357000},"page":"231-256","source":"Crossref","is-referenced-by-count":976,"title":"The Jahn-Teller effect and magnetism: transition metal compounds","prefix":"10.1070","volume":"25","author":[{"given":"Kliment I","family":"Kugel'","sequence":"first","affiliation":[]},{"given":"D I","family":"Khomskiĭ","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2007,10,18]]},"container-title":"Soviet Physics Uspekhi","original-title":[],"link":[{"URL":"http://stacks.iop.org/0038-5670/25/i=4/a=R03/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2018,3,7]],"date-time":"2018-03-07T09:22:28Z","timestamp":1520414548000},"score":1,"resource":{"primary":{"URL":"http://stacks.iop.org/0038-5670/25/i=4/a=R03?key=crossref.1b66cf258ac8ee13f109957b04145a7c"},"secondary":[{"URL":"http://www.turpion.org/abstract/pu4537"}]},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,4,30]]},"references-count":0,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1982,4,30]]}},"URL":"http://dx.doi.org/10.1070/PU1982v025n04ABEH004537","relation":{},"ISSN":["0038-5670"],"container-title-short":"Sov. 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However, the two-qubit error-correction technique, known as ‘flagged’ syndrome extraction, works slowly, measuring only one syndrome at a time. This is a disadvantage in experiments with high qubit rest error rates. We extend the technique to extract multiple syndromes at once, without needing more qubits. Qubits for different syndromes can flag errors in each other. This gives equally fast and more qubit-efficient alternatives to Steane’s error-correction method, and also conforms to planar geometry constraints. We further show that Steane’s code and some others can be error-corrected with no extra qubits, provided there are at least two code blocks. The rough idea is that two seven-qubit codewords can be temporarily joined into a twelve-qubit code, freeing two qubits for flagged syndrome measurement.</jats:p>","DOI":"10.1088/2058-9565/abc6f4","type":"journal-article","created":{"date-parts":[[2020,12,4]],"date-time":"2020-12-04T13:40:28Z","timestamp":1607089228000},"page":"015007","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":12,"title":"Fault-tolerant quantum error correction for Steane’s seven-qubit color code with few or no extra qubits","prefix":"10.1088","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0002-4934-8732","authenticated-orcid":false,"given":"Ben W","family":"Reichardt","sequence":"first","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2020,11,24]]},"reference":[],"container-title":"Quantum Science and Technology","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4","content-type":"text/html","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,23]],"date-time":"2022-01-23T02:12:06Z","timestamp":1642903926000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abc6f4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,24]]},"references-count":20,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,11,24]]},"published-print":{"date-parts":[[2021,1,1]]}},"URL":"http://dx.doi.org/10.1088/2058-9565/abc6f4","relation":{},"ISSN":["2058-9565"],"subject":["Electrical and Electronic Engineering","Physics and Astronomy (miscellaneous)","Materials Science (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum Sci. 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We believe that the construction may be of interest for hardware in which one wants to use both long-range two-qubit gates as well as short-range gates between small clusters of qubits.</jats:p>","DOI":"10.26421/qic16.15-16-1","type":"journal-article","created":{"date-parts":[[2021,2,25]],"date-time":"2021-02-25T02:02:10Z","timestamp":1614218530000},"page":"1261-1281","source":"Crossref","is-referenced-by-count":5,"title":"Noise thresholds for the [4,2,2]-concatenated toric code","prefix":"10.26421","volume":"16","author":[{"given":"Ben","family":"Criger","sequence":"first","affiliation":[]},{"given":"Barbara","family":"Terhal","sequence":"additional","affiliation":[]}],"member":"10955","published-online":{"date-parts":[[2016,11]]},"container-title":"Quantum Information and 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Manipulation of spin quantum memory generally requires a magnetic field, which hinders the integration with superconducting qubits. Completely zero-field operation is desirable for scaling up a quantum computer based on superconducting qubits. Here we demonstrate quantum error correction to protect the nuclear spin of the nitrogen as a quantum memory in a diamond nitrogen-vacancy center with two nuclear spins of the surrounding carbon isotopes under a zero magnetic field. 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[1996], has placed the PCP theorem at the heart of the area of inapproximability.</jats:p>\n          <jats:p>In this work, we present a new proof of the PCP theorem that draws on this equivalence. We give a combinatorial proof for the NP-hardness of approximating a certain constraint satisfaction problem, which can then be reinterpreted to yield the PCP theorem.</jats:p>\n          <jats:p>\n            Our approach is to consider the\n            <jats:italic>unsat value</jats:italic>\n            of a constraint system, which is the smallest fraction of unsatisfied constraints, ranging over all possible assignments for the underlying variables. We describe a new combinatorial amplification transformation that doubles the unsat-value of a constraint-system, with only a linear blowup in the size of the system. The amplification step causes an increase in alphabet-size that is corrected by a (standard) PCP composition step. Iterative application of these two steps yields a proof for the PCP theorem.\n          </jats:p>\n          <jats:p>The amplification lemma relies on a new notion of “graph powering” that can be applied to systems of binary constraints. This powering amplifies the unsat-value of a constraint system provided that the underlying graph structure is an expander.</jats:p>\n          <jats:p>\n            We also extend our amplification lemma towards construction of assignment testers (alternatively, PCPs of Proximity) which are slightly stronger objects than PCPs. We then construct PCPs and locally-testable codes whose length is linear up to a\n            <jats:italic>polylog</jats:italic>\n            factor, and whose correctness can be probabilistically verified by making a\n            <jats:italic>constant</jats:italic>\n            number of queries. Namely, we prove\n            <jats:italic>SAT</jats:italic>\n            ∈\n            <jats:italic>PCP</jats:italic>\n            <jats:sub>1/2,1</jats:sub>\n            [log\n            <jats:sub>2</jats:sub>\n            (\n            <jats:italic>n</jats:italic>\n            ⋅poly log\n            <jats:italic>n</jats:italic>\n            ),\n            <jats:italic>O</jats:italic>\n            (1)].\n          </jats:p>","DOI":"10.1145/1236457.1236459","type":"journal-article","created":{"date-parts":[[2007,9,14]],"date-time":"2007-09-14T13:44:55Z","timestamp":1189777495000},"page":"12","source":"Crossref","is-referenced-by-count":197,"title":"The PCP theorem by gap amplification","prefix":"10.1145","volume":"54","author":[{"given":"Irit","family":"Dinur","sequence":"first","affiliation":[{"name":"The Hebrew University, Jerusalem, Israel"}]}],"member":"320","reference":[],"container-title":"Journal of the ACM","original-title":[],"language":"en","link":[{"URL":"https://dl.acm.org/doi/pdf/10.1145/1236457.1236459","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,20]],"date-time":"2021-02-20T04:41:34Z","timestamp":1613796094000},"score":1,"resource":{"primary":{"URL":"https://dl.acm.org/doi/10.1145/1236457.1236459"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,6]]},"references-count":33,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2007,6]]}},"alternative-id":["10.1145/1236457.1236459"],"URL":"http://dx.doi.org/10.1145/1236457.1236459","relation":{},"ISSN":["0004-5411","1557-735X"],"subject":["Artificial Intelligence","Hardware and Architecture","Information Systems","Control and Systems Engineering","Software"],"container-title-short":"J. 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Comput.","published":{"date-parts":[[1998,6]]},"id":"doi:10.1137/S0097539796302531","_hash":"d7bee88b3b8ac5bdcecfc446cb891b28a894ba9b06770d0f3306877217519a58"},"expire":1701130699284},"doi:10.1145/502090.502098":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T19:12:55Z","timestamp":1670008375741},"reference-count":32,"publisher":"Association for Computing Machinery (ACM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2001,7]]},"abstract":"<jats:p>\n            We prove optimal, up to an arbitrary ε &gt; 0, inapproximability results for Max-E\n            <jats:italic>k</jats:italic>\n            -Sat for\n            <jats:italic>k</jats:italic>\n            ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime\n            <jats:italic>p</jats:italic>\n            and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-di-Cut, and Vertex cover.\n          </jats:p>","DOI":"10.1145/502090.502098","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T11:26:13Z","timestamp":1027769173000},"page":"798-859","source":"Crossref","is-referenced-by-count":705,"title":"Some optimal inapproximability results","prefix":"10.1145","volume":"48","author":[{"given":"Johan","family":"Håstad","sequence":"first","affiliation":[{"name":"Royal Institute of Technology, Stockholm, Sweden"}]}],"member":"320","reference":[],"container-title":"Journal of the ACM","original-title":[],"language":"en","link":[{"URL":"https://dl.acm.org/doi/pdf/10.1145/502090.502098","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,4]],"date-time":"2021-03-04T17:21:46Z","timestamp":1614878506000},"score":1,"resource":{"primary":{"URL":"https://dl.acm.org/doi/10.1145/502090.502098"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,7]]},"references-count":32,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2001,7]]}},"alternative-id":["10.1145/502090.502098"],"URL":"http://dx.doi.org/10.1145/502090.502098","relation":{},"ISSN":["0004-5411","1557-735X"],"subject":["Artificial Intelligence","Hardware and Architecture","Information Systems","Control and Systems Engineering","Software"],"container-title-short":"J. 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Inform. Theory","published":{"date-parts":[[2018,8]]},"id":"doi:10.1109/TIT.2018.2809788","_hash":"aa0983d9ef733ab3aa7a71690eee4e2ca9051e37d39434c29d12508cb5eb407b"},"expire":1701130701372},"doi:10.1145/3051093":{"value":{"indexed":{"date-parts":[[2022,9,4]],"date-time":"2022-09-04T23:48:45Z","timestamp":1662335325255},"reference-count":69,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2018,5,25]],"date-time":"2018-05-25T00:00:00Z","timestamp":1527206400000},"content-version":"vor","delay-in-days":357,"URL":"http://www.acm.org/publications/policies/copyright_policy#Background"}],"funder":[{"DOI":"10.13039/100000001","name":"NSF","doi-asserted-by":"publisher","award":["CCF-1253886, CCF-1412958, CCF-1445755 and CCF-1350572"]},{"name":"Sloan Fellowship, Rothschild Fellowship"},{"name":"ERC","award":["239986"]},{"DOI":"10.13039/501100003977","name":"Israel Science Foundation","doi-asserted-by":"crossref","award":["460/05"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2017,6,2]]},"abstract":"<jats:p>\n            Locally correctable codes (LCCs) and locally testable codes (LTCs) are error-correcting codes that admit\n            <jats:italic>local</jats:italic>\n            algorithms for correction and detection of errors. Those algorithms are local in the sense that they only query a small number of entries of the corrupted codeword. The fundamental question about LCCs and LTCs is to determine the optimal tradeoff among their rate, distance, and query complexity.\n          </jats:p>\n          <jats:p>\n            In this work, we construct the first LCCs and LTCs with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist LCCs and LTCs with block length\n            <jats:italic>n</jats:italic>\n            , constant rate (which can even be taken arbitrarily close to 1), and constant relative distance, whose query complexity is exp(Õ(√log\n            <jats:italic>n</jats:italic>\n            )) (for LCCs) and (log\n            <jats:italic>n</jats:italic>\n            )\n            <jats:sup>\n              <jats:italic>O</jats:italic>\n              (log log\n              <jats:italic>n</jats:italic>\n              )\n            </jats:sup>\n            (for LTCs).\n          </jats:p>\n          <jats:p>\n            In addition to having small query complexity, our codes also achieve better tradeoffs between the rate and the relative distance than were previously known to be achievable by LCCs or LTCs. Specifically, over large (but constant size) alphabet, our codes approach the Singleton bound, that is, they have almost the best-possible relationship between their rate and distance. Over the binary alphabet, our codes meet the Zyablov bound. Such tradeoffs between the rate and the relative distance were previously not known for any\n            <jats:italic>o</jats:italic>\n            (\n            <jats:italic>n</jats:italic>\n            ) query complexity. Our results on LCCs also immediately give locally decodable codes with the same parameters.\n          </jats:p>","DOI":"10.1145/3051093","type":"journal-article","created":{"date-parts":[[2017,5,25]],"date-time":"2017-05-25T16:16:45Z","timestamp":1495729005000},"page":"1-42","source":"Crossref","is-referenced-by-count":16,"title":"High-Rate Locally Correctable and Locally Testable Codes with Sub-Polynomial Query Complexity","prefix":"10.1145","volume":"64","author":[{"given":"Swastik","family":"Kopparty","sequence":"first","affiliation":[{"name":"Department of Mathematics 8 Department of Computer Science, Rutgers University, Piscataway NJ, USA"}]},{"given":"Or","family":"Meir","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Haifa University, Haifa, Israel"}]},{"given":"Noga","family":"Ron-Zewi","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Ben-Gurion University, Be’er Sheva, Israel"}]},{"given":"Shubhangi","family":"Saraf","sequence":"additional","affiliation":[{"name":"Department of Mathematics 8 Department of Computer Science, Rutgers University, Piscataway NJ, USA"}]}],"member":"320","reference":[],"container-title":"Journal of the ACM","original-title":[],"language":"en","link":[{"URL":"https://dl.acm.org/doi/pdf/10.1145/3051093","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://dl.acm.org/doi/pdf/10.1145/3051093","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,2]],"date-time":"2021-03-02T23:56:12Z","timestamp":1614729372000},"score":1,"resource":{"primary":{"URL":"https://dl.acm.org/doi/10.1145/3051093"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,6,2]]},"references-count":69,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2017,6,2]]}},"alternative-id":["10.1145/3051093"],"URL":"http://dx.doi.org/10.1145/3051093","relation":{},"ISSN":["0004-5411","1557-735X"],"subject":["Artificial Intelligence","Hardware and Architecture","Information Systems","Control and Systems Engineering","Software"],"container-title-short":"J. ACM","published":{"date-parts":[[2017,6,2]]},"id":"doi:10.1145/3051093","_hash":"4b068138d8842495ac031af2d85cb046e927cb169cb9042515d2b60ce1c64b8a"},"expire":1701130702279},"doi:10.1145/1162349.1162351":{"value":{"indexed":{"date-parts":[[2022,11,22]],"date-time":"2022-11-22T03:29:30Z","timestamp":1669087770977},"reference-count":32,"publisher":"Association for Computing Machinery (ACM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2006,7]]},"abstract":"<jats:p>\n            We initiate a systematic study of locally testable codes; that is, error-correcting codes that admit very efficient membership tests. Specifically, these are codes accompanied with tests that make a constant number of (random) queries into any given word and reject non-codewords with probability proportional to their distance from the code.Locally testable codes are believed to be the combinatorial core of PCPs. However, the relation is less immediate than commonly believed. Nevertheless, we show that certain PCP systems can be modified to yield locally testable codes. On the other hand, we adapt techniques that we develop for the construction of the latter to yield new PCPs.Our main results are locally testable codes and PCPs of almost-linear length. Specifically, we prove the existence of the following constructs:---Locally testable binary (linear) codes in which\n            <jats:italic>k</jats:italic>\n            information bits are encoded by a codeword of length\n            <jats:italic>k</jats:italic>\n            ⋅ exp(Õ(√(log\n            <jats:italic>k</jats:italic>\n            ))). This improves over previous results that either yield codewords of exponential length or obtained almost quadratic length codewords for sufficiently large nonbinary alphabet.---PCP systems of almost-linear length for SAT. The length of the proof is\n            <jats:italic>n</jats:italic>\n            ⋅ exp(Õ(√(log\n            <jats:italic>n</jats:italic>\n            ))) and verification in performed by a constant number (i.e., 19) of queries, as opposed to previous results that used proof length\n            <jats:italic>n</jats:italic>\n            <jats:sup>\n              (1 +\n              <jats:italic>O</jats:italic>\n              (1/\n              <jats:italic>q</jats:italic>\n              ))\n            </jats:sup>\n            for verification by\n            <jats:italic>q</jats:italic>\n            queries.The novel techniques in use include a random projection of certain codewords and PCP-oracles that preserves local-testability, an adaptation of PCP constructions to obtain “linear PCP-oracles” for proving conjunctions of linear conditions, and design of PCPs with some new soundness properties---a direct construction of locally testable (linear) codes of subexponential length.\n          </jats:p>","DOI":"10.1145/1162349.1162351","type":"journal-article","created":{"date-parts":[[2006,10,18]],"date-time":"2006-10-18T18:11:32Z","timestamp":1161195092000},"page":"558-655","source":"Crossref","is-referenced-by-count":78,"title":"Locally testable codes and PCPs of almost-linear length","prefix":"10.1145","volume":"53","author":[{"given":"Oded","family":"Goldreich","sequence":"first","affiliation":[{"name":"Weizmann Institute of Science, Rehovot, Israel"}]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, MA"}]}],"member":"320","reference":[],"container-title":"Journal of the ACM","original-title":[],"language":"en","link":[{"URL":"https://dl.acm.org/doi/pdf/10.1145/1162349.1162351","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,20]],"date-time":"2021-02-20T06:46:19Z","timestamp":1613803579000},"score":1,"resource":{"primary":{"URL":"https://dl.acm.org/doi/10.1145/1162349.1162351"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,7]]},"references-count":32,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2006,7]]}},"alternative-id":["10.1145/1162349.1162351"],"URL":"http://dx.doi.org/10.1145/1162349.1162351","relation":{},"ISSN":["0004-5411","1557-735X"],"subject":["Artificial Intelligence","Hardware and Architecture","Information Systems","Control and Systems Engineering","Software"],"container-title-short":"J. 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The following tight spherical <jats:italic>t</jats:italic>-designs are unique: the 5-design in Ω<jats:sub>7</jats:sub>, the 7-designs in Ω<jats:sub>8</jats:sub> and Ω<jats:sub>23</jats:sub>, and the 11-design in Ω<jats:sub>24</jats:sub>. It was shown in <jats:bold>[20]</jats:bold> that the maximum number of nonoverlapping unit spheres in <jats:bold>R</jats:bold><jats:sup>8</jats:sup> (resp. <jats:bold>R</jats:bold><jats:sup>24</jats:sup>) that can touch another unit sphere is 240 (resp. 196560). Arrangements of spheres meeting these bounds can be obtained from the E<jats:sub>8</jats:sub> and Leech lattices, respectively. 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Others are given in [2<jats:sup><jats:italic>m</jats:italic></jats:sup>], <jats:italic>m</jats:italic> ⩾ 5, with the same density as those of §1.6, but in which each sphere touches fewer other spheres than in the earlier packings.</jats:p>","DOI":"10.4153/cjm-1967-017-0","type":"journal-article","created":{"date-parts":[[2010,12,7]],"date-time":"2010-12-07T23:24:44Z","timestamp":1291764284000},"page":"251-267","source":"Crossref","is-referenced-by-count":123,"title":"Notes on Sphere Packings","prefix":"10.4153","volume":"19","author":[{"given":"John","family":"Leech","sequence":"first","affiliation":[]}],"member":"2643","published-online":{"date-parts":[[2018,11,20]]},"reference":[],"container-title":"Canadian Journal of Mathematics","original-title":[],"language":"en","link":[{"URL":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00054250","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,7]],"date-time":"2019-06-07T18:58:25Z","timestamp":1559933905000},"score":1,"resource":{"primary":{"URL":"https://www.cambridge.org/core/product/identifier/S0008414X00054250/type/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1967]]},"references-count":5,"alternative-id":["S0008414X00054250"],"URL":"http://dx.doi.org/10.4153/CJM-1967-017-0","relation":{},"ISSN":["0008-414X","1496-4279"],"subject":["General Mathematics"],"container-title-short":"Can. j. math.","published":{"date-parts":[[1967]]},"id":"doi:10.4153/CJM-1967-017-0","_hash":"8f3c20b864a438e7f994eeb556b77230d92d5bf65fa6e6c9e88a9a51447ffc6b"},"expire":1701130793327},"doi:10.1017/S1446788700025064":{"value":{"indexed":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T22:54:53Z","timestamp":1668120893903},"reference-count":3,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2009,4,9]],"date-time":"2009-04-09T00:00:00Z","timestamp":1239235200000},"content-version":"unspecified","delay-in-days":18149,"URL":"https://www.cambridge.org/core/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[1959,8]]},"abstract":"<jats:p>Let <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1446788700025064_inline1\" /> be a positive definite quadratic form of determinant <jats:italic>D</jats:italic>, and let <jats:italic>M</jats:italic> be the minimum of <jats:italic>f</jats:italic>(<jats:italic>x</jats:italic>) for integral x ≠ 0. Then we set <jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1446788700025064_eqn1\" /></jats:disp-formula> and <jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1446788700025064_eqn2\" /></jats:disp-formula> the maximum being over all positive forms <jats:italic>f</jats:italic> in <jats:italic>n</jats:italic> variables. <jats:italic>f</jats:italic> is said to be extreme if <jats:italic>y</jats:italic> γ<jats:sub>n</jats:sub>(<jats:italic>f</jats:italic>) is a local maximum for varying <jats:italic>f</jats:italic>, <jats:italic>absolutely extreme</jats:italic> if <jats:italic>y</jats:italic> γ(<jats:italic>f</jats:italic>) is an absolute maximum, i.e. if <jats:italic>y</jats:italic> γ(f) = γ<jats:sub>n</jats:sub>.</jats:p>","DOI":"10.1017/s1446788700025064","type":"journal-article","created":{"date-parts":[[2009,4,14]],"date-time":"2009-04-14T09:25:31Z","timestamp":1239701131000},"page":"47-63","source":"Crossref","is-referenced-by-count":44,"title":"Some extreme forms defined in terms of Abelian groups","prefix":"10.1017","volume":"1","author":[{"given":"E. 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We give a complete description of the groups of coincidence rotations with their associated indices and encapsulate their statistics by means of generating functions.</jats:p>","DOI":"10.1515/zna-1995-0802","type":"journal-article","created":{"date-parts":[[2015,2,21]],"date-time":"2015-02-21T11:25:16Z","timestamp":1424517916000},"page":"711-717","source":"Crossref","is-referenced-by-count":7,"title":"Algebraic Solution of the Coincidence Problem in Two and Three Dimensions","prefix":"10.1515","volume":"50","author":[{"given":"Michael","family":"Baake","sequence":"first","affiliation":[{"name":"Institut für Theoretische Physik, Universität Tübingen, D-72076 Tübingen, Germany"}]},{"given":"Peter A. 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Only degenerate codes with the maximal stabilizer weight much smaller than their minimum distance are considered. It is shown that with the help of OSD-like post-processing the performance of the standard belief propagation (BP) decoder on many QLDPC codes can be improved by several orders of magnitude. Using this new BP-OSD decoder we study the performance of several known classes of degenerate QLDPC codes including hypergraph product codes, hyperbicycle codes, homological product codes, and Haah's cubic codes. We also construct several interesting examples of short generalized bicycle codes. Some of them have an additional property that their syndromes are protected by small BCH codes, which may be useful for the fault-tolerant syndrome measurement. We also propose a new large family of QLDPC codes that contains the class of hypergraph product codes, where one of the used parity-check matrices is square. 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xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>&amp;#x03C0;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>N</mml:mi></mml:math> rotations around the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math> axis for some integer <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math>. The resulting generalised stabiliser formalism – denoted the XP stabiliser formalism – allows for a wider range of states and codespaces to be represented. We describe the states which arise in the formalism, and demonstrate an equivalence between XP stabiliser states and 'weighted hypergraph states' – a generalisation of both hypergraph and weighted graph states. Given an arbitrary set of XP operators, we present algorithms for determining the codespace and logical operators for an XP code. Finally, we consider whether measurements of XP operators on XP codes can be classically simulated.</jats:p>","DOI":"10.22331/q-2022-09-22-815","type":"journal-article","created":{"date-parts":[[2022,9,22]],"date-time":"2022-09-22T15:25:34Z","timestamp":1663860334000},"page":"815","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"The XP Stabiliser Formalism: a Generalisation of the Pauli Stabiliser Formalism with Arbitrary Phases","prefix":"10.22331","volume":"6","author":[{"given":"Mark A.","family":"Webster","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia"},{"name":"Sydney Quantum Academy, Sydney, NSW, Australia"}]},{"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia"},{"name":"Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, University of Copenhagen, 2100 Copenhagen, Denmark"}]},{"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia"}]}],"member":"9598","published-online":{"date-parts":[[2022,9,22]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-09-22-815/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,9,22]],"date-time":"2022-09-22T15:25:38Z","timestamp":1663860338000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-09-22-815/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,22]]},"references-count":26,"URL":"http://dx.doi.org/10.22331/q-2022-09-22-815","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,9,22]]},"article-number":"815","id":"doi:10.22331/q-2022-09-22-815","_hash":"2b2091a06ee33bc0171c65fbc5ffeaca0d45cc9813ce098f5bdc2af69796c40e"},"expire":1701130830769},"doi:10.1103/PhysRevA.105.042616":{"value":{"indexed":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T14:09:28Z","timestamp":1651068568729},"reference-count":74,"publisher":"American Physical Society (APS)","issue":"4","license":[{"start":{"date-parts":[[2022,4,25]],"date-time":"2022-04-25T00:00:00Z","timestamp":1650844800000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/501100004063","name":"Knut och Alice Wallenbergs 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A","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevA.105.042616","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.105.042616/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,25]],"date-time":"2022-04-25T14:12:27Z","timestamp":1650895947000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.105.042616"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,25]]},"references-count":74,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,4]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.105.042616","relation":{},"ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","published":{"date-parts":[[2022,4,25]]},"article-number":"042616","id":"doi:10.1103/PhysRevA.105.042616","_hash":"ec3a9523ee1980ec3427944eb2268584f81b11521fa2f8ebc0121cb1b95d3642"},"expire":1701130831783},"doi:10.22331/q-2022-04-27-698":{"value":{"indexed":{"date-parts":[[2022,5,4]],"date-time":"2022-05-04T22:54:32Z","timestamp":1651704872137},"reference-count":67,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T00:00:00Z","timestamp":1651017600000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>We consider a topological stabilizer code on a honeycomb grid, the \"XYZ<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi /><mml:mn>2</mml:mn></mml:msup></mml:math>\" code. The code is inspired by the Kitaev honeycomb model and is a simple realization of a \"matching code\" discussed by Wootton [J. Phys. A: Math. Theor. 48, 215302 (2015)], with a specific implementation of the boundary. It utilizes weight-six (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi><mml:mi>Y</mml:mi><mml:mi>Z</mml:mi><mml:mi>X</mml:mi><mml:mi>Y</mml:mi><mml:mi>Z</mml:mi></mml:math>) plaquette stabilizers and weight-two (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi><mml:mi>X</mml:mi></mml:math>) link stabilizers on a planar hexagonal grid composed of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> qubits for code distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>, with weight-three stabilizers at the boundary, stabilizing one logical qubit. We study the properties of the code using maximum-likelihood decoding, assuming perfect stabilizer measurements. For pure <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi></mml:math>, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Y</mml:mi></mml:math>, or <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math> noise, we can solve for the logical failure rate analytically, giving a threshold of 50%. In contrast to the rotated surface code and the XZZX code, which have code distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> only for pure <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Y</mml:mi></mml:math> noise, here the code distance is <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> for both pure <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math> and pure <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Y</mml:mi></mml:math> noise. Thresholds for noise with finite <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math> bias are similar to the XZZX code, but with markedly lower sub-threshold logical failure rates. The code possesses distinctive syndrome properties with unidirectional pairs of plaquette defects along the three directions of the triangular lattice for isolated errors, which may be useful for efficient matching-based or other approximate decoding.</jats:p>","DOI":"10.22331/q-2022-04-27-698","type":"journal-article","created":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T13:48:13Z","timestamp":1651067293000},"page":"698","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"The XYZ<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi /><mml:mn>2</mml:mn></mml:msup></mml:math> hexagonal stabilizer code","prefix":"10.22331","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0002-4972-4216","authenticated-orcid":false,"given":"Basudha","family":"Srivastava","sequence":"first","affiliation":[{"name":"Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden"}]},{"ORCID":"http://orcid.org/0000-0002-2534-3021","authenticated-orcid":false,"given":"Anton","family":"Frisk Kockum","sequence":"additional","affiliation":[{"name":"Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-41296 Gothenburg, Sweden"}]},{"ORCID":"http://orcid.org/0000-0003-3185-2014","authenticated-orcid":false,"given":"Mats","family":"Granath","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden"}]}],"member":"9598","published-online":{"date-parts":[[2022,4,27]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-04-27-698/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T13:48:38Z","timestamp":1651067318000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-04-27-698/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,27]]},"references-count":67,"URL":"http://dx.doi.org/10.22331/q-2022-04-27-698","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,4,27]]},"article-number":"698","id":"doi:10.22331/q-2022-04-27-698","_hash":"40cc448b9d448b6c51f9e2b28d0aadd66f8f0bf5ef94dd4543ef701b413c7127"},"expire":1701130832618},"doi:10.1088/1751-8121/ac7a75":{"value":{"indexed":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T12:27:04Z","timestamp":1663763224619},"reference-count":24,"publisher":"IOP Publishing","issue":"29","license":[{"start":{"date-parts":[[2022,7,4]],"date-time":"2022-07-04T00:00:00Z","timestamp":1656892800000},"content-version":"vor","delay-in-days":0,"URL":"https://iopscience.iop.org/page/copyright"},{"start":{"date-parts":[[2022,7,4]],"date-time":"2022-07-04T00:00:00Z","timestamp":1656892800000},"content-version":"tdm","delay-in-days":0,"URL":"https://iopscience.iop.org/info/page/text-and-data-mining"}],"funder":[{"DOI":"10.13039/501100001711","name":"Swiss National Science Foundation","doi-asserted-by":"crossref"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2022,7,22]]},"abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Matching codes are stabilizer codes based on Kitaev’s honeycomb lattice model. The hexagonal form of these codes are particularly well-suited to the heavy-hexagon device layouts currently pursued in the hardware of IBM quantum. Here we show how the stabilizers of the code can be measured solely through two-body measurements that are native to the architecture. Though the subsystem code formed by these measurements has a trivial code space, the sequence in which they are measured allows the desired logical subspace to be preserved. This therefore achieves a result similar to the recently introduced Floquet codes, but via a completely different method. The process is then run on 27 and 65 qubit devices, to compare results with simulations for a standard error model. It is found that the results correspond well to simulations where the noise strength is similar to that found in the benchmarking of the devices. The best devices show results consistent with a noise model with an error probability of around 1.5%–2%.</jats:p>","DOI":"10.1088/1751-8121/ac7a75","type":"journal-article","created":{"date-parts":[[2022,6,20]],"date-time":"2022-06-20T22:24:54Z","timestamp":1655763894000},"page":"295302","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":"Hexagonal matching codes with two-body measurements","prefix":"10.1088","volume":"55","author":[{"ORCID":"http://orcid.org/0000-0003-1943-5306","authenticated-orcid":true,"given":"James R","family":"Wootton","sequence":"first","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2022,7,4]]},"reference":[],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,30]],"date-time":"2022-06-30T16:59:58Z","timestamp":1656608398000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7a75"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,4]]},"references-count":24,"journal-issue":{"issue":"29","published-online":{"date-parts":[[2022,7,4]]},"published-print":{"date-parts":[[2022,7,22]]}},"URL":"http://dx.doi.org/10.1088/1751-8121/ac7a75","relation":{},"ISSN":["1751-8113","1751-8121"],"subject":["General Physics and Astronomy","Mathematical Physics","Modeling and Simulation","Statistics and Probability","Statistical and Nonlinear Physics"],"container-title-short":"J. Phys. A: Math. Theor.","published":{"date-parts":[[2022,7,4]]},"assertion":[{"value":"Hexagonal matching codes with two-body measurements","name":"article_title","label":"Article Title"},{"value":"Journal of Physics A: Mathematical and Theoretical","name":"journal_title","label":"Journal Title"},{"value":"paper","name":"article_type","label":"Article Type"},{"value":"© 2022 IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2022-01-26","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2022-06-20","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2022-07-04","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1751-8121/ac7a75","_hash":"f7c2e5cc374d39f344d1471a419bff1e3ac6ef092c0aea0ce7890f583a00fcd8"},"expire":1701130833686},"doi:10.1103/PhysRevLett.90.016803":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T12:22:53Z","timestamp":1669983773394},"reference-count":19,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[2003,1,10]],"date-time":"2003-01-10T00:00:00Z","timestamp":1042156800000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physrevlett.90.016803","type":"journal-article","created":{"date-parts":[[2003,1,11]],"date-time":"2003-01-11T04:18:41Z","timestamp":1042258721000},"source":"Crossref","is-referenced-by-count":245,"title":"Quantum Orders in an Exact Soluble Model","prefix":"10.1103","volume":"90","author":[{"given":"Xiao-Gang","family":"Wen","sequence":"first","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2003,1,10]]},"reference":[],"container-title":"Physical Review Letters","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevLett.90.016803","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.90.016803/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,15]],"date-time":"2017-06-15T17:18:43Z","timestamp":1497547123000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevLett.90.016803"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1,10]]},"references-count":19,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2003,1]]}},"URL":"http://dx.doi.org/10.1103/PhysRevLett.90.016803","relation":{},"ISSN":["0031-9007","1079-7114"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. Rev. Lett.","published":{"date-parts":[[2003,1,10]]},"article-number":"016803","id":"doi:10.1103/PhysRevLett.90.016803","_hash":"f429aff3ca8cd29e6a08c68b5fbb9ec1e07f89380d564410f93bd28660292862"},"expire":1701130834707},"doi:10.1103/PhysRevResearch.2.013303":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T12:20:19Z","timestamp":1669983619742},"reference-count":60,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[2020,3,12]],"date-time":"2020-03-12T00:00:00Z","timestamp":1583971200000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["CE170100009"]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevresearch.2.013303","type":"journal-article","created":{"date-parts":[[2020,3,13]],"date-time":"2020-03-13T14:41:59Z","timestamp":1584110519000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":23,"title":"Parallelized quantum error correction with fracton topological codes","prefix":"10.1103","volume":"2","author":[{"ORCID":"http://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"first","affiliation":[]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2020,3,12]]},"reference":[],"container-title":"Physical Review Research","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevResearch.2.013303","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevResearch.2.013303/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,13]],"date-time":"2020-03-13T14:42:02Z","timestamp":1584110522000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevResearch.2.013303"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,12]]},"references-count":60,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,3]]}},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.013303","relation":{},"ISSN":["2643-1564"],"subject":["General Engineering"],"container-title-short":"Phys. Rev. Research","published":{"date-parts":[[2020,3,12]]},"article-number":"013303","id":"doi:10.1103/PhysRevResearch.2.013303","_hash":"e3f03f3e4ba4a1f7bb4b760255d436e175644210f06c3956b59985e66d5285d9"},"expire":1701130835812},"doi:10.1038/s41467-022-32094-6":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T16:26:21Z","timestamp":1669998381874},"reference-count":72,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,8,9]],"date-time":"2022-08-09T00:00:00Z","timestamp":1660003200000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,8,9]],"date-time":"2022-08-09T00:00:00Z","timestamp":1660003200000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["OMA-2016136","OMA-2120757","OMA-2120757"]},{"DOI":"10.13039/100000183","name":"United States Department of Defense | United States Army | U.S. Army Research, Development and Engineering Command | Army Research Office","doi-asserted-by":"publisher","award":["W911NF-21-1-0012","W911NF-18-10215"]},{"DOI":"10.13039/100007297","name":"United States Department of Defense | United States Navy | ONR | Office of Naval Research Global","doi-asserted-by":"publisher","award":["N00014-20-1-2426"]},{"DOI":"10.13039/100000185","name":"United States Department of Defense | Defense Advanced Research Projects Agency","doi-asserted-by":"publisher","award":["W911NF-20-10021"]},{"DOI":"10.13039/100000879","name":"Alfred P. Sloan Foundation","doi-asserted-by":"publisher"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"<jats:title>Abstract</jats:title><jats:p>Executing quantum algorithms on error-corrected logical qubits is a critical step for scalable quantum computing, but the requisite numbers of qubits and physical error rates are demanding for current experimental hardware. Recently, the development of error correcting codes tailored to particular physical noise models has helped relax these requirements. In this work, we propose a qubit encoding and gate protocol for <jats:sup>171</jats:sup>Yb neutral atom qubits that converts the dominant physical errors into erasures, that is, errors in known locations. The key idea is to encode qubits in a metastable electronic level, such that gate errors predominantly result in transitions to disjoint subspaces whose populations can be continuously monitored via fluorescence. We estimate that 98% of errors can be converted into erasures. We quantify the benefit of this approach via circuit-level simulations of the surface code, finding a threshold increase from 0.937% to 4.15%. We also observe a larger code distance near the threshold, leading to a faster decrease in the logical error rate for the same number of physical qubits, which is important for near-term implementations. Erasure conversion should benefit any error correcting code, and may also be applied to design new gates and encodings in other qubit platforms.</jats:p>","DOI":"10.1038/s41467-022-32094-6","type":"journal-article","created":{"date-parts":[[2022,8,9]],"date-time":"2022-08-09T10:03:15Z","timestamp":1660039395000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":"Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays","prefix":"10.1038","volume":"13","author":[{"given":"Yue","family":"Wu","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-7095-1547","authenticated-orcid":false,"given":"Shimon","family":"Kolkowitz","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-5194-0591","authenticated-orcid":false,"given":"Shruti","family":"Puri","sequence":"additional","affiliation":[]},{"given":"Jeff D.","family":"Thompson","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,8,9]]},"reference":[],"container-title":"Nature Communications","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41467-022-32094-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-022-32094-6","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-022-32094-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T18:31:22Z","timestamp":1669314682000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41467-022-32094-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,9]]},"references-count":72,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["32094"],"URL":"http://dx.doi.org/10.1038/s41467-022-32094-6","relation":{},"ISSN":["2041-1723"],"subject":["General Physics and Astronomy","General Biochemistry, Genetics and Molecular Biology","General Chemistry","Multidisciplinary"],"container-title-short":"Nat Commun","published":{"date-parts":[[2022,8,9]]},"assertion":[{"value":"11 January 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 July 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 August 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"4657","id":"doi:10.1038/s41467-022-32094-6","_hash":"ff3ca173e53c88f92a182b4910339911fe10dba552b8f4d4770ac397cd002be0"},"expire":1701130836780},"doi:10.1103/PhysRevLett.124.130501":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T12:22:44Z","timestamp":1669983764930},"reference-count":35,"publisher":"American Physical Society (APS)","issue":"13","license":[{"start":{"date-parts":[[2020,3,30]],"date-time":"2020-03-30T00:00:00Z","timestamp":1585526400000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"funder":[{"DOI":"10.13039/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["CE170100009"]},{"DOI":"10.13039/501100001774","name":"University of Sydney","doi-asserted-by":"publisher"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevlett.124.130501","type":"journal-article","created":{"date-parts":[[2020,3,30]],"date-time":"2020-03-30T16:01:23Z","timestamp":1585584083000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":36,"title":"Fault-Tolerant Thresholds for the Surface Code in Excess of \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mn>5</mml:mn><mml:mo>%</mml:mo></mml:math>\n Under Biased Noise","prefix":"10.1103","volume":"124","author":[{"ORCID":"http://orcid.org/0000-0002-3776-2864","authenticated-orcid":true,"given":"David K.","family":"Tuckett","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4387-670X","authenticated-orcid":true,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-3975-0226","authenticated-orcid":true,"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2020,3,30]]},"reference":[],"container-title":"Physical Review Letters","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevLett.124.130501","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.124.130501/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,30]],"date-time":"2020-03-30T16:06:09Z","timestamp":1585584369000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevLett.124.130501"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,30]]},"references-count":35,"journal-issue":{"issue":"13","published-print":{"date-parts":[[2020,3]]}},"URL":"http://dx.doi.org/10.1103/PhysRevLett.124.130501","relation":{},"ISSN":["0031-9007","1079-7114"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. Rev. 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The challenge is to design practical quantum error-correcting codes that perform well against realistic noise using modest resources. Here we show that a variant of the surface code—the XZZX code—offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favourable sub-threshold resource scaling that can be obtained by specialising a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation.</jats:p>","DOI":"10.1038/s41467-021-22274-1","type":"journal-article","created":{"date-parts":[[2021,4,12]],"date-time":"2021-04-12T10:09:07Z","timestamp":1618222147000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":"The XZZX surface code","prefix":"10.1038","volume":"12","author":[{"ORCID":"http://orcid.org/0000-0001-5518-7907","authenticated-orcid":false,"given":"J. 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Call details: Consolidator Grant (CoG), PE2, ERC-2015-CoG"},{"DOI":"10.13039/100011039","name":"ODNI | Intelligence Advanced Research Projects Activity","doi-asserted-by":"publisher","award":["W911NF-16-1-0071"]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2020,12]]},"abstract":"<jats:title>Abstract</jats:title><jats:p>Leakage outside of the qubit computational subspace, present in many leading experimental platforms, constitutes a threatening error for quantum error correction (QEC) for qubits. We develop a leakage-detection scheme via Hidden Markov models (HMMs) for transmon-based implementations of the surface code. By performing realistic density-matrix simulations of the distance-3 surface code (Surface-17), we observe that leakage is sharply projected and leads to an increase in the surface-code defect probability of neighboring stabilizers. Together with the analog readout of the ancilla qubits, this increase enables the accurate detection of the time and location of leakage. We restore the logical error rate below the memory break-even point by post-selecting out leakage, discarding less than half of the data for the given noise parameters. Leakage detection via HMMs opens the prospect for near-term QEC demonstrations, targeted leakage reduction and leakage-aware decoding and is applicable to other experimental platforms.</jats:p>","DOI":"10.1038/s41534-020-00330-w","type":"journal-article","created":{"date-parts":[[2020,12,14]],"date-time":"2020-12-14T11:13:36Z","timestamp":1607944416000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":"Leakage detection for a transmon-based surface code","prefix":"10.1038","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0001-7124-8933","authenticated-orcid":false,"given":"Boris Mihailov","family":"Varbanov","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4800-2518","authenticated-orcid":false,"given":"Francesco","family":"Battistel","sequence":"additional","affiliation":[]},{"given":"Brian Michael","family":"Tarasinski","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-9892-3759","authenticated-orcid":false,"given":"Viacheslav Petrovych","family":"Ostroukh","sequence":"additional","affiliation":[]},{"given":"Thomas Eugene","family":"O’Brien","sequence":"additional","affiliation":[]},{"given":"Leonardo","family":"DiCarlo","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-0218-6614","authenticated-orcid":false,"given":"Barbara Maria","family":"Terhal","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,12,14]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"http://www.nature.com/articles/s41534-020-00330-w.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/s41534-020-00330-w","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/s41534-020-00330-w.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T16:52:59Z","timestamp":1638463979000},"score":1,"resource":{"primary":{"URL":"http://www.nature.com/articles/s41534-020-00330-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12]]},"references-count":55,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,12]]}},"alternative-id":["330"],"URL":"http://dx.doi.org/10.1038/s41534-020-00330-w","relation":{},"ISSN":["2056-6387"],"subject":["Computational Theory and Mathematics","Computer Networks and Communications","Statistical and Nonlinear Physics","Computer Science (miscellaneous)"],"container-title-short":"npj Quantum Inf","published":{"date-parts":[[2020,12]]},"assertion":[{"value":"6 March 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 October 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 December 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"102","id":"doi:10.1038/s41534-020-00330-w","_hash":"40192785a1c8ba60d1a57ef903462648dd3374155d4d987c4da0cd6aa8580913"},"expire":1701130843814},"doi:10.1038/s41534-017-0039-x":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T11:52:40Z","timestamp":1669981960917},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2017,9,25]],"date-time":"2017-09-25T00:00:00Z","timestamp":1506297600000},"content-version":"unspecified","delay-in-days":0,"URL":"http://creativecommons.org/licenses/by/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2017,12]]},"DOI":"10.1038/s41534-017-0039-x","type":"journal-article","created":{"date-parts":[[2017,9,18]],"date-time":"2017-09-18T14:28:20Z","timestamp":1505744900000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":48,"title":"Density-matrix simulation of small surface codes under current and projected experimental noise","prefix":"10.1038","volume":"3","author":[{"given":"T. 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In most state-of-the-art approaches, qubits interact locally, constrained by the connectivity associated with their fixed spatial layout. Here we demonstrate a quantum processor with dynamic, non-local connectivity, in which entangled qubits are coherently transported in a highly parallel manner across two spatial dimensions, between layers of single- and two-qubit operations. Our approach makes use of neutral atom arrays trapped and transported by optical tweezers; hyperfine states are used for robust quantum information storage, and excitation into Rydberg states is used for entanglement generation<jats:sup>3–5</jats:sup>. We use this architecture to realize programmable generation of entangled graph states, such as cluster states and a seven-qubit Steane code state<jats:sup>6,7</jats:sup>. Furthermore, we shuttle entangled ancilla arrays to realize a surface code state with thirteen data and six ancillary qubits<jats:sup>8</jats:sup> and a toric code state on a torus with sixteen data and eight ancillary qubits<jats:sup>9</jats:sup>. Finally, we use this architecture to realize a hybrid analogue–digital evolution<jats:sup>2</jats:sup> and use it for measuring entanglement entropy in quantum simulations<jats:sup>10–12</jats:sup>, experimentally observing non-monotonic entanglement dynamics associated with quantum many-body scars<jats:sup>13,14</jats:sup>. Realizing a long-standing goal, these results provide a route towards scalable quantum processing and enable applications ranging from simulation to metrology.</jats:p>","DOI":"10.1038/s41586-022-04592-6","type":"journal-article","created":{"date-parts":[[2022,4,20]],"date-time":"2022-04-20T16:03:41Z","timestamp":1650470621000},"page":"451-456","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":"A quantum processor based on coherent transport of entangled atom arrays","prefix":"10.1038","volume":"604","author":[{"given":"Dolev","family":"Bluvstein","sequence":"first","affiliation":[]},{"given":"Harry","family":"Levine","sequence":"additional","affiliation":[]},{"given":"Giulia","family":"Semeghini","sequence":"additional","affiliation":[]},{"given":"Tout T.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Sepehr","family":"Ebadi","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-0605-8791","authenticated-orcid":false,"given":"Marcin","family":"Kalinowski","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Keesling","sequence":"additional","affiliation":[]},{"given":"Nishad","family":"Maskara","sequence":"additional","affiliation":[]},{"given":"Hannes","family":"Pichler","sequence":"additional","affiliation":[]},{"given":"Markus","family":"Greiner","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-9786-0538","authenticated-orcid":false,"given":"Vladan","family":"Vuletić","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-8658-1007","authenticated-orcid":false,"given":"Mikhail D.","family":"Lukin","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,20]]},"reference":[],"container-title":"Nature","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41586-022-04592-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41586-022-04592-6","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41586-022-04592-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,21]],"date-time":"2022-04-21T13:21:12Z","timestamp":1650547272000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41586-022-04592-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,20]]},"references-count":75,"journal-issue":{"issue":"7906","published-print":{"date-parts":[[2022,4,21]]}},"alternative-id":["4592"],"URL":"http://dx.doi.org/10.1038/s41586-022-04592-6","relation":{},"ISSN":["0028-0836","1476-4687"],"subject":["Multidisciplinary"],"container-title-short":"Nature","published":{"date-parts":[[2022,4,20]]},"assertion":[{"value":"6 December 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 February 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 April 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"M.G., V.V. and M.D.L. are co-founders and shareholders of QuEra Computing. A.K. is an executive at and shareholder of QuEra Computing. All other authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"id":"doi:10.1038/s41586-022-04592-6","_hash":"797c58ffc8b7e194efc4f248f0ab36f269ac304df683cbddc6189ff28c6f87de"},"expire":1701130856136},"doi:10.1016/j.nuclphysb.2004.07.003":{"value":{"indexed":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T15:55:28Z","timestamp":1669823728488},"reference-count":22,"publisher":"Elsevier BV","issue":"3","license":[{"start":{"date-parts":[[2004,10,1]],"date-time":"2004-10-01T00:00:00Z","timestamp":1096588800000},"content-version":"tdm","delay-in-days":0,"URL":"https://www.elsevier.com/tdm/userlicense/1.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2004,10]]},"DOI":"10.1016/j.nuclphysb.2004.07.003","type":"journal-article","created":{"date-parts":[[2004,8,5]],"date-time":"2004-08-05T11:25:56Z","timestamp":1091705156000},"page":"462-480","source":"Crossref","is-referenced-by-count":38,"title":"Phase structure of the random-plaquette gauge model: accuracy threshold for a toric quantum memory","prefix":"10.1016","volume":"697","author":[{"given":"Takuya","family":"Ohno","sequence":"first","affiliation":[]},{"given":"Gaku","family":"Arakawa","sequence":"additional","affiliation":[]},{"given":"Ikuo","family":"Ichinose","sequence":"additional","affiliation":[]},{"given":"Tetsuo","family":"Matsui","sequence":"additional","affiliation":[]}],"member":"78","reference":[],"container-title":"Nuclear Physics B","original-title":[],"language":"en","link":[{"URL":"https://api.elsevier.com/content/article/PII:S055032130400481X?httpAccept=text/xml","content-type":"text/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https://api.elsevier.com/content/article/PII:S055032130400481X?httpAccept=text/plain","content-type":"text/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2019,2,4]],"date-time":"2019-02-04T22:21:45Z","timestamp":1549318905000},"score":1,"resource":{"primary":{"URL":"https://linkinghub.elsevier.com/retrieve/pii/S055032130400481X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,10]]},"references-count":22,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2004,10]]}},"alternative-id":["S055032130400481X"],"URL":"http://dx.doi.org/10.1016/j.nuclphysb.2004.07.003","relation":{},"ISSN":["0550-3213"],"subject":["Nuclear and High Energy Physics"],"container-title-short":"Nuclear Physics B","published":{"date-parts":[[2004,10]]},"id":"doi:10.1016/j.nuclphysb.2004.07.003","_hash":"1fb53b5c9e17ed07fd57a35577a4faf7c3419bbcb0adfb80fb217aa9d4268af0"},"expire":1701130856771},"doi:10.1103/PhysRevLett.102.200501":{"value":{"indexed":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T18:49:39Z","timestamp":1669315779691},"reference-count":21,"publisher":"American Physical Society (APS)","issue":"20","license":[{"start":{"date-parts":[[2009,5,18]],"date-time":"2009-05-18T00:00:00Z","timestamp":1242604800000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physrevlett.102.200501","type":"journal-article","created":{"date-parts":[[2009,5,18]],"date-time":"2009-05-18T18:24:41Z","timestamp":1242671081000},"source":"Crossref","is-referenced-by-count":77,"title":"Thresholds for Topological Codes in the Presence of Loss","prefix":"10.1103","volume":"102","author":[{"given":"Thomas M.","family":"Stace","sequence":"first","affiliation":[]},{"given":"Sean D.","family":"Barrett","sequence":"additional","affiliation":[]},{"given":"Andrew C.","family":"Doherty","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2009,5,18]]},"reference":[],"container-title":"Physical Review Letters","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevLett.102.200501","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.102.200501/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,18]],"date-time":"2017-06-18T19:59:39Z","timestamp":1497815979000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevLett.102.200501"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,5,18]]},"references-count":21,"journal-issue":{"issue":"20","published-print":{"date-parts":[[2009,5]]}},"URL":"http://dx.doi.org/10.1103/PhysRevLett.102.200501","relation":{},"ISSN":["0031-9007","1079-7114"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. 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Lett.","published":{"date-parts":[[2012,11,2]]},"article-number":"180502","id":"doi:10.1103/PhysRevLett.109.180502","_hash":"4dd839b473c728fa2f8dfe6d7a9cd6b2f6fdb80badf169673834cb48bdc4bcc9"},"expire":1701130868747},"doi:10.1103/PhysRevA.86.032324":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T22:50:14Z","timestamp":1670021414162},"reference-count":50,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2012,9,18]],"date-time":"2012-09-18T00:00:00Z","timestamp":1347926400000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.86.032324","type":"journal-article","created":{"date-parts":[[2012,9,21]],"date-time":"2012-09-21T20:23:30Z","timestamp":1348259010000},"source":"Crossref","is-referenced-by-count":1183,"title":"Surface codes: Towards practical large-scale quantum computation","prefix":"10.1103","volume":"86","author":[{"given":"Austin G.","family":"Fowler","sequence":"first","affiliation":[]},{"given":"Matteo","family":"Mariantoni","sequence":"additional","affiliation":[]},{"given":"John M.","family":"Martinis","sequence":"additional","affiliation":[]},{"given":"Andrew N.","family":"Cleland","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2012,9,18]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.86.032324","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.86.032324/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,7,4]],"date-time":"2019-07-04T04:09:16Z","timestamp":1562213356000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.86.032324"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,9,18]]},"references-count":50,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,9]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.86.032324","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. Rev. A","published":{"date-parts":[[2012,9,18]]},"article-number":"032324","id":"doi:10.1103/PhysRevA.86.032324","_hash":"74ffac12650d924b8b632407ecd96716bedf02e73798868fa56c7374d99ca7b2"},"expire":1701130869812},"doi:10.22331/q-2022-05-24-721":{"value":{"indexed":{"date-parts":[[2022,10,21]],"date-time":"2022-10-21T18:11:42Z","timestamp":1666375902060},"reference-count":50,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T00:00:00Z","timestamp":1653350400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"University College London and the Engineering and Physical Sciences Research Council","award":["EP/L015242/1"]},{"name":"Australian Research Council via the Centre of Excellence in Engineered Quantum Systems","award":["CE170100009"]},{"DOI":"10.13039/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"crossref","award":["EP/R043647/1"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>We build on recent work by B. Brown (Sci. Adv. 6, eaay4929 (2020)) to develop and simulate an explicit recipe for a just-in-time decoding scheme in three 3D surface codes, which can be used to implement a transversal (non-Clifford) <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mover><mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi></mml:mrow><mml:mo accent=\"false\">&amp;#x00AF;</mml:mo></mml:mover></mml:math> between three 2D surface codes in time linear in the code distance. We present a fully detailed set of bounded-height lattice slices through the 3D codes which retain the code distance and measurement-error detecting properties of the full 3D code and admit a dimension-jumping process which expands from/collapses to 2D surface codes supported on the boundaries of each slice. At each timestep of the procedure the slices agree on a common set of overlapping qubits on which <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi></mml:math> should be applied. We use these slices to simulate the performance of a simple JIT decoder against stochastic <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi></mml:math> and measurement errors and find evidence for a threshold <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>p</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>&amp;#x223C;</mml:mo><mml:mn>0.1</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> in all three codes. We expect that this threshold could be improved by optimisation of the decoder.</jats:p>","DOI":"10.22331/q-2022-05-24-721","type":"journal-article","created":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T12:33:35Z","timestamp":1653395615000},"page":"721","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":"Numerical Implementation of Just-In-Time Decoding in Novel Lattice Slices Through the Three-Dimensional Surface Code","prefix":"10.22331","volume":"6","author":[{"given":"T. R.","family":"Scruby","sequence":"first","affiliation":[{"name":"Okinawa Institute of Science and Technology, Okinawa, 904-0495, Japan"},{"name":"Dept. of Physics and Astronomy, University College London, London, WC1E 6BT, UK"}]},{"given":"D. E.","family":"Browne","sequence":"additional","affiliation":[{"name":"Dept. of Physics and Astronomy, University College London, London, WC1E 6BT, UK"}]},{"given":"P.","family":"Webster","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia"}]},{"given":"M.","family":"Vasmer","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada"},{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada"}]}],"member":"9598","published-online":{"date-parts":[[2022,5,24]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-05-24-721/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T12:33:45Z","timestamp":1653395625000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-05-24-721/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,24]]},"references-count":50,"URL":"http://dx.doi.org/10.22331/q-2022-05-24-721","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,5,24]]},"article-number":"721","id":"doi:10.22331/q-2022-05-24-721","_hash":"e8cecb316100746bebb54ca43a1cff72dcf29b65b8ac192e5c1094f6743b88af"},"expire":1701130870972},"doi:10.1126/sciadv.aay4929":{"value":{"indexed":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T08:31:49Z","timestamp":1669797109931},"reference-count":41,"publisher":"American Association for the Advancement of Science (AAAS)","issue":"21","funder":[{"name":"Australian Research Council via the Centre of Excellence in Engineered Quantum System","award":["CE170100009"]},{"name":"University of Sydney Fellowship Programme"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2020,5,22]]},"abstract":"<jats:p>We complete a universal set of fault-tolerant quantum logic gates for a two-dimensional surface code architecture.</jats:p>","DOI":"10.1126/sciadv.aay4929","type":"journal-article","created":{"date-parts":[[2020,5,22]],"date-time":"2020-05-22T23:08:48Z","timestamp":1590188928000},"source":"Crossref","is-referenced-by-count":27,"title":"A fault-tolerant non-Clifford gate for the surface code in two dimensions","prefix":"10.1126","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia."}]}],"member":"221","reference":[],"container-title":"Science Advances","original-title":[],"language":"en","link":[{"URL":"https://syndication.highwire.org/content/doi/10.1126/sciadv.aay4929","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,14]],"date-time":"2022-01-14T01:21:12Z","timestamp":1642123272000},"score":1,"resource":{"primary":{"URL":"https://www.science.org/doi/10.1126/sciadv.aay4929"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,22]]},"references-count":41,"journal-issue":{"issue":"21","published-print":{"date-parts":[[2020,5,22]]}},"alternative-id":["10.1126/sciadv.aay4929"],"URL":"http://dx.doi.org/10.1126/sciadv.aay4929","relation":{},"ISSN":["2375-2548"],"subject":["Multidisciplinary"],"container-title-short":"Sci. Adv.","published":{"date-parts":[[2020,5,22]]},"id":"doi:10.1126/sciadv.aay4929","_hash":"e953ac20ee6cc8a14d5fbae4d4c6914fe5e1903efe7675bcbd5e2322118b04c4"},"expire":1701130871850},"doi:10.1103/PhysRevA.94.042316":{"value":{"indexed":{"date-parts":[[2022,10,26]],"date-time":"2022-10-26T11:31:25Z","timestamp":1666783885119},"reference-count":40,"publisher":"American Physical Society (APS)","issue":"4","license":[{"start":{"date-parts":[[2016,10,12]],"date-time":"2016-10-12T00:00:00Z","timestamp":1476230400000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"},{"start":{"date-parts":[[2017,10,12]],"date-time":"2017-10-12T00:00:00Z","timestamp":1507766400000},"content-version":"am","delay-in-days":365,"URL":"http://link.aps.org/licenses/aps-default-accepted-manuscript-license"}],"funder":[{"DOI":"10.13039/100006234","name":"Sandia National Laboratories","doi-asserted-by":"publisher"},{"DOI":"10.13039/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-AC04-94AL85000"]},{"DOI":"10.13039/100006168","name":"National Nuclear Security Administration","doi-asserted-by":"publisher"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.94.042316","type":"journal-article","created":{"date-parts":[[2016,10,12]],"date-time":"2016-10-12T22:08:25Z","timestamp":1476310105000},"source":"Crossref","is-referenced-by-count":11,"title":"Transversal Clifford gates on folded surface codes","prefix":"10.1103","volume":"94","author":[{"given":"Jonathan E.","family":"Moussa","sequence":"first","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2016,10,12]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/accepted/10.1103/PhysRevA.94.042316","content-type":"application/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"http://link.aps.org/article/10.1103/PhysRevA.94.042316","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.94.042316/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,4,5]],"date-time":"2017-04-05T17:27:46Z","timestamp":1491413266000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.94.042316"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,10,12]]},"references-count":40,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2016,10]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.94.042316","relation":{},"ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","published":{"date-parts":[[2016,10,12]]},"article-number":"042316","id":"doi:10.1103/PhysRevA.94.042316","_hash":"a3fd2d44dc9830e22fa1f732603c0ab7f0a0aae1d1f39947528394c57476d102"},"expire":1701130872818},"doi:10.22331/q-2021-12-02-595":{"value":{"indexed":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T15:36:34Z","timestamp":1669822594294},"reference-count":68,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T00:00:00Z","timestamp":1638403200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>In order to build a large scale quantum computer, one must be able to correct errors extremely fast. We design a fast decoding algorithm for topological codes to correct for Pauli errors and erasure and combination of both errors and erasure. Our algorithm has a worst case complexity of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math>, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math> is the number of physical qubits and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>α</mml:mi></mml:math> is the inverse of Ackermann's function, which is very slowly growing. For all practical purposes, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≤</mml:mo><mml:mn>3</mml:mn></mml:math>. We prove that our algorithm performs optimally for errors of weight up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>d</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:math> and for loss of up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math> qubits, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> is the minimum distance of the code. Numerically, we obtain a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>9.9</mml:mn><mml:mi mathvariant=\"normal\">%</mml:mi></mml:math> for the 2d-toric code with perfect syndrome measurements and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2.6</mml:mn><mml:mi mathvariant=\"normal\">%</mml:mi></mml:math> with faulty measurements.</jats:p>","DOI":"10.22331/q-2021-12-02-595","type":"journal-article","created":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T16:07:00Z","timestamp":1638461220000},"page":"595","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":14,"title":"Almost-linear time decoding algorithm for topological codes","prefix":"10.22331","volume":"5","author":[{"given":"Nicolas","family":"Delfosse","sequence":"first","affiliation":[{"name":"IQIM, California Institute of Technology, Pasadena, CA, USA"},{"name":"Department of Physics and Astronomy, University of California, Riverside, CA, USA"},{"name":"Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Naomi H.","family":"Nickerson","sequence":"additional","affiliation":[{"name":"Quantum Optics and Laser Science, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom"}]}],"member":"9598","published-online":{"date-parts":[[2021,12,2]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2021-12-02-595/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T16:07:08Z","timestamp":1638461228000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2021-12-02-595/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,2]]},"references-count":68,"URL":"http://dx.doi.org/10.22331/q-2021-12-02-595","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2021,12,2]]},"article-number":"595","id":"doi:10.22331/q-2021-12-02-595","_hash":"05711d600aa93726e31fefce01d82984d2c9f7f2ae49d049c5b1470a3c1cf2ae"},"expire":1701130873816},"doi:10.1088/2632-2153/abc609":{"value":{"indexed":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T19:20:55Z","timestamp":1669317655037},"reference-count":69,"publisher":"IOP Publishing","issue":"2","license":[{"start":{"date-parts":[[2020,12,28]],"date-time":"2020-12-28T00:00:00Z","timestamp":1609113600000},"content-version":"vor","delay-in-days":0,"URL":"http://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2020,12,28]],"date-time":"2020-12-28T00:00:00Z","timestamp":1609113600000},"content-version":"tdm","delay-in-days":0,"URL":"https://iopscience.iop.org/info/page/text-and-data-mining"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2021,6,1]]},"abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Topological error correcting codes, and particularly the surface code, currently provide the most feasible road-map towards large-scale fault-tolerant quantum computation. As such, obtaining fast and flexible decoding algorithms for these codes, within the experimentally realistic and challenging context of faulty syndrome measurements, without requiring any final read-out of the physical qubits, is of critical importance. In this work, we show that the problem of decoding such codes can be naturally reformulated as a process of repeated interactions between a decoding agent and a code environment, to which the machinery of reinforcement learning can be applied to obtain decoding agents. 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In this paper, we discuss strategies for surface-code quantum computing on small, intermediate and large scales. They are strategies for space-time trade-offs, going from slow computations using few qubits to fast computations using many qubits. Our schemes are based on surface-code patches, which not only feature a low space cost compared to other surface-code schemes, but are also conceptually simple~--~simple enough that they can be described as a tile-based game with a small set of rules. 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Bravyi et al. (2006) showed that encoding a state in the surface code using local unitary operations requires time at least linear in the lattice size <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>L</mml:mi></mml:math>, however the most efficient known method for encoding an unknown state, introduced by Dennis et al. (2002), has <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math> time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>L</mml:mi></mml:math> time steps to encode an unknown state in a distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>L</mml:mi></mml:math> planar code. We further show how an <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> complexity local unitary encoder for the toric code can be found by enforcing locality in the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math>-depth non-local renormalisation encoder. We relate these techniques by providing an <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> local unitary circuit to convert between a toric code and a planar code, and also provide optimal encoders for the rectangular, rotated and 3D surface codes. Furthermore, we show how our encoding circuit for the planar code can be used to prepare fermionic states in the compact mapping, a recently introduced fermion to qubit mapping that has a stabiliser structure similar to that of the surface code and is particularly efficient for simulating the Fermi-Hubbard model.</jats:p>","DOI":"10.22331/q-2021-08-05-517","type":"journal-article","created":{"date-parts":[[2021,8,5]],"date-time":"2021-08-05T19:19:03Z","timestamp":1628191143000},"page":"517","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":"Optimal local unitary encoding circuits for the surface code","prefix":"10.22331","volume":"5","author":[{"given":"Oscar","family":"Higgott","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"}]},{"given":"Matthew","family":"Wilson","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"},{"name":"Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom"}]},{"given":"James","family":"Hefford","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"},{"name":"Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom"}]},{"given":"James","family":"Dborin","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"},{"name":"London Centre for Nanotechnology, University College London, Gordon St., London WC1H 0AH, United Kingdom"}]},{"given":"Farhan","family":"Hanif","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"}]},{"given":"Simon","family":"Burton","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"}]},{"given":"Dan E.","family":"Browne","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"}]}],"member":"9598","published-online":{"date-parts":[[2021,8,5]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2021-08-05-517/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2021,8,19]],"date-time":"2021-08-19T16:13:34Z","timestamp":1629389614000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2021-08-05-517/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,5]]},"references-count":57,"URL":"http://dx.doi.org/10.22331/q-2021-08-05-517","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2021,8,5]]},"article-number":"517","id":"doi:10.22331/q-2021-08-05-517","_hash":"e7fa6105f83455c0425ea135567b797934e877954c633e8d2e61cffc5a39c3d6"},"expire":1701130895831},"doi:10.1103/PhysRevLett.104.050503":{"value":{"indexed":{"date-parts":[[2022,10,26]],"date-time":"2022-10-26T11:31:09Z","timestamp":1666783869708},"reference-count":20,"publisher":"American Physical Society (APS)","issue":"5","license":[{"start":{"date-parts":[[2010,2,5]],"date-time":"2010-02-05T00:00:00Z","timestamp":1265328000000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physrevlett.104.050503","type":"journal-article","created":{"date-parts":[[2010,3,19]],"date-time":"2010-03-19T17:17:32Z","timestamp":1269019052000},"source":"Crossref","is-referenced-by-count":79,"title":"Tradeoffs for Reliable Quantum Information Storage in 2D Systems","prefix":"10.1103","volume":"104","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]},{"given":"Barbara","family":"Terhal","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2010,2,5]]},"reference":[],"container-title":"Physical Review 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Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In particular, we show that for the toric code subject to such independent coherent noise, and for minimal-weight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and fault-tolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is held constant as the code block grows, and we recount the difficulties that prevented us from extending the result to that case. Nevertheless our work supports the idea that fault-tolerant quantum computing schemes will work effectively against coherent noise, providing encouraging news for quantum hardware builders who worry about the damaging effects of control errors and coherent interactions with the environment.</jats:p>","DOI":"10.1088/1367-2630/ab8e5c","type":"journal-article","created":{"date-parts":[[2020,4,29]],"date-time":"2020-04-29T22:24:15Z","timestamp":1588199055000},"page":"073066","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":13,"title":"Coherence in logical quantum channels","prefix":"10.1088","volume":"22","author":[{"ORCID":"http://orcid.org/0000-0003-4665-8839","authenticated-orcid":false,"given":"Joseph K","family":"Iverson","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-2421-4762","authenticated-orcid":false,"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2020,7,31]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c","content-type":"text/html","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,27]],"date-time":"2021-11-27T16:44:20Z","timestamp":1638031460000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab8e5c"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,1]]},"references-count":37,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7,31]]},"published-print":{"date-parts":[[2020,7,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/ab8e5c","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. 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Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft","name":"copyright_information","label":"Copyright Information"},{"value":"2020-02-22","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-04-29","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-07-31","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/ab8e5c","_hash":"d7cba2a02c85835b48c99a33621b6d8e5a5a5255521194295cd3eaec662d4b5a"},"expire":1701130898842},"doi:10.1103/PhysRevX.10.011022":{"value":{"indexed":{"date-parts":[[2022,11,26]],"date-time":"2022-11-26T05:28:26Z","timestamp":1669440506295},"reference-count":43,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[2020,1,31]],"date-time":"2020-01-31T00:00:00Z","timestamp":1580428800000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevx.10.011022","type":"journal-article","created":{"date-parts":[[2020,1,31]],"date-time":"2020-01-31T15:54:25Z","timestamp":1580486065000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":63,"title":"Topological and Subsystem Codes on Low-Degree Graphs with Flag Qubits","prefix":"10.1103","volume":"10","author":[{"ORCID":"http://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-3375-4445","authenticated-orcid":true,"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-9614-2836","authenticated-orcid":true,"given":"Theodore J.","family":"Yoder","sequence":"additional","affiliation":[]},{"given":"Jared B.","family":"Hertzberg","sequence":"additional","affiliation":[]},{"given":"Andrew W.","family":"Cross","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2020,1,31]]},"reference":[],"container-title":"Physical Review X","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevX.10.011022","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevX.10.011022/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,31]],"date-time":"2020-01-31T15:55:37Z","timestamp":1580486137000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevX.10.011022"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,31]]},"references-count":43,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,1]]}},"URL":"http://dx.doi.org/10.1103/PhysRevX.10.011022","relation":{},"ISSN":["2160-3308"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. 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Compared to independent depolarizing noise, leaked qubits may produce many more configurations of harmful correlated errors during error-correction. In this work, we investigate different local codes in the low-error regime of a leakage gate error model. When restricting to bare-ancilla extraction, we observe that subsystem codes are good candidates for handling leakage, as their locality can limit damaging correlated errors. As a case study, we compare subspace surface codes to the subsystem surface codes introduced by Bravyi <jats:italic>et al</jats:italic>. In contrast to depolarizing noise, subsystem surface codes outperform same-distance subspace surface codes below error rates as high as ⪅ 7.5 × 10<jats:sup>−4</jats:sup> while offering better per-qubit distance protection. Furthermore, we show that at low to intermediate distances, Bacon–Shor codes offer better per-qubit error protection against leakage in an ion-trap motivated error model below error rates as high as ⪅ 1.2 × 10<jats:sup>−3</jats:sup>. For restricted leakage models, this advantage can be extended to higher distances by relaxing to unverified two-qubit cat state extraction in the surface code. These results highlight an intrinsic benefit of subsystem code locality to error-corrective performance.</jats:p>","DOI":"10.1088/1367-2630/ab3372","type":"journal-article","created":{"date-parts":[[2019,7,18]],"date-time":"2019-07-18T17:46:25Z","timestamp":1563471985000},"page":"073055","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":15,"title":"Handling leakage with subsystem codes","prefix":"10.1088","volume":"21","author":[{"given":"Natalie C","family":"Brown","sequence":"first","affiliation":[]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[]},{"given":"Kenneth R","family":"Brown","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2019,7,30]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"link":[{"URL":"http://iopscience.iop.org/article/10.1088/1367-2630/ab3372","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"http://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372","content-type":"text/html","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,18]],"date-time":"2021-11-18T22:38:02Z","timestamp":1637275082000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab3372"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,1]]},"references-count":102,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2019,7,30]]},"published-print":{"date-parts":[[2019,7,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/ab3372","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. 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Attempts to process and manipulate quantum states can destroy the encoded information. Nigg\n            <jats:italic>et al.</jats:italic>\n            encoded the quantum state of a single qubit (in this case, a trapped ion) over the global properties of a series of trapped ions. These so-called stabilizers protected the information against noise sources that can degrade the single qubit. The protocol provides a route to fault-tolerant quantum computing.\n          </jats:p>\n          <jats:p>\n            <jats:italic>Science</jats:italic>\n            , this issue p.\n            <jats:related-article xmlns:xlink=\"http://www.w3.org/1999/xlink\" ext-link-type=\"doi\" issue=\"6194\" page=\"302\" related-article-type=\"in-this-issue\" vol=\"345\" xlink:href=\"10.1126/science.1253742\">302</jats:related-article>\n          </jats:p>","DOI":"10.1126/science.1253742","type":"journal-article","created":{"date-parts":[[2014,6,13]],"date-time":"2014-06-13T07:43:13Z","timestamp":1402645393000},"page":"302-305","source":"Crossref","is-referenced-by-count":255,"title":"Quantum computations on a topologically encoded qubit","prefix":"10.1126","volume":"345","author":[{"given":"D.","family":"Nigg","sequence":"first","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."}]},{"given":"M.","family":"Müller","sequence":"additional","affiliation":[{"name":"Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain."}]},{"given":"E. A.","family":"Martinez","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."}]},{"given":"P.","family":"Schindler","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."}]},{"given":"M.","family":"Hennrich","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."}]},{"given":"T.","family":"Monz","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."}]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[{"name":"Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain."}]},{"given":"R.","family":"Blatt","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."},{"name":"Institut für Quantenoptik und Quanteninformation der Österreichischen Akademie der Wissenschaften, A-6020 Innsbruck, Austria."}]}],"member":"221","reference":[],"container-title":"Science","original-title":[],"language":"en","link":[{"URL":"https://syndication.highwire.org/content/doi/10.1126/science.1253742","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,14]],"date-time":"2022-01-14T08:38:13Z","timestamp":1642149493000},"score":1,"resource":{"primary":{"URL":"https://www.science.org/doi/10.1126/science.1253742"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,7,18]]},"references-count":28,"journal-issue":{"issue":"6194","published-print":{"date-parts":[[2014,7,18]]}},"alternative-id":["10.1126/science.1253742"],"URL":"http://dx.doi.org/10.1126/science.1253742","relation":{},"ISSN":["0036-8075","1095-9203"],"subject":["Multidisciplinary"],"container-title-short":"Science","published":{"date-parts":[[2014,7,18]]},"id":"doi:10.1126/science.1253742","_hash":"3d42a5648eb362e7ec14a2d66488f0a194a4bfb508fbffb7bd9c94aeef20eb2e"},"expire":1701130939923},"doi:10.1103/PhysRevLett.121.050502":{"value":{"indexed":{"date-parts":[[2022,12,1]],"date-time":"2022-12-01T18:26:05Z","timestamp":1669919165484},"reference-count":24,"publisher":"American Physical Society (APS)","issue":"5","license":[{"start":{"date-parts":[[2018,8,1]],"date-time":"2018-08-01T00:00:00Z","timestamp":1533081600000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"},{"start":{"date-parts":[[2019,8,1]],"date-time":"2019-08-01T00:00:00Z","timestamp":1564617600000},"content-version":"am","delay-in-days":365,"URL":"https://link.aps.org/licenses/aps-default-accepted-manuscript-license"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1254119"]},{"DOI":"10.13039/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["W911NF-12-1-0541"]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevlett.121.050502","type":"journal-article","created":{"date-parts":[[2018,8,1]],"date-time":"2018-08-01T14:01:31Z","timestamp":1533132091000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":78,"title":"Quantum Error Correction with Only Two Extra Qubits","prefix":"10.1103","volume":"121","author":[{"given":"Rui","family":"Chao","sequence":"first","affiliation":[]},{"given":"Ben W.","family":"Reichardt","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2018,8,1]]},"reference":[],"container-title":"Physical Review Letters","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/accepted/10.1103/PhysRevLett.121.050502","content-type":"application/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"https://link.aps.org/article/10.1103/PhysRevLett.121.050502","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.121.050502/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,28]],"date-time":"2022-08-28T16:00:00Z","timestamp":1661702400000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevLett.121.050502"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,8,1]]},"references-count":24,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2018,8]]}},"URL":"http://dx.doi.org/10.1103/PhysRevLett.121.050502","relation":{},"ISSN":["0031-9007","1079-7114"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. Rev. Lett.","published":{"date-parts":[[2018,8,1]]},"article-number":"050502","id":"doi:10.1103/PhysRevLett.121.050502","_hash":"88832226f861bba82985d4694b4abf513e1dbfa2102a8c91ed97549812e0a806"},"expire":1701130940940},"doi:10.22331/q-2017-04-25-2":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T12:21:28Z","timestamp":1669983688752},"reference-count":45,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2017,4,25]],"date-time":"2017-04-25T00:00:00Z","timestamp":1493078400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"<jats:p>The surface code is one of the most successful approaches to topological quantum error-correction. It boasts the smallest known syndrome extraction circuits and correspondingly largest thresholds. Defect-based logical encodings of a new variety called twists have made it possible to implement the full Clifford group without state distillation. Here we investigate a patch-based encoding involving a modified twist. In our modified formulation, the resulting codes, called triangle codes for the shape of their planar layout, have only weight-four checks and relatively simple syndrome extraction circuits that maintain a high, near surface-code-level threshold. They also use 25% fewer physical qubits per logical qubit than the surface code. Moreover, benefiting from the twist, we can implement all Clifford gates by lattice surgery without the need for state distillation. By a surgical transformation to the surface code, we also develop a scheme of doing all Clifford gates on surface code patches in an atypical planar layout, though with less qubit efficiency than the triangle code. Finally, we remark that logical qubits encoded in triangle codes are naturally amenable to logical tomography, and the smallest triangle code can demonstrate high-pseudothreshold fault-tolerance to depolarizing noise using just 13 physical qubits.</jats:p>","DOI":"10.22331/q-2017-04-25-2","type":"journal-article","created":{"date-parts":[[2017,4,25]],"date-time":"2017-04-25T08:44:49Z","timestamp":1493109889000},"page":"2","source":"Crossref","is-referenced-by-count":57,"title":"The surface code with a twist","prefix":"10.22331","volume":"1","author":[{"ORCID":"http://orcid.org/0000-0001-9614-2836","authenticated-orcid":false,"given":"Theodore J.","family":"Yoder","sequence":"first","affiliation":[{"name":"Department of Physics, Massachusetts Institute of Technology"}]},{"given":"Isaac H.","family":"Kim","sequence":"additional","affiliation":[{"name":"IBM, Thomas J. Watson Research Center"}]}],"member":"9598","published-online":{"date-parts":[[2017,4,25]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","deposited":{"date-parts":[[2017,6,7]],"date-time":"2017-06-07T12:30:04Z","timestamp":1496838604000},"score":1,"resource":{"primary":{"URL":"http://quantum-journal.org/papers/q-2017-04-25-2/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,25]]},"references-count":45,"URL":"http://dx.doi.org/10.22331/q-2017-04-25-2","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2017,4,25]]},"article-number":"2","id":"doi:10.22331/q-2017-04-25-2","_hash":"7194ed8f454d3aa458d6dc260ccd4d7b836910a84e12f13b8fc08da46921d1ce"},"expire":1701130941937},"doi:10.1103/PhysRevX.6.031039":{"value":{"indexed":{"date-parts":[[2022,7,8]],"date-time":"2022-07-08T11:11:59Z","timestamp":1657278719949},"reference-count":44,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2016,9,13]],"date-time":"2016-09-13T00:00:00Z","timestamp":1473724800000},"content-version":"vor","delay-in-days":0,"URL":"http://creativecommons.org/licenses/by/3.0/"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1111337"]},{"DOI":"10.13039/100000183","name":"Army Research Office","doi-asserted-by":"publisher"},{"DOI":"10.13039/100014037","name":"National Defense Science and Engineering Graduate","doi-asserted-by":"crossref"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physrevx.6.031039","type":"journal-article","created":{"date-parts":[[2016,9,13]],"date-time":"2016-09-13T22:08:34Z","timestamp":1473804514000},"source":"Crossref","is-referenced-by-count":33,"title":"Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes","prefix":"10.1103","volume":"6","author":[{"given":"Theodore J.","family":"Yoder","sequence":"first","affiliation":[]},{"given":"Ryuji","family":"Takagi","sequence":"additional","affiliation":[]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2016,9,13]]},"reference":[],"container-title":"Physical Review X","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevX.6.031039","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevX.6.031039/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,7,8]],"date-time":"2022-07-08T10:44:03Z","timestamp":1657277043000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevX.6.031039"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,9,13]]},"references-count":44,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2016,9]]}},"URL":"http://dx.doi.org/10.1103/PhysRevX.6.031039","relation":{},"ISSN":["2160-3308"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. Rev. X","published":{"date-parts":[[2016,9,13]]},"article-number":"031039","id":"doi:10.1103/PhysRevX.6.031039","_hash":"66678c69609fa6dd75baebe3ea8409918169e03a642fdc2f3d016847034678d9"},"expire":1701130943023},"doi:10.1038/s41586-022-04819-6":{"value":{"indexed":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T08:31:41Z","timestamp":1669797101488},"reference-count":52,"publisher":"Springer Science and Business Media LLC","issue":"7916","license":[{"start":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T00:00:00Z","timestamp":1651708800000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T00:00:00Z","timestamp":1651708800000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2022,6,30]]},"abstract":"<jats:title>Abstract</jats:title><jats:p>Solid-state spin qubits is a promising platform for quantum computation and quantum networks<jats:sup>1,2</jats:sup>. Recent experiments have demonstrated high-quality control over multi-qubit systems<jats:sup>3–8</jats:sup>, elementary quantum algorithms<jats:sup>8–11</jats:sup> and non-fault-tolerant error correction<jats:sup>12–14</jats:sup>. Large-scale systems will require using error-corrected logical qubits that are operated fault tolerantly, so that reliable computation becomes possible despite noisy operations<jats:sup>15–18</jats:sup>. Overcoming imperfections in this way remains an important outstanding challenge for quantum science<jats:sup>15,19–27</jats:sup>. Here, we demonstrate fault-tolerant operations on a logical qubit using spin qubits in diamond. Our approach is based on the five-qubit code with a recently discovered flag protocol that enables fault tolerance using a total of seven qubits<jats:sup>28–30</jats:sup>. We encode the logical qubit using a new protocol based on repeated multi-qubit measurements and show that it outperforms non-fault-tolerant encoding schemes. We then fault-tolerantly manipulate the logical qubit through a complete set of single-qubit Clifford gates. Finally, we demonstrate flagged stabilizer measurements with real-time processing of the outcomes. Such measurements are a primitive for fault-tolerant quantum error correction. Although future improvements in fidelity and the number of qubits will be required to suppress logical error rates below the physical error rates, our realization of fault-tolerant protocols on the logical-qubit level is a key step towards quantum information processing based on solid-state spins.</jats:p>","DOI":"10.1038/s41586-022-04819-6","type":"journal-article","created":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T16:08:44Z","timestamp":1651766924000},"page":"884-889","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":"Fault-tolerant operation of a logical qubit in a diamond quantum processor","prefix":"10.1038","volume":"606","author":[{"ORCID":"http://orcid.org/0000-0001-8205-8166","authenticated-orcid":false,"given":"M. H.","family":"Abobeih","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-5483-0339","authenticated-orcid":false,"given":"Y.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"J.","family":"Randall","sequence":"additional","affiliation":[]},{"given":"S. J. H.","family":"Loenen","sequence":"additional","affiliation":[]},{"given":"C. E.","family":"Bradley","sequence":"additional","affiliation":[]},{"given":"M.","family":"Markham","sequence":"additional","affiliation":[]},{"given":"D. J.","family":"Twitchen","sequence":"additional","affiliation":[]},{"given":"B. M.","family":"Terhal","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-2355-727X","authenticated-orcid":false,"given":"T. H.","family":"Taminiau","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,5,5]]},"reference":[],"container-title":"Nature","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41586-022-04819-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41586-022-04819-6","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41586-022-04819-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,29]],"date-time":"2022-06-29T16:14:20Z","timestamp":1656519260000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41586-022-04819-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,5]]},"references-count":52,"journal-issue":{"issue":"7916","published-print":{"date-parts":[[2022,6,30]]}},"alternative-id":["4819"],"URL":"http://dx.doi.org/10.1038/s41586-022-04819-6","relation":{},"ISSN":["0028-0836","1476-4687"],"subject":["Multidisciplinary"],"container-title-short":"Nature","published":{"date-parts":[[2022,5,5]]},"assertion":[{"value":"10 August 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 April 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 May 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"id":"doi:10.1038/s41586-022-04819-6","_hash":"feb8f5e8b0491c6decae984a614e15edb37fb7eb77324f14d77503509c701387"},"expire":1701130943946},"doi:10.1093/nsr/nwab011":{"value":{"indexed":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T10:27:20Z","timestamp":1669372040640},"reference-count":31,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T00:00:00Z","timestamp":1611187200000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["21931001","21871121","21971097"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2022,1,19]]},"abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error-correcting code, with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state and state decoding. To address this challenge, we experimentally realise the [5, 1, 3] code, the so-called smallest perfect code that permits corrections of generic single-qubit errors. In the experiment, having optimised the encoding circuit, we employ an array of superconducting qubits to realise the [5, 1, 3] code for several typical logical states including the magic state, an indispensable resource for realising non-Clifford gates. The encoded states are prepared with an average fidelity of $57.1(3)\\%$ while with a high fidelity of $98.6(1)\\%$ in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilisers. We further implement logical Pauli operations with a fidelity of $97.2(2)\\%$ within the code space. Finally, we realise the decoding circuit and recover the input state with an overall fidelity of $74.5(6)\\%$, in total with 92 gates. Our work demonstrates each key aspect of the [5, 1, 3] code and verifies the viability of experimental realisation of quantum error-correcting codes with superconducting qubits.</jats:p>","DOI":"10.1093/nsr/nwab011","type":"journal-article","created":{"date-parts":[[2021,1,15]],"date-time":"2021-01-15T12:20:13Z","timestamp":1610713213000},"source":"Crossref","is-referenced-by-count":12,"title":"Experimental exploration of five-qubit quantum error-correcting code with superconducting qubits","prefix":"10.1093","volume":"9","author":[{"given":"Ming","family":"Gong","sequence":"first","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Xiao","family":"Yuan","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University , Beijing 100084, China"},{"name":"Department of Materials, University of Oxford , Oxford OX1 3PH, UK"}]},{"given":"Shiyu","family":"Wang","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Yulin","family":"Wu","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Youwei","family":"Zhao","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Chen","family":"Zha","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Shaowei","family":"Li","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Zhen","family":"Zhang","sequence":"additional","affiliation":[{"name":"Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University , Beijing 100084, China"}]},{"given":"Qi","family":"Zhao","sequence":"additional","affiliation":[{"name":"Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University , Beijing 100084, China"}]},{"given":"Yunchao","family":"Liu","sequence":"additional","affiliation":[{"name":"Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University , Beijing 100084, China"}]},{"given":"Futian","family":"Liang","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Jin","family":"Lin","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum 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Many applications call for error rates as low as 10<jats:sup>−15</jats:sup> (refs. <jats:sup>2–9</jats:sup>), but state-of-the-art quantum platforms typically have physical error rates near 10<jats:sup>−3</jats:sup> (refs. <jats:sup>10–14</jats:sup>). Quantum error correction<jats:sup>15–17</jats:sup> promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device<jats:sup>18,19</jats:sup> and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. 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This error has been corrected online.","order":6,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing financial interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"id":"doi:10.1038/nature23460","_hash":"f91184c126fccbc0c2acff16893e1ad33fb8840d103e2aa9cacef7013dd42755"},"expire":1701130964068},"doi:10.1103/PhysRevA.71.022316":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T22:32:45Z","timestamp":1670020365199},"reference-count":21,"publisher":"American Physical Society (APS)","issue":"2","license":[{"start":{"date-parts":[[2005,2,22]],"date-time":"2005-02-22T00:00:00Z","timestamp":1109030400000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.71.022316","type":"journal-article","created":{"date-parts":[[2005,2,23]],"date-time":"2005-02-23T00:08:13Z","timestamp":1109117293000},"source":"Crossref","is-referenced-by-count":660,"title":"Universal quantum computation with ideal Clifford gates and noisy ancillas","prefix":"10.1103","volume":"71","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Alexei","family":"Kitaev","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2005,2,22]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.71.022316","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.71.022316/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,16]],"date-time":"2017-06-16T16:09:36Z","timestamp":1497629376000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.71.022316"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,2,22]]},"references-count":21,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2005,2]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.71.022316","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. Rev. A","published":{"date-parts":[[2005,2,22]]},"article-number":"022316","id":"doi:10.1103/PhysRevA.71.022316","_hash":"3a11db61d225b7ff0135dacc040de04605323391ead05bfc15a0c9df00fab4ad"},"expire":1701130964947},"doi:10.22331/q-2017-10-03-31":{"value":{"indexed":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T19:15:48Z","timestamp":1669317348147},"reference-count":53,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2017,10,3]],"date-time":"2017-10-03T00:00:00Z","timestamp":1506988800000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"<jats:p>We present an infinite family of protocols to distill magic states for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>-gates that has a low space overhead and uses an asymptotic number of input magic states to achieve a given target error that is conjectured to be optimal. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>-gate depth of the circuit scale linearly in the logarithm of the target error <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>δ</mml:mi></mml:math> (up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>δ</mml:mi></mml:math>). Unlike other distillation protocols, this protocol achieves this performance without concatenation and the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. The protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy, with the same asymptotic performance. The protocol relies on the construction of weakly self-dual CSS codes with many logical qubits and large distance, allowing us to implement control-SWAPs on multiple qubits. We call this code the \"inner code\". The control-SWAPs are then used to measure properties of the magic state and detect errors, using another code that we call the \"outer code\". Alternatively, we use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. We present several specific small examples of this protocol.</jats:p>","DOI":"10.22331/q-2017-10-03-31","type":"journal-article","created":{"date-parts":[[2017,10,3]],"date-time":"2017-10-03T11:34:08Z","timestamp":1507030448000},"page":"31","source":"Crossref","is-referenced-by-count":38,"title":"Magic state distillation with low space overhead and optimal asymptotic input count","prefix":"10.22331","volume":"1","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Quantum Architectures and Computation, Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"name":"Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA"},{"name":"Quantum Architectures and Computation, Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"D.","family":"Poulin","sequence":"additional","affiliation":[{"name":"Département de Physique & Institut Quantique, Université de Sherbrooke, Quebec, Canada"}]},{"given":"D.","family":"Wecker","sequence":"additional","affiliation":[{"name":"Quantum Architectures and Computation, Microsoft Research, Redmond, WA 98052, USA"}]}],"member":"9598","published-online":{"date-parts":[[2017,10,3]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","deposited":{"date-parts":[[2018,1,16]],"date-time":"2018-01-16T15:52:13Z","timestamp":1516117933000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2017-10-03-31/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,3]]},"references-count":53,"URL":"http://dx.doi.org/10.22331/q-2017-10-03-31","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2017,10,3]]},"article-number":"31","id":"doi:10.22331/q-2017-10-03-31","_hash":"941e3f690d13015cb506b1a17e4a6ab862fe353878a92edb40cc35fe275e9d59"},"expire":1701130966073},"doi:10.1364/OPTICA.439170":{"value":{"indexed":{"date-parts":[[2022,11,14]],"date-time":"2022-11-14T18:29:16Z","timestamp":1668450556202},"reference-count":35,"publisher":"Optica Publishing Group","issue":"2","license":[{"start":{"date-parts":[[2022,2,2]],"date-time":"2022-02-02T00:00:00Z","timestamp":1643760000000},"content-version":"vor","delay-in-days":0,"URL":"https://doi.org/10.1364/OA_License_v1#VOR-OA"}],"funder":[{"name":"National Key Research and Development (R&D) Plan of China","award":["2018YFA0306501"]},{"DOI":"10.13039/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11425417","12104444","11975222","U1738140"]},{"name":"Shanghai Municipal Science and Technology Major Project","award":["2019SHZDZX01"]},{"name":"Anhui Initiative in Quantum Information Technologies"},{"name":"Chinese Academy of Sciences and the Shanghai Science and Technology Development Funds","award":["18JC1414700"]},{"name":"Key R&D Program of Guangdong province","award":["2018B030325001"]},{"DOI":"10.13039/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2021M693093"]}],"content-domain":{"domain":["opg.optica.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2022,2,20]]},"abstract":"<jats:p>The all-photonic quantum repeater (APQR) is a promising repeater scheme to realize long-distance quantum communication. For a practical APQR, an indispensable requirement is the robustness of the repeater graph state (RGS) against photon loss. We propose a new loss-tolerant scheme by applying the generalized Shor code to RGS, which can be experimentally demonstrated with current technology. Experimentally, we first prepare and verify the nine-qubit Shor code. Then, by applying the generalized Shor code to APQR and preparing a simplified encoded RGS with the structure of <jats:inline-formula>\n      <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n\t<mml:mrow class=\"MJX-TeXAtom-ORD\">\n\t  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n\t    <mml:mn>1</mml:mn>\n\t  </mml:mrow>\n\t</mml:mrow>\n\t<mml:mo>×<!-- × --></mml:mo>\n\t<mml:mrow class=\"MJX-TeXAtom-ORD\">\n\t  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n\t    <mml:mn>2</mml:mn>\n\t  </mml:mrow>\n\t</mml:mrow>\n      </mml:math>\n    </jats:inline-formula> based on the Shor code state, the effectiveness of our loss-tolerant scheme and the loss tolerance of the encoded RGS are respectively verified. Our results make an essential step toward a practical APQR and enrich the research of quantum error correction code.</jats:p>","DOI":"10.1364/optica.439170","type":"journal-article","created":{"date-parts":[[2021,12,21]],"date-time":"2021-12-21T17:02:33Z","timestamp":1640106153000},"page":"152","update-policy":"http://dx.doi.org/10.1364/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":"Loss-tolerant all-photonic quantum repeater with generalized Shor code","prefix":"10.1364","volume":"9","author":[{"ORCID":"http://orcid.org/0000-0001-5687-5576","authenticated-orcid":true,"given":"Rui","family":"Zhang","sequence":"first","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Li-Zheng","family":"Liu","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Zheng-Da","family":"Li","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Yue-Yang","family":"Fei","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Xu-Fei","family":"Yin","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Li","family":"Li","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Nai-Le","family":"Liu","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Yingqiu","family":"Mao","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Yu-Ao","family":"Chen","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Jian-Wei","family":"Pan","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]}],"member":"285","published-online":{"date-parts":[[2022,2,2]]},"reference":[],"container-title":"Optica","original-title":[],"language":"en","link":[{"URL":"https://opg.optica.org/viewmedia.cfm?URI=optica-9-2-152&seq=0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,9]],"date-time":"2022-08-09T18:28:48Z","timestamp":1660069728000},"score":1,"resource":{"primary":{"URL":"https://opg.optica.org/abstract.cfm?URI=optica-9-2-152"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,2]]},"references-count":35,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022]]},"published-print":{"date-parts":[[2022]]}},"URL":"http://dx.doi.org/10.1364/OPTICA.439170","relation":{},"ISSN":["2334-2536"],"subject":["Atomic and Molecular Physics, and Optics","Electronic, Optical and Magnetic Materials"],"container-title-short":"Optica","published":{"date-parts":[[2022,2,2]]},"assertion":[{"value":"Optica Publishing Group","name":"publisher","label":"This article is maintained by"},{"value":"https://doi.org/10.1364/OPTICA.439170","name":"articlelink","label":"Crossref DOI link to publisher maintained version"},{"value":"research-article","name":"content_type","label":"Article type"},{"value":"Screened by Similarity Check","name":"cross_check","label":"Similarity check"},{"value":"Yes","order":0,"name":"peer_reviewed","label":"Peer reviewed","group":{"name":"peer_review","label":"Peer review"}},{"value":"Single blind","order":1,"name":"review_process","label":"Review process","group":{"name":"peer_review","label":"Peer review"}},{"value":"29 July 2021","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"19 December 2021","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2 February 2022","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}},{"value":"© 2022 Optical Society of America under the terms of the OSA Open Access Publishing Agreement","name":"copyright","label":"Copyright"}],"id":"doi:10.1364/OPTICA.439170","_hash":"9169e646d5a63dfe5876af424ebed319ddb4499a54042932a90c3961204bd72e"},"expire":1701130966966},"doi:10.1073/pnas.2026250118":{"value":{"indexed":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T19:00:22Z","timestamp":1669316422209},"reference-count":41,"publisher":"Proceedings of the National Academy of Sciences","issue":"36","license":[{"start":{"date-parts":[[2021,9,3]],"date-time":"2021-09-03T00:00:00Z","timestamp":1630627200000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by-nc-nd/4.0/"}],"content-domain":{"domain":["www.pnas.org"],"crossmark-restriction":true},"published-print":{"date-parts":[[2021,9,7]]},"abstract":"<jats:title>Significance</jats:title>\n          <jats:p>Quantum teleportation and quantum error correction play crucial roles in fault-tolerant quantum computing. Here, we implemented error-correctable quantum teleportation to manipulate a logical qubit and observed the protection of quantum information. Our work presents a useful technology for scalable quantum computing and can serve as a quantum simulator for holographic quantum gravity.</jats:p>","DOI":"10.1073/pnas.2026250118","type":"journal-article","created":{"date-parts":[[2021,9,14]],"date-time":"2021-09-14T16:47:29Z","timestamp":1631638049000},"update-policy":"http://dx.doi.org/10.1073/pnas.cm10313","source":"Crossref","is-referenced-by-count":9,"title":"Quantum teleportation of physical qubits into logical code spaces","prefix":"10.1073","volume":"118","author":[{"given":"Yi-Han","family":"Luo","sequence":"first","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Chinese Academy of Sciences Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, Hefei, Anhui 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Among the possible candidate platforms for realizing quantum computing devices, the compatibility with mature nanofabrication technologies of silicon-based spin qubits offers promise to overcome the challenges in scaling up device sizes from the prototypes of today to large-scale computers<jats:sup>3–5</jats:sup>. Recent advances in silicon-based qubits have enabled the implementations of high-quality one-qubit and two-qubit systems<jats:sup>6–8</jats:sup>. However, the demonstration of QEC, which requires three or more coupled qubits<jats:sup>1</jats:sup>, and involves a three-qubit gate<jats:sup>9–11</jats:sup> or measurement-based feedback, remains an open challenge. Here we demonstrate a three-qubit phase-correcting code in silicon, in which an encoded three-qubit state is protected against any phase-flip error on one of the three qubits. The correction to this encoded state is performed by a three-qubit conditional rotation, which we implement by an efficient single-step resonantly driven iToffoli gate. As expected, the error correction mitigates the errors owing to one-qubit phase-flip, as well as the intrinsic dephasing mainly owing to quasi-static phase noise. These results show successful implementation of QEC and the potential of a silicon-based platform for large-scale quantum computing.</jats:p>","DOI":"10.1038/s41586-022-04986-6","type":"journal-article","created":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T16:04:14Z","timestamp":1661357054000},"page":"682-686","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":"Quantum error correction with silicon spin qubits","prefix":"10.1038","volume":"608","author":[{"ORCID":"http://orcid.org/0000-0003-1240-1103","authenticated-orcid":false,"given":"Kenta","family":"Takeda","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-9145-0303","authenticated-orcid":false,"given":"Akito","family":"Noiri","sequence":"additional","affiliation":[]},{"given":"Takashi","family":"Nakajima","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-2841-8129","authenticated-orcid":false,"given":"Takashi","family":"Kobayashi","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-7465-0135","authenticated-orcid":false,"given":"Seigo","family":"Tarucha","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,8,24]]},"reference":[],"container-title":"Nature","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41586-022-04986-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41586-022-04986-6","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41586-022-04986-6.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,12]],"date-time":"2022-11-12T19:09:14Z","timestamp":1668280154000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41586-022-04986-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,24]]},"references-count":45,"journal-issue":{"issue":"7924","published-print":{"date-parts":[[2022,8,25]]}},"alternative-id":["4986"],"URL":"http://dx.doi.org/10.1038/s41586-022-04986-6","relation":{},"ISSN":["0028-0836","1476-4687"],"subject":["Multidisciplinary"],"container-title-short":"Nature","published":{"date-parts":[[2022,8,24]]},"assertion":[{"value":"21 January 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 June 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 August 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"id":"doi:10.1038/s41586-022-04986-6","_hash":"97a1c7c63bfb638cf92682dbc825579431ed9098c6176451c0e0ed95fe21a190"},"expire":1701130972987},"doi:10.1038/s41467-022-29906-0":{"value":{"indexed":{"date-parts":[[2022,11,26]],"date-time":"2022-11-26T05:33:18Z","timestamp":1669440798295},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,4,28]],"date-time":"2022-04-28T00:00:00Z","timestamp":1651104000000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,4,28]],"date-time":"2022-04-28T00:00:00Z","timestamp":1651104000000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/100006754","name":"United States Department of Defense | United States Army | U.S. Army Research, Development and Engineering Command | Army Research Laboratory","doi-asserted-by":"publisher","award":["W911NF-17-S-0008","W911NF-17-S-0008","W911NF-17-S-0008","W911NF-17-S-0008","W911NF-17-S-0008"]},{"DOI":"10.13039/501100001742","name":"United States-Israel Binational Science Foundation","doi-asserted-by":"publisher","award":["735/18"]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"<jats:title>Abstract</jats:title><jats:p>The storage and processing of quantum information are susceptible to external noise, resulting in computational errors. A powerful method to suppress these effects is quantum error correction. Typically, quantum error correction is executed in discrete rounds, using entangling gates and projective measurement on ancillary qubits to complete each round of error correction. Here we use direct parity measurements to implement a continuous quantum bit-flip correction code in a resource-efficient manner, eliminating entangling gates, ancillary qubits, and their associated errors. An FPGA controller actively corrects errors as they are detected, achieving an average bit-flip detection efficiency of up to 91%. Furthermore, the protocol increases the relaxation time of the protected logical qubit by a factor of 2.7 over the relaxation times of the bare comprising qubits. Our results showcase resource-efficient stabilizer measurements in a multi-qubit architecture and demonstrate how continuous error correction codes can address challenges in realizing a fault-tolerant system.</jats:p>","DOI":"10.1038/s41467-022-29906-0","type":"journal-article","created":{"date-parts":[[2022,4,28]],"date-time":"2022-04-28T10:04:48Z","timestamp":1651140288000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":"Experimental demonstration of continuous quantum error correction","prefix":"10.1038","volume":"13","author":[{"ORCID":"http://orcid.org/0000-0001-8399-0975","authenticated-orcid":false,"given":"William P.","family":"Livingston","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-9736-4158","authenticated-orcid":false,"given":"Machiel S.","family":"Blok","sequence":"additional","affiliation":[]},{"given":"Emmanuel","family":"Flurin","sequence":"additional","affiliation":[]},{"given":"Justin","family":"Dressel","sequence":"additional","affiliation":[]},{"given":"Andrew N.","family":"Jordan","sequence":"additional","affiliation":[]},{"given":"Irfan","family":"Siddiqi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,28]]},"reference":[],"container-title":"Nature Communications","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41467-022-29906-0.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-022-29906-0","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-022-29906-0.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T17:00:23Z","timestamp":1669309223000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41467-022-29906-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,28]]},"references-count":45,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["29906"],"URL":"http://dx.doi.org/10.1038/s41467-022-29906-0","relation":{},"ISSN":["2041-1723"],"subject":["General Physics and Astronomy","General Biochemistry, Genetics and Molecular Biology","General Chemistry","Multidisciplinary"],"container-title-short":"Nat Commun","published":{"date-parts":[[2022,4,28]]},"assertion":[{"value":"31 January 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 April 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 April 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"2307","id":"doi:10.1038/s41467-022-29906-0","_hash":"0c6435c6396560802a82b003e827bd2511595c7e5cf423ce4fa600e28202eaa3"},"expire":1701130973998},"doi:10.1038/nature14270":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T22:50:19Z","timestamp":1670021419168},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"7541","license":[{"start":{"date-parts":[[2015,3,4]],"date-time":"2015-03-04T00:00:00Z","timestamp":1425427200000},"content-version":"tdm","delay-in-days":0,"URL":"https://www.springer.com/tdm"},{"start":{"date-parts":[[2015,3,4]],"date-time":"2015-03-04T00:00:00Z","timestamp":1425427200000},"content-version":"vor","delay-in-days":0,"URL":"https://www.springer.com/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2015,3,5]]},"DOI":"10.1038/nature14270","type":"journal-article","created":{"date-parts":[[2015,3,3]],"date-time":"2015-03-03T16:48:51Z","timestamp":1425401331000},"page":"66-69","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":606,"title":"State preservation by repetitive error detection in a superconducting quantum circuit","prefix":"10.1038","volume":"519","author":[{"given":"J.","family":"Kelly","sequence":"first","affiliation":[]},{"given":"R.","family":"Barends","sequence":"additional","affiliation":[]},{"given":"A. 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This includes a need to determine how well these devices support the techniques required for quantum error correction. In this paper we introduce the <jats:monospace>topological_codes</jats:monospace> module of Qiskit-Ignis, which is designed to provide the tools necessary to perform such tests. Specifically, we use the <jats:monospace>RepetitionCode</jats:monospace> and <jats:monospace>GraphDecoder</jats:monospace> classes to run tests based on the repetition code and process the results. As an example, data from a 43 qubit code running on IBM’s <jats:italic>Rochester</jats:italic> device is presented.</jats:p>","DOI":"10.1088/2058-9565/aba038","type":"journal-article","created":{"date-parts":[[2020,6,26]],"date-time":"2020-06-26T16:27:09Z","timestamp":1593188829000},"page":"044004","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":14,"title":"Benchmarking near-term devices with quantum error correction","prefix":"10.1088","volume":"5","author":[{"ORCID":"http://orcid.org/0000-0003-1943-5306","authenticated-orcid":false,"given":"James R","family":"Wootton","sequence":"first","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2020,7,31]]},"reference":[],"container-title":"Quantum Science and Technology","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038","content-type":"text/html","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,27]],"date-time":"2021-11-27T16:33:56Z","timestamp":1638030836000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/aba038"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,31]]},"references-count":29,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,7,31]]},"published-print":{"date-parts":[[2020,10,1]]}},"URL":"http://dx.doi.org/10.1088/2058-9565/aba038","relation":{},"ISSN":["2058-9565"],"subject":["Electrical and Electronic Engineering","Physics and Astronomy (miscellaneous)","Materials Science (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum Sci. 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Published by IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2020-01-28","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-06-25","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-07-31","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/2058-9565/aba038","_hash":"fc272c23a0d0870f08909343789cf9a7e4a6084a35fa50f339d121a474f25345"},"expire":1701130975989},"doi:10.1103/PhysRevA.97.052313":{"value":{"indexed":{"date-parts":[[2022,12,1]],"date-time":"2022-12-01T18:33:39Z","timestamp":1669919619321},"reference-count":21,"publisher":"American Physical Society (APS)","issue":"5","license":[{"start":{"date-parts":[[2018,5,10]],"date-time":"2018-05-10T00:00:00Z","timestamp":1525910400000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"funder":[{"DOI":"10.13039/501100001711","name":"Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung","doi-asserted-by":"publisher"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physreva.97.052313","type":"journal-article","created":{"date-parts":[[2018,5,10]],"date-time":"2018-05-10T10:03:46Z","timestamp":1525946626000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":33,"title":"Repetition code of 15 qubits","prefix":"10.1103","volume":"97","author":[{"given":"James R.","family":"Wootton","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Loss","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2018,5,10]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevA.97.052313","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.97.052313/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2018,5,10]],"date-time":"2018-05-10T10:03:50Z","timestamp":1525946630000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.97.052313"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,10]]},"references-count":21,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2018,5]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.052313","relation":{},"ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. 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When instantiated with 3 classical LDPC codes, this \"XYZ product\" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Denise Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes also depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved.Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes with constant rate might display a minimum distance as large as <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x0398;</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>N</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>2</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes that can be embedded in three dimensions with local interaction. Some of these codes seem to share properties with Haah's cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier.</jats:p>","DOI":"10.22331/q-2022-07-14-766","type":"journal-article","created":{"date-parts":[[2022,7,14]],"date-time":"2022-07-14T13:42:11Z","timestamp":1657806131000},"page":"766","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Quantum XYZ Product Codes","prefix":"10.22331","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0002-6707-1458","authenticated-orcid":false,"given":"Anthony","family":"Leverrier","sequence":"first","affiliation":[{"name":"Inria, France"}]},{"ORCID":"http://orcid.org/0000-0003-3823-6804","authenticated-orcid":false,"given":"Simon","family":"Apers","sequence":"additional","affiliation":[{"name":"CNRS, IRIF, Université Paris 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Inform. Theory","published":{"date-parts":[[2014,2]]},"id":"doi:10.1109/TIT.2013.2292061","_hash":"eb497befcd1f372b1a4167b5dcf4853e21f8b766f460fe79d8d6803b52d6da50"},"expire":1701131024699},"doi:10.22331/q-2019-02-04-120":{"value":{"indexed":{"date-parts":[[2022,7,28]],"date-time":"2022-07-28T14:29:59Z","timestamp":1659018599702},"reference-count":51,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2019,2,4]],"date-time":"2019-02-04T00:00:00Z","timestamp":1549238400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"<jats:p>A method for the implementation of a universal set of fault-tolerant logical gates is presented using homological product codes. In particular, it is shown that one can fault-tolerantly map between different encoded representations of a given logical state, enabling the application of different classes of transversal gates belonging to the underlying quantum codes. This allows for the circumvention of no-go results pertaining to universal sets of transversal gates and provides a general scheme for fault-tolerant computation while keeping the stabilizer generators of the code sparse.</jats:p>","DOI":"10.22331/q-2019-02-04-120","type":"journal-article","created":{"date-parts":[[2019,2,4]],"date-time":"2019-02-04T11:50:45Z","timestamp":1549281045000},"page":"120","source":"Crossref","is-referenced-by-count":8,"title":"Fault-tolerant gates via homological product codes","prefix":"10.22331","volume":"3","author":[{"ORCID":"http://orcid.org/0000-0002-8265-4774","authenticated-orcid":false,"given":"Tomas","family":"Jochym-O'Connor","sequence":"first","affiliation":[{"name":"Walter Burke Institute for Theoretical Physics, Institute for Quantum Information & Matter, California Institute of Technology, Pasadena, CA 91125, USA"}]}],"member":"9598","published-online":{"date-parts":[[2019,2,4]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2019-02-04-120/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2019,2,4]],"date-time":"2019-02-04T11:51:47Z","timestamp":1549281107000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2019-02-04-120/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,4]]},"references-count":51,"URL":"http://dx.doi.org/10.22331/q-2019-02-04-120","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2019,2,4]]},"article-number":"120","id":"doi:10.22331/q-2019-02-04-120","_hash":"f6048c598de9bcf066faac478d879468aa749da59b5966f0297490a3c36d1744"},"expire":1701131025757},"doi:10.22331/q-2021-03-10-406":{"value":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T13:19:00Z","timestamp":1648559940083},"reference-count":17,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2021,3,10]],"date-time":"2021-03-10T00:00:00Z","timestamp":1615334400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Homological product codes are a class of codes that can have improved distance while retaining relatively low stabilizer weight. We show how to build union-find decoders for these codes, using a union-find decoder for one of the codes in the product and a brute force decoder for the other code. We apply this construction to the specific case of the product of a surface code with a small code such as a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:math> code, which we call an augmented surface code. The distance of the augmented surface code is the product of the distance of the surface code with that of the small code, and the union-find decoder, with slight modifications, can decode errors up to half the distance. We present numerical simulations, showing that while the threshold of these augmented codes is lower than that of the surface code, the low noise performance is improved.</jats:p>","DOI":"10.22331/q-2021-03-10-406","type":"journal-article","created":{"date-parts":[[2021,3,10]],"date-time":"2021-03-10T13:12:34Z","timestamp":1615381954000},"page":"406","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Union-Find Decoders For Homological Product Codes","prefix":"10.22331","volume":"5","author":[{"given":"Nicolas","family":"Delfosse","sequence":"first","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"name":"Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA"},{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"}]}],"member":"9598","published-online":{"date-parts":[[2021,3,10]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2021-03-10-406/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2021,3,10]],"date-time":"2021-03-10T13:13:11Z","timestamp":1615381991000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2021-03-10-406/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,10]]},"references-count":17,"URL":"http://dx.doi.org/10.22331/q-2021-03-10-406","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2021,3,10]]},"article-number":"406","id":"doi:10.22331/q-2021-03-10-406","_hash":"4afecabbcf6e66825b5834af24aa90b5c51c60753d93295160813f53e75c59ed"},"expire":1701131026798},"doi:10.22331/q-2022-02-24-661":{"value":{"indexed":{"date-parts":[[2022,12,1]],"date-time":"2022-12-01T13:18:53Z","timestamp":1669900733329},"reference-count":38,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,2,24]],"date-time":"2022-02-24T00:00:00Z","timestamp":1645660800000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/anr","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["QCDA"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>We introduce the hemicubic codes, a family of quantum codes obtained by associating qubits with the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>p</mml:mi></mml:math>-faces of the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>-cube (for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi><mml:mo>&amp;#x003E;</mml:mo><mml:mi>p</mml:mi></mml:math>) and stabilizer constraints with faces of dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>&amp;#x00B1;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. The quantum code obtained by identifying antipodal faces of the resulting complex encodes one logical qubit into <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>n</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mi>p</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mrow><mml:mrow class=\"MJX-TeXAtom-OPEN\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mfrac linethickness=\"0\"><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:mfrac><mml:mrow class=\"MJX-TeXAtom-CLOSE\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math> physical qubits and displays local testability with a soundness of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x03A9;</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math> beating the current state-of-the-art of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>log</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> due to Hastings. We exploit this local testability to devise an efficient decoding algorithm that corrects arbitrary errors of size less than the minimum distance, up to polylog factors.We then extend this code family by considering the quotient of the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>-cube by arbitrary linear classical codes of length <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>. We establish the parameters of these generalized hemicubic codes. Interestingly, if the soundness of the hemicubic code could be shown to be constant, similarly to the ordinary <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>-cube, then the generalized hemicubic codes could yield quantum locally testable codes of length not exceeding an exponential or even polynomial function of the code dimension.</jats:p>","DOI":"10.22331/q-2022-02-24-661","type":"journal-article","created":{"date-parts":[[2022,2,24]],"date-time":"2022-02-24T19:54:29Z","timestamp":1645732469000},"page":"661","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":"Towards local testability for quantum coding","prefix":"10.22331","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0002-6707-1458","authenticated-orcid":false,"given":"Anthony","family":"Leverrier","sequence":"first","affiliation":[{"name":"Inria, France"}]},{"given":"Vivien","family":"Londe","sequence":"additional","affiliation":[{"name":"Microsoft, France"}]},{"ORCID":"http://orcid.org/0000-0002-6041-9554","authenticated-orcid":false,"given":"Gilles","family":"Zémor","sequence":"additional","affiliation":[{"name":"Institut de Mathématiques de Bordeaux, UMR 5251, France"}]}],"member":"9598","published-online":{"date-parts":[[2022,2,24]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-02-24-661/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,3,9]],"date-time":"2022-03-09T20:59:53Z","timestamp":1646859593000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-02-24-661/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,24]]},"references-count":38,"URL":"http://dx.doi.org/10.22331/q-2022-02-24-661","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,2,24]]},"article-number":"661","id":"doi:10.22331/q-2022-02-24-661","_hash":"21c9fc410af34064e140ff2180d8fa6221b0a16fbfc514db2259e4936c8558c2"},"expire":1701131027732},"doi:10.1103/PhysRevA.97.032346":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T22:50:17Z","timestamp":1670021417165},"reference-count":135,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2018,3,30]],"date-time":"2018-03-30T00:00:00Z","timestamp":1522368000000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"},{"start":{"date-parts":[[2019,3,30]],"date-time":"2019-03-30T00:00:00Z","timestamp":1553904000000},"content-version":"am","delay-in-days":365,"URL":"https://link.aps.org/licenses/aps-default-accepted-manuscript-license"}],"funder":[{"DOI":"10.13039/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["W911NF-14-1-0011","W911NF-14-1-0563","W911NF-16-1-0349","W911NF1410011"]},{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["EFMA-1640959","DMR-1609326"]},{"DOI":"10.13039/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-14-1-0052","FA9550-15-1-0015"]},{"DOI":"10.13039/100000879","name":"Alfred P. 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T.","family":"Brierley","sequence":"additional","affiliation":[]},{"given":"Philip","family":"Reinhold","sequence":"additional","affiliation":[]},{"given":"Christophe","family":"Vuillot","sequence":"additional","affiliation":[]},{"given":"Linshu","family":"Li","sequence":"additional","affiliation":[]},{"given":"Chao","family":"Shen","sequence":"additional","affiliation":[]},{"given":"S. M.","family":"Girvin","sequence":"additional","affiliation":[]},{"given":"Barbara M.","family":"Terhal","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2018,3,30]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/accepted/10.1103/PhysRevA.97.032346","content-type":"application/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"https://link.aps.org/article/10.1103/PhysRevA.97.032346","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.97.032346/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,10,30]],"date-time":"2020-10-30T16:07:03Z","timestamp":1604074023000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.97.032346"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,3,30]]},"references-count":135,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2018,3]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.032346","relation":{},"ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. 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Rev. A","published":{"date-parts":[[2014,12,10]]},"article-number":"062317","id":"doi:10.1103/PhysRevA.90.062317","_hash":"cad75a55f074cbb12dc36312461aac46cfad2d79c971b18f2c780f0ff269ede5"},"expire":1701131030839},"doi:10.1007/JHEP08(2019)152":{"value":{"indexed":{"date-parts":[[2022,4,7]],"date-time":"2022-04-07T14:04:42Z","timestamp":1649340282215},"reference-count":53,"publisher":"Springer Science and Business Media LLC","issue":"8","license":[{"start":{"date-parts":[[2019,8,1]],"date-time":"2019-08-01T00:00:00Z","timestamp":1564617600000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"},{"start":{"date-parts":[[2019,8,27]],"date-time":"2019-08-27T00:00:00Z","timestamp":1566864000000},"content-version":"vor","delay-in-days":26,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2019,8]]},"abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>The eigenstate thermalization hypothesis (ETH) is a powerful conjecture for understanding how statistical mechanics emerges in a large class of many-body quantum systems. It has also been interpreted in a CFT context, and, in particular, holographic CFTs are expected to satisfy ETH. Recently, it was observed that the ETH condition corresponds to a necessary and sufficient condition for an approximate quantum error correcting code (AQECC), implying the presence of AQECCs in systems satisfying ETH. In this paper, we explore the properties of ETH as an error correcting code and show that there exists an explicit universal recovery channel for the code. Based on the analysis, we discuss a generalization that all chaotic theories contain error correcting codes. We then specialize to AdS/CFT to demonstrate the possibility of total bulk reconstruction in black holes with a well-defined macroscopic geometry. When combined with the existing AdS/CFT error correction story, this shows that black holes are enormously robust against erasure errors.</jats:p>","DOI":"10.1007/jhep08(2019)152","type":"journal-article","created":{"date-parts":[[2019,8,28]],"date-time":"2019-08-28T14:05:27Z","timestamp":1567001127000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":"Eigenstate thermalization hypothesis and approximate quantum error correction","prefix":"10.1007","volume":"2019","author":[{"given":"Ning","family":"Bao","sequence":"first","affiliation":[]},{"given":"Newton","family":"Cheng","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,8,27]]},"reference":[],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","link":[{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP08(2019)152.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/article/10.1007/JHEP08(2019)152/fulltext.html","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP08(2019)152.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,3]],"date-time":"2021-09-03T17:59:42Z","timestamp":1630691982000},"score":1,"resource":{"primary":{"URL":"https://link.springer.com/10.1007/JHEP08(2019)152"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8]]},"references-count":53,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2019,8]]}},"alternative-id":["11156"],"URL":"http://dx.doi.org/10.1007/JHEP08(2019)152","relation":{},"ISSN":["1029-8479"],"subject":["Nuclear and High Energy Physics"],"container-title-short":"J. High Energ. Phys.","published":{"date-parts":[[2019,8]]},"assertion":[{"value":"25 June 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 August 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 August 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"152","id":"doi:10.1007/JHEP08(2019)152","_hash":"2e4136e0c2c5498ff6a58be5cbe6ea9202297305c01862b534911f6e40db4de7"},"expire":1701131031987},"doi:10.1103/PhysRevLett.123.110502":{"value":{"indexed":{"date-parts":[[2022,11,18]],"date-time":"2022-11-18T20:18:06Z","timestamp":1668802686404},"reference-count":35,"publisher":"American Physical Society (APS)","issue":"11","license":[{"start":{"date-parts":[[2019,9,11]],"date-time":"2019-09-11T00:00:00Z","timestamp":1568160000000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"funder":[{"DOI":"10.13039/100000936","name":"Gordon and Betty Moore Foundation","doi-asserted-by":"publisher","award":["GBMF-12500028"]},{"DOI":"10.13039/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["385612"]},{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["PHY-1125565","PHY-1125915"]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevlett.123.110502","type":"journal-article","created":{"date-parts":[[2019,9,11]],"date-time":"2019-09-11T14:36:37Z","timestamp":1568212597000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":23,"title":"Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains","prefix":"10.1103","volume":"123","author":[{"given":"Fernando G. S. L.","family":"Brandão","sequence":"first","affiliation":[]},{"given":"Elizabeth","family":"Crosson","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4488-2300","authenticated-orcid":true,"given":"M. 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Conventional schemes for error correction with the surface code place qubits on a planar grid and assume native CNOT gates between the data qubits with nearest-neighbor ancilla qubits.Here, we present surface code error-correction schemes using <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">only</mml:mtext></mml:mrow></mml:math> Pauli measurements on single qubits and on pairs of nearest-neighbor qubits. In particular, we provide several qubit layouts that offer favorable trade-offs between qubit overhead, circuit depth and connectivity degree. We also develop minimized measurement sequences for syndrome extraction, enabling reduced logical error rates and improved fault-tolerance thresholds.Our work applies to topologically protected qubits realized with Majorana zero modes and to similar systems in which multi-qubit Pauli measurements rather than CNOT gates are the native operations.</jats:p>","DOI":"10.22331/q-2020-10-28-352","type":"journal-article","created":{"date-parts":[[2020,10,28]],"date-time":"2020-10-28T11:06:16Z","timestamp":1603883176000},"page":"352","source":"Crossref","is-referenced-by-count":12,"title":"Optimization of the surface code design for Majorana-based qubits","prefix":"10.22331","volume":"4","author":[{"given":"Rui","family":"Chao","sequence":"first","affiliation":[{"name":"University of Southern California, Los Angeles, CA, USA"}]},{"given":"Michael E.","family":"Beverland","sequence":"additional","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA, USA"}]},{"given":"Nicolas","family":"Delfosse","sequence":"additional","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA, USA"}]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA, USA"}]}],"member":"9598","published-online":{"date-parts":[[2020,10,28]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2020-10-28-352/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2020,10,28]],"date-time":"2020-10-28T11:06:19Z","timestamp":1603883179000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2020-10-28-352/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,28]]},"references-count":20,"URL":"http://dx.doi.org/10.22331/q-2020-10-28-352","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2020,10,28]]},"article-number":"352","id":"doi:10.22331/q-2020-10-28-352","_hash":"b93bb89e6c6b7920278591eba87518774c6211b44614fbc0a8691cd6eb3f0dce"},"expire":1701131047322},"doi:10.22331/q-2022-09-21-813":{"value":{"indexed":{"date-parts":[[2022,9,22]],"date-time":"2022-09-22T05:45:36Z","timestamp":1663825536452},"reference-count":17,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T00:00:00Z","timestamp":1663718400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>We improve the planar honeycomb code by describing boundaries that need no additional physical connectivity, and by optimizing the shape of the qubit patch. We then benchmark the code using Monte Carlo sampling to estimate logical error rates and derive metrics including thresholds, lambdas, and teraquop qubit counts. We determine that the planar honeycomb code can create a logical qubit with one-in-a-trillion logical error rates using 7000 physical qubits at a 0.1% gate-level error rate (or 900 physical qubits given native two-qubit parity measurements). Our results cement the honeycomb code as a promising candidate for two-dimensional qubit architectures with sparse connectivity.</jats:p>","DOI":"10.22331/q-2022-09-21-813","type":"journal-article","created":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T12:09:46Z","timestamp":1663762186000},"page":"813","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Benchmarking the Planar Honeycomb Code","prefix":"10.22331","volume":"6","author":[{"given":"Craig","family":"Gidney","sequence":"first","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Matt","family":"McEwen","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"},{"name":"University of California, Santa Barbara, 93106, USA"}]}],"member":"9598","published-online":{"date-parts":[[2022,9,21]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-09-21-813/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T12:09:53Z","timestamp":1663762193000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-09-21-813/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,21]]},"references-count":17,"URL":"http://dx.doi.org/10.22331/q-2022-09-21-813","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,9,21]]},"article-number":"813","id":"doi:10.22331/q-2022-09-21-813","_hash":"7dbaa53da2454a63694155ce21078e26bff1866178dd006ddb892be692355450"},"expire":1701131048167},"doi:10.22331/q-2021-12-20-605":{"value":{"indexed":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T08:31:37Z","timestamp":1669797097092},"reference-count":44,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2021,12,20]],"date-time":"2021-12-20T00:00:00Z","timestamp":1639958400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Recently, Hastings &amp; Haah introduced a quantum memory defined on the honeycomb lattice. Remarkably, this honeycomb code assembles weight-six parity checks using only two-local measurements. The sparse connectivity and two-local measurements are desirable features for certain hardware, while the weight-six parity checks enable robust performance in the circuit model.In this work, we quantify the robustness of logical qubits preserved by the honeycomb code using a correlated minimum-weight perfect-matching decoder. Using Monte Carlo sampling, we estimate the honeycomb code's threshold in different error models, and project how efficiently it can reach the \"teraquop regime\" where trillions of quantum logical operations can be executed reliably. We perform the same estimates for the rotated surface code, and find a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.2</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>0.3</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> for the honeycomb code compared to a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.5</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>0.7</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> for the surface code in a controlled-not circuit model. In a circuit model with native two-body measurements, the honeycomb code achieves a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1.5</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi><mml:mo>&amp;#x003C;</mml:mo><mml:mi>p</mml:mi><mml:mo>&amp;#x003C;</mml:mo><mml:mn>2.0</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>p</mml:mi></mml:math> is the collective error rate of the two-body measurement gate - including both measurement and correlated data depolarization error processes. With such gates at a physical error rate of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mn>10</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>, we project that the honeycomb code can reach the teraquop regime with only <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>600</mml:mn></mml:math> physical qubits. </jats:p>","DOI":"10.22331/q-2021-12-20-605","type":"journal-article","created":{"date-parts":[[2022,1,30]],"date-time":"2022-01-30T10:03:32Z","timestamp":1643537012000},"page":"605","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":10,"title":"A Fault-Tolerant Honeycomb Memory","prefix":"10.22331","volume":"5","author":[{"given":"Craig","family":"Gidney","sequence":"first","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Austin","family":"Fowler","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Michael","family":"Broughton","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]}],"member":"9598","published-online":{"date-parts":[[2021,12,20]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2021-12-20-605/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,1,30]],"date-time":"2022-01-30T10:03:35Z","timestamp":1643537015000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2021-12-20-605/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,20]]},"references-count":44,"URL":"http://dx.doi.org/10.22331/q-2021-12-20-605","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and 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B","published":{"date-parts":[[2022,8,15]]},"article-number":"085122","id":"doi:10.1103/PhysRevB.106.085122","_hash":"b32e7efe9dac609e9c38cbf437aaac3746acf5e1802e13484e4c6a4e3cec0b0d"},"expire":1701131050130},"doi:10.22331/q-2022-04-21-693":{"value":{"indexed":{"date-parts":[[2022,11,26]],"date-time":"2022-11-26T05:28:41Z","timestamp":1669440521129},"reference-count":14,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,4,21]],"date-time":"2022-04-21T00:00:00Z","timestamp":1650499200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>We introduce a simple construction of boundary conditions for the honeycomb code \\cite{honeycomb} that uses only pairwise checks and allows parallelogram geometries at the cost of modifying the bulk measurement sequence. We discuss small instances of the code.</jats:p>","DOI":"10.22331/q-2022-04-21-693","type":"journal-article","created":{"date-parts":[[2022,4,21]],"date-time":"2022-04-21T08:03:42Z","timestamp":1650528222000},"page":"693","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":"Boundaries for the Honeycomb Code","prefix":"10.22331","volume":"6","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"},{"name":"Station Q, Microsoft Quantum, Santa Barbara, CA 93106-6105, USA"}]}],"member":"9598","published-online":{"date-parts":[[2022,4,21]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-04-21-693/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,4,21]],"date-time":"2022-04-21T08:03:51Z","timestamp":1650528231000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-04-21-693/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,21]]},"references-count":14,"URL":"http://dx.doi.org/10.22331/q-2022-04-21-693","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,4,21]]},"article-number":"693","id":"doi:10.22331/q-2022-04-21-693","_hash":"5197165b1ed7a1fe723bc3dda2c98ae516d9fa3fc4ed1d0cd60aab54aa6df835"},"expire":1701131050970},"doi:10.1142/S1230161208000043":{"value":{"indexed":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T19:18:43Z","timestamp":1669317523408},"reference-count":14,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2008,3]]},"abstract":"<jats:p> We give a short proof that the coherent information is an achievable rate for the transmission of quantum information through a noisy quantum channel. Our method is to produce random codes by performing a unitarily covariant projective measurement on a typical subspace of a tensor power state. We show that, provided the rank of each measurement operator is sufficiently small, the transmitted data will, with high probability, be decoupled from the channel environment. We also show that our construction leads to random codes whose average input is close to a product state and outline a modification yielding unitarily invariant ensembles of maximally entangled codes. </jats:p>","DOI":"10.1142/s1230161208000043","type":"journal-article","created":{"date-parts":[[2008,3,26]],"date-time":"2008-03-26T04:30:06Z","timestamp":1206505806000},"page":"7-19","source":"Crossref","is-referenced-by-count":87,"title":"A Decoupling Approach to the Quantum Capacity","prefix":"10.1142","volume":"15","author":[{"given":"Patrick","family":"Hayden","sequence":"first","affiliation":[{"name":"School of Computer Science, McGill University, Montreal, Quebec, Canada"}]},{"given":"Michał","family":"Horodecki","sequence":"additional","affiliation":[{"name":"Institute of Theoretical Physics and Astrophysics, University of Gdańsk, Poland"},{"name":"National Quantum Information Centre of Gdańsk, Gen. W. Andersa 27, 81 – 824 Sopot, Poland"}]},{"given":"Andreas","family":"Winter","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bristol, UK"}]},{"given":"Jon","family":"Yard","sequence":"additional","affiliation":[{"name":"CNLS (Center for Nonlinear Studies), CCS-3 (Computer, Computational and Statistical Sciences), Los Alamos National Laboratories, Los Alamos, NM, USA"},{"name":"Institute for Quantum Information, Caltech, Pasadena, CA, USA"}]}],"member":"219","published-online":{"date-parts":[[2012,1,25]]},"reference":[],"container-title":"Open Systems &amp; Information Dynamics","original-title":[],"language":"en","link":[{"URL":"https://www.worldscientific.com/doi/pdf/10.1142/S1230161208000043","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T21:56:35Z","timestamp":1565128595000},"score":1,"resource":{"primary":{"URL":"https://www.worldscientific.com/doi/abs/10.1142/S1230161208000043"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,3]]},"references-count":14,"journal-issue":{"issue":"01","published-online":{"date-parts":[[2012,1,25]]},"published-print":{"date-parts":[[2008,3]]}},"alternative-id":["10.1142/S1230161208000043"],"URL":"http://dx.doi.org/10.1142/S1230161208000043","relation":{},"ISSN":["1230-1612","1793-7191"],"subject":["Mathematical Physics","Statistics and Probability","Statistical and Nonlinear Physics"],"container-title-short":"Open Syst. 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The ultimate end-to-end rates of quantum communication networks are known to be achievable by an optimal entanglement distillation protocol followed by teleportation. In this work, we give a practical design for this achievability. Our ultimate design is an iterative approach, where each purification step operates on shared entangled states and detects loss errors at the highest rates allowed by physics. As a simpler design, we show that the first round of iterations can purify completely at high rates. We propose an experimental implementation using linear optics and photon-number measurements which is robust to inefficient operations and measurements, showcasing its near-term potential for real-world practical applications.</jats:p>","DOI":"10.1038/s41534-022-00641-0","type":"journal-article","created":{"date-parts":[[2022,11,8]],"date-time":"2022-11-08T03:02:32Z","timestamp":1667876552000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":"Achieving the ultimate end-to-end rates of lossy quantum communication networks","prefix":"10.1038","volume":"8","author":[{"ORCID":"http://orcid.org/0000-0003-3457-4451","authenticated-orcid":false,"given":"Matthew S.","family":"Winnel","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-9990-6341","authenticated-orcid":false,"given":"Joshua J.","family":"Guanzon","sequence":"additional","affiliation":[]},{"given":"Nedasadat","family":"Hosseinidehaj","sequence":"additional","affiliation":[]},{"given":"Timothy C.","family":"Ralph","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,11,8]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41534-022-00641-0.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-022-00641-0","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-022-00641-0.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,8]],"date-time":"2022-11-08T03:06:26Z","timestamp":1667876786000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-022-00641-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,8]]},"references-count":62,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["641"],"URL":"http://dx.doi.org/10.1038/s41534-022-00641-0","relation":{},"ISSN":["2056-6387"],"subject":["Computational Theory and Mathematics","Computer Networks and Communications","Statistical and Nonlinear Physics","Computer Science (miscellaneous)"],"container-title-short":"npj Quantum Inf","published":{"date-parts":[[2022,11,8]]},"assertion":[{"value":"25 March 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 October 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 November 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"129","id":"doi:10.1038/s41534-022-00641-0","_hash":"2005456e99b1fa2b5ea95b41e47a6596f58c05a86a2dd03690e8ab21ac06eefe"},"expire":1701131111120},"doi:10.1103/PhysRevLett.82.4556":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T22:56:07Z","timestamp":1670021767715},"reference-count":19,"publisher":"American Physical Society (APS)","issue":"22","license":[{"start":{"date-parts":[[1999,5,31]],"date-time":"1999-05-31T00:00:00Z","timestamp":928108800000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physrevlett.82.4556","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:16:54Z","timestamp":1027721814000},"page":"4556-4559","source":"Crossref","is-referenced-by-count":226,"title":"Concatenating Decoherence-Free Subspaces with Quantum Error Correcting Codes","prefix":"10.1103","volume":"82","author":[{"given":"D. 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In situations where the Heisenberg scaling is achievable, we provide a semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cramér-Rao (HCR) bound for pure states. We provide examples of significant advantages offered by our joint-QEC protocols, that sense all the parameters utilizing a single error-corrected subspace, over separate-QEC protocols where each parameter is effectively sensed in a separate subspace.</jats:p>","DOI":"10.22331/q-2020-07-02-288","type":"journal-article","created":{"date-parts":[[2020,7,2]],"date-time":"2020-07-02T17:47:57Z","timestamp":1593712077000},"page":"288","source":"Crossref","is-referenced-by-count":18,"title":"Optimal probes and error-correction schemes in multi-parameter quantum metrology","prefix":"10.22331","volume":"4","author":[{"ORCID":"http://orcid.org/0000-0001-9912-9186","authenticated-orcid":false,"given":"Wojciech","family":"Górecki","sequence":"first","affiliation":[{"name":"Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, 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The need for understanding the limits of covariant quantum error correction arises in various realms of physics including fault-tolerant quantum computation, condensed matter physics and quantum gravity. Here, we explore covariant quantum error correction with respect to continuous symmetries from the perspectives of quantum metrology and quantum resource theory, establishing solid connections between these formerly disparate fields. We prove new and powerful lower bounds on the infidelity of covariant quantum error correction, which not only extend the scope of previous no-go results but also provide a substantial improvement over existing bounds. Explicit lower bounds are derived for both erasure and depolarizing noises. 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In particular, several researchers have considered the individual code qubits being encoded with the continuous variable GottesmanKitaev-Preskill (GKP) code, and then imposed an outer discrete-variable code such as the surface code on these GKP qubits. Under such a concatenation scheme, the analog information from the inner GKP error correction improves the noise threshold of the outer code. However, the surface code has vanishing rate and demands a lot of resources with growing distance. In this work, we concatenate the GKP code with generic quantum low-density parity-check (QLDPC) codes and demonstrate a natural way to exploit the GKP analog information in iterative decoding algorithms. We first show the noise thresholds for two lifted product QLDPC code families, and then show the improvements of noise thresholds when the iterative decoder – a hardware-friendly min-sum algorithm (MSA) – utilizes the GKP analog information. We also show that, when the GKP analog information is combined with a sequential update schedule for MSA, the scheme surpasses the well-known CSS Hamming bound for these code families. Furthermore, we observe that the GKP analog information helps the iterative decoder in escaping harmful trapping sets in the Tanner graph of the QLDPC code, thereby eliminating or significantly lowering the error floor of the logical error rate curves. Finally, we discuss new fundamental and practical questions that arise from this work on channel capacity under GKP analog information, and on improving decoder design and analysis.</jats:p>","DOI":"10.22331/q-2022-07-20-767","type":"journal-article","created":{"date-parts":[[2022,7,20]],"date-time":"2022-07-20T13:39:18Z","timestamp":1658324358000},"page":"767","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":"Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound","prefix":"10.22331","volume":"6","author":[{"given":"Nithin","family":"Raveendran","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]},{"given":"Narayanan","family":"Rengaswamy","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]},{"given":"Filip","family":"Rozpędek","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Chicago, IL 60637, USA"}]},{"given":"Ankur","family":"Raina","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462066, India"}]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Chicago, IL 60637, USA"}]},{"given":"Bane","family":"Vasić","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]}],"member":"9598","published-online":{"date-parts":[[2022,7,20]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-07-20-767/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,7,20]],"date-time":"2022-07-20T13:39:45Z","timestamp":1658324385000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-07-20-767/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,20]]},"references-count":61,"URL":"http://dx.doi.org/10.22331/q-2022-07-20-767","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,7,20]]},"article-number":"767","id":"doi:10.22331/q-2022-07-20-767","_hash":"ca5bd48b8ce93634d7f330cfc52b3ed2807a4b0ea66d8b9ed78b62bc66d86df1"},"expire":1701131137992},"doi:10.22331/q-2022-02-10-648":{"value":{"indexed":{"date-parts":[[2022,5,20]],"date-time":"2022-05-20T14:49:34Z","timestamp":1653058174421},"reference-count":53,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,2,10]],"date-time":"2022-02-10T00:00:00Z","timestamp":1644451200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"BMBF","award":["RealistiQ"]},{"name":"BMBF","award":["PhoQuant"]},{"name":"BMBF","award":["QPIC-1"]},{"name":"BMBF","award":["QSolid"]},{"name":"DFG","award":["CRC 183"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>We examine general Gottesman-Kitaev-Preskill (GKP) codes for continuous-variable quantum error correction, including concatenated GKP codes, through the lens of lattice theory, in order to better understand the structure of this class of stabilizer codes. 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Publishing Services by Elsevier B.V. on behalf of KeAi Communications Co. Ltd.","name":"copyright","label":"Copyright"}],"id":"doi:10.1016/j.fmre.2020.12.006","_hash":"e370db8b702a3ebb97756bb142bb401a416b70e03783251f875aa5e2ecf2b8bb"},"expire":1701131164178},"doi:10.1088/2058-9565/abe989":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T22:53:31Z","timestamp":1670021611865},"reference-count":246,"publisher":"IOP Publishing","issue":"3","license":[{"start":{"date-parts":[[2021,4,1]],"date-time":"2021-04-01T00:00:00Z","timestamp":1617235200000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"},{"start":{"date-parts":[[2021,4,1]],"date-time":"2021-04-01T00:00:00Z","timestamp":1617235200000},"content-version":"tdm","delay-in-days":0,"URL":"https://iopscience.iop.org/info/page/text-and-data-mining"}],"funder":[{"DOI":"10.13039/501100001381","name":"Singapore National Research Foundation","doi-asserted-by":"crossref","award":["Fellowship (Class of 2020)"]}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2021,7,1]]},"abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>The unique features of quantum theory offer a powerful new paradigm for information processing. Translating these mathematical abstractions into useful algorithms and applications requires quantum systems with significant complexity and sufficiently low error rates. Such quantum systems must be made from robust hardware that can coherently store, process, and extract the encoded information, as well as possess effective quantum error correction (QEC) protocols to detect and correct errors. Circuit quantum electrodynamics (cQED) provides a promising hardware platform for implementing robust quantum devices. In particular, bosonic encodings in cQED that use multi-photon states of superconducting cavities to encode information have shown success in realizing hardware-efficient QEC. Here, we review recent developments in the theory and implementation of QEC with bosonic codes and report the progress made toward realizing fault-tolerant quantum information processing with cQED devices.</jats:p>","DOI":"10.1088/2058-9565/abe989","type":"journal-article","created":{"date-parts":[[2021,2,24]],"date-time":"2021-02-24T22:31:28Z","timestamp":1614205888000},"page":"033001","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":30,"title":"Quantum information processing with bosonic qubits in circuit QED","prefix":"10.1088","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0003-1445-2086","authenticated-orcid":false,"given":"Atharv","family":"Joshi","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-6318-8472","authenticated-orcid":false,"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-6360-9679","authenticated-orcid":false,"given":"Yvonne Y","family":"Gao","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2021,4,1]]},"reference":[],"container-title":"Quantum Science and Technology","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,19]],"date-time":"2022-01-19T03:08:42Z","timestamp":1642561722000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/abe989"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,1]]},"references-count":246,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,4,1]]},"published-print":{"date-parts":[[2021,7,1]]}},"URL":"http://dx.doi.org/10.1088/2058-9565/abe989","relation":{},"ISSN":["2058-9565"],"subject":["Electrical and Electronic Engineering","Physics and Astronomy (miscellaneous)","Materials Science (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum Sci. 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Published by IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2020-09-01","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2021-02-24","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2021-04-01","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/2058-9565/abe989","_hash":"f7bd44fb534df433e88a770c13e3ceb30164ff8146662519a785c8959990b073"},"expire":1701131165301},"doi:10.1103/PhysRevA.75.042316":{"value":{"indexed":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T22:57:08Z","timestamp":1670021828977},"reference-count":28,"publisher":"American Physical Society (APS)","issue":"4","license":[{"start":{"date-parts":[[2007,4,16]],"date-time":"2007-04-16T00:00:00Z","timestamp":1176681600000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.75.042316","type":"journal-article","created":{"date-parts":[[2007,4,16]],"date-time":"2007-04-16T21:02:25Z","timestamp":1176757345000},"source":"Crossref","is-referenced-by-count":36,"title":"Protecting an optical qubit against photon loss","prefix":"10.1103","volume":"75","author":[{"given":"Wojciech","family":"Wasilewski","sequence":"first","affiliation":[]},{"given":"Konrad","family":"Banaszek","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2007,4,16]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.75.042316","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.75.042316/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,17]],"date-time":"2017-06-17T16:24:55Z","timestamp":1497716695000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.75.042316"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,4,16]]},"references-count":28,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2007,4]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.75.042316","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. 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Despite its simplicity compared with the conventional measurement-based quantum error correction, it is still a far from practical technique because of significant hardware overhead. We propose an autonomous quantum error correction scheme for a rotational symmetric bosonic code in a four-photon Kerr parametric oscillator. Our scheme is the simplest possible error correction scheme that can surpass the break-even point—it requires only a single continuous microwave tone. We also introduce an unconditional reset scheme that requires one more continuous microwave tone in addition to that for the error correction. The key properties underlying this simplicity are protected quasienergy states of a four-photon Kerr parametric oscillator and the degeneracy in its quasienergy level structure. These properties eliminate the need for state-by-state correction in the Fock basis. 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Here we consider qubits encoded with constant space overhead (i.e. finite encoding rate) in the limit of arbitrarily large code distance<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>through the use of topological codes associated to triangulations of hyperbolic surfaces. We introduce explicit protocols to demonstrate how Dehn twists of the hyperbolic surface can be implemented on the code through constant depth unitary circuits, without increasing the space overhead. The circuit for a given Dehn twist consists of a permutation of physical qubits, followed by a constant depth local unitary circuit, where locality here is defined with respect to a hyperbolic metric that defines the code. Applying our results to the hyperbolic Fibonacci Turaev-Viro code implies the possibility of applying universal logical gate sets on encoded qubits through constant depth unitary circuits and with constant space overhead. Our circuits are inherently protected from errors as they map local operators to local operators while changing the size of their support by at most a constant factor; in the presence of noisy syndrome measurements, our results suggest the possibility of universal fault tolerant quantum computation with constant space overhead and time overhead of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>d</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. For quantum circuits that allow parallel gate operations, this yields the optimal scaling of space-time overhead known to date.</jats:p>","DOI":"10.22331/q-2019-08-26-180","type":"journal-article","created":{"date-parts":[[2019,8,26]],"date-time":"2019-08-26T14:31:27Z","timestamp":1566829887000},"page":"180","source":"Crossref","is-referenced-by-count":14,"title":"Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes","prefix":"10.22331","volume":"3","author":[{"given":"Ali","family":"Lavasani","sequence":"first","affiliation":[{"name":"Department of Physics, Condensed Matter Theory Center, University of Maryland, College Park, Maryland 20742, USA and Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA"}]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[{"name":"Department of Physics, Condensed Matter Theory Center, University of Maryland, College Park, Maryland 20742, USA and Joint 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A transistor, despite its roots in energy band structure, does not invoke quantum mechanically coherent transmission between terminals. The impressive progress in the past decade in mesoscopic physics, when combined with studies that have analysed a totally quantum mechanical computational process, suggest that we may be ready to move toward more quantum mechanical procedures for information processing. This paper is a warning signal; this possibility is beset by problems. The case will be made via two separate but complementary arguments. First, by summarizing this author's published comments on computation via totally quantum mechanical coherent Hamiltonians. The computation is likely to suffer from\n            <jats:italic>localization</jats:italic>\n            , i.e. from reflection of the computational trajectory, causing the computation to turn around. Additionally, small errors will accumulate and cause the computation to go off track. This is supplemented by analysis of specific proposals that suggest more detailed machinery than invoked in the general literature on quantum mechanical Hamiltonian computation.\n          </jats:p>","DOI":"10.1098/rsta.1995.0106","type":"journal-article","created":{"date-parts":[[2006,12,15]],"date-time":"2006-12-15T19:15:05Z","timestamp":1166210105000},"page":"367-376","source":"Crossref","is-referenced-by-count":101,"title":"Is quantum mechanics useful?","prefix":"10.1098","volume":"353","member":"175","published-online":{"date-parts":[[1997,1]]},"reference":[],"container-title":"Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences","original-title":[],"language":"en","link":[{"URL":"https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1995.0106","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1995.0106","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,20]],"date-time":"2021-02-20T19:21:17Z","timestamp":1613848877000},"score":1,"resource":{"primary":{"URL":"https://royalsocietypublishing.org/doi/10.1098/rsta.1995.0106"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,12,15]]},"references-count":45,"journal-issue":{"issue":"1703","published-print":{"date-parts":[[1995,12,15]]}},"alternative-id":["10.1098/rsta.1995.0106"],"URL":"http://dx.doi.org/10.1098/rsta.1995.0106","relation":{},"ISSN":["0962-8428","2054-0299"],"subject":["Pharmacology (medical)","Complementary and alternative medicine","Pharmaceutical Science"],"container-title-short":"Phil. Trans. R. Soc. Lond. A","published":{"date-parts":[[1995,12,15]]},"id":"doi:10.1098/rsta.1995.0106","_hash":"a2007c279c67eb3f5e8e2b9527d42ef5ee7bb7f90fb0234b3a7246ddc9ae5068"},"expire":1702831892965},"doi:10.1145/1568318.1568324":{"value":{"indexed":{"date-parts":[[2022,12,21]],"date-time":"2022-12-21T19:31:48Z","timestamp":1671651108260},"reference-count":38,"publisher":"Association for Computing Machinery (ACM)","issue":"6","funder":[{"DOI":"10.13039/501100004965","name":"Sixth Framework Programme","doi-asserted-by":"publisher","award":["15848"]},{"DOI":"10.13039/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["DAAD19-03-1-0082"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2009,9]]},"abstract":"<jats:p>\n            Our main result is a reduction from worst-case lattice problems such as GapSVP and SIVP to a certain learning problem. This learning problem is a natural extension of the “learning from parity with error” problem to higher moduli. It can also be viewed as the problem of decoding from a random linear code. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quantum. Hence, an efficient solution to the learning problem implies a\n            <jats:italic>quantum</jats:italic>\n            algorithm for GapSVP and SIVP. A main open question is whether this reduction can be made classical (i.e., nonquantum).\n          </jats:p>\n          <jats:p>\n            We also present a (classical) public-key cryptosystem whose security is based on the hardness of the learning problem. By the main result, its security is also based on the worst-case quantum hardness of GapSVP and SIVP. The new cryptosystem is much more efficient than previous lattice-based cryptosystems: the public key is of size Õ(\n            <jats:italic>n</jats:italic>\n            <jats:sup>2</jats:sup>\n            ) and encrypting a message increases its size by a factor of Õ(\n            <jats:italic>n</jats:italic>\n            ) (in previous cryptosystems these values are Õ(\n            <jats:italic>n</jats:italic>\n            <jats:sup>4</jats:sup>\n            ) and Õ(\n            <jats:italic>n</jats:italic>\n            <jats:sup>2</jats:sup>\n            ), respectively). In fact, under the assumption that all parties share a random bit string of length Õ(\n            <jats:italic>n</jats:italic>\n            <jats:sup>2</jats:sup>\n            ), the size of the public key can be reduced to Õ(\n            <jats:italic>n</jats:italic>\n            ).\n          </jats:p>","DOI":"10.1145/1568318.1568324","type":"journal-article","created":{"date-parts":[[2009,9,8]],"date-time":"2009-09-08T12:53:03Z","timestamp":1252414383000},"page":"1-40","source":"Crossref","is-referenced-by-count":844,"title":"On lattices, learning with errors, random linear codes, and cryptography","prefix":"10.1145","volume":"56","author":[{"given":"Oded","family":"Regev","sequence":"first","affiliation":[{"name":"Tel Aviv University, Tel Aviv, Israel"}]}],"member":"320","published-online":{"date-parts":[[2009,9,8]]},"reference":[],"container-title":"Journal of the ACM","original-title":[],"language":"en","link":[{"URL":"https://dl.acm.org/doi/pdf/10.1145/1568318.1568324","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,5]],"date-time":"2022-12-05T17:11:00Z","timestamp":1670260260000},"score":1,"resource":{"primary":{"URL":"https://dl.acm.org/doi/10.1145/1568318.1568324"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,9]]},"references-count":38,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2009,9]]}},"alternative-id":["10.1145/1568318.1568324"],"URL":"http://dx.doi.org/10.1145/1568318.1568324","relation":{},"ISSN":["0004-5411","1557-735X"],"subject":["Artificial Intelligence","Hardware and Architecture","Information Systems","Control and Systems Engineering","Software"],"container-title-short":"J. ACM","published":{"date-parts":[[2009,9]]},"id":"doi:10.1145/1568318.1568324","_hash":"63ad40b554ff88226cc2cd4b9579a30f40ccf5937a3fe759a360ae28351ab486"},"expire":1702831894234},"doi:10.1112/blms/19.2.113":{"value":{"indexed":{"date-parts":[[2022,11,16]],"date-time":"2022-11-16T05:54:49Z","timestamp":1668578089412},"reference-count":0,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[1987,3,1]],"date-time":"1987-03-01T00:00:00Z","timestamp":541555200000},"content-version":"tdm","delay-in-days":0,"URL":"http://doi.wiley.com/10.1002/tdm_license_1.1"},{"start":{"date-parts":[[2016,12,23]],"date-time":"2016-12-23T00:00:00Z","timestamp":1482451200000},"content-version":"vor","delay-in-days":10890,"URL":"http://onlinelibrary.wiley.com/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[1987,3]]},"DOI":"10.1112/blms/19.2.113","type":"journal-article","created":{"date-parts":[[2007,2,14]],"date-time":"2007-02-14T02:46:39Z","timestamp":1171421199000},"page":"113-134","source":"Crossref","is-referenced-by-count":143,"title":"From Groups to Groupoids: a Brief Survey","prefix":"10.1112","volume":"19","author":[{"given":"Ronald","family":"Brown","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2016,12,23]]},"container-title":"Bulletin of the London Mathematical Society","original-title":[],"language":"en","link":[{"URL":"https://api.wiley.com/onlinelibrary/tdm/v1/articles/10.1112%2Fblms%2F19.2.113","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://onlinelibrary.wiley.com/wol1/doi/10.1112/blms/19.2.113/fullpdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,20]],"date-time":"2021-07-20T11:22:08Z","timestamp":1626780128000},"score":1,"resource":{"primary":{"URL":"http://doi.wiley.com/10.1112/blms/19.2.113"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1987,3]]},"references-count":0,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1987,3]]}},"URL":"http://dx.doi.org/10.1112/blms/19.2.113","relation":{},"ISSN":["0024-6093"],"subject":["General Mathematics"],"container-title-short":"Bulletin of the London Mathematical Society","published":{"date-parts":[[1987,3]]},"reference":[],"id":"doi:10.1112/blms/19.2.113","_hash":"f37e1d70eb08e42ab422726cc78e280e11dc1fcb612548299c23b2d0d042ebf5"},"expire":1702831895003},"doi:10.1038/nphoton.2014.280":{"value":{"indexed":{"date-parts":[[2022,12,21]],"date-time":"2022-12-21T02:48:06Z","timestamp":1671590886702},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2014,11,17]],"date-time":"2014-11-17T00:00:00Z","timestamp":1416182400000},"content-version":"tdm","delay-in-days":0,"URL":"http://www.springer.com/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2015,1]]},"DOI":"10.1038/nphoton.2014.280","type":"journal-article","created":{"date-parts":[[2014,11,17]],"date-time":"2014-11-17T16:10:48Z","timestamp":1416240648000},"page":"48-53","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":83,"title":"Photon number resolution enables quantum receiver for realistic coherent optical communications","prefix":"10.1038","volume":"9","author":[{"given":"F. E.","family":"Becerra","sequence":"first","affiliation":[]},{"given":"J.","family":"Fan","sequence":"additional","affiliation":[]},{"given":"A.","family":"Migdall","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2014,11,17]]},"reference":[],"container-title":"Nature Photonics","original-title":[],"language":"en","link":[{"URL":"http://www.nature.com/articles/nphoton.2014.280.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/nphoton.2014.280","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/nphoton.2014.280.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T07:03:43Z","timestamp":1638428623000},"score":1,"resource":{"primary":{"URL":"http://www.nature.com/articles/nphoton.2014.280"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,11,17]]},"references-count":45,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2015,1]]}},"alternative-id":["BFnphoton2014280"],"URL":"http://dx.doi.org/10.1038/nphoton.2014.280","relation":{},"ISSN":["1749-4885","1749-4893"],"subject":["Atomic and Molecular Physics, and Optics","Electronic, Optical and Magnetic Materials"],"container-title-short":"Nature Photon","published":{"date-parts":[[2014,11,17]]},"assertion":[{"value":"26 May 2014","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 October 2014","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 November 2014","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing financial interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"id":"doi:10.1038/nphoton.2014.280","_hash":"2fbda0eeb355708ef57f0447af4ac91308df07d069683d5e2d8bef61ef7b1e85"},"expire":1702831896099},"doi:10.1038/s41534-022-00573-9":{"value":{"indexed":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T05:59:32Z","timestamp":1669355972248},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,5,27]],"date-time":"2022-05-27T00:00:00Z","timestamp":1653609600000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,5,27]],"date-time":"2022-05-27T00:00:00Z","timestamp":1653609600000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["ECCS 1927674"]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"<jats:title>Abstract</jats:title><jats:p>We present a systematic study of quantum receivers and modulation methods enabling resource efficient quantum-enhanced optical communication. We introduce quantum-inspired modulation schemes that theoretically yield a better resource efficiency than legacy protocols. Experimentally, we demonstrate below the shot-noise limit symbol error rates for <jats:italic>M</jats:italic> ≤ 16 legacy and quantum-inspired communication alphabets using software-configurable optical communication time-resolving quantum receiver testbed. Further, we experimentally verify that our quantum-inspired modulation schemes boost the accuracy of practical quantum measurements and significantly optimize the combined use of energy and bandwidth for communication alphabets that are longer than <jats:italic>M</jats:italic> = 4 symbols.</jats:p>","DOI":"10.1038/s41534-022-00573-9","type":"journal-article","created":{"date-parts":[[2022,5,27]],"date-time":"2022-05-27T10:03:49Z","timestamp":1653645829000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":"Energy and bandwidth efficiency optimization of quantum-enabled optical communication channels","prefix":"10.1038","volume":"8","author":[{"ORCID":"http://orcid.org/0000-0001-6022-6085","authenticated-orcid":false,"given":"M. V.","family":"Jabir","sequence":"first","affiliation":[]},{"given":"N. Fajar R.","family":"Annafianto","sequence":"additional","affiliation":[]},{"given":"I. A.","family":"Burenkov","sequence":"additional","affiliation":[]},{"given":"A.","family":"Battou","sequence":"additional","affiliation":[]},{"given":"S. V.","family":"Polyakov","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,5,27]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41534-022-00573-9.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-022-00573-9","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-022-00573-9.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T18:50:50Z","timestamp":1669315850000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-022-00573-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,27]]},"references-count":39,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["573"],"URL":"http://dx.doi.org/10.1038/s41534-022-00573-9","relation":{},"ISSN":["2056-6387"],"subject":["Computational Theory and Mathematics","Computer Networks and Communications","Statistical and Nonlinear Physics","Computer Science (miscellaneous)"],"container-title-short":"npj Quantum Inf","published":{"date-parts":[[2022,5,27]]},"assertion":[{"value":"15 September 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 April 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 May 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"63","id":"doi:10.1038/s41534-022-00573-9","_hash":"a5e53a90a575e7b1aa3b46e135e13b50b4f816b39c66d53cace69203b22a8608"},"expire":1702831896946},"doi:10.1364/OSAC.409200":{"value":{"indexed":{"date-parts":[[2022,11,24]],"date-time":"2022-11-24T19:20:10Z","timestamp":1669317610502},"reference-count":29,"publisher":"Optica Publishing Group","issue":"12","license":[{"start":{"date-parts":[[2020,11,19]],"date-time":"2020-11-19T00:00:00Z","timestamp":1605744000000},"content-version":"vor","delay-in-days":0,"URL":"https://doi.org/10.1364/OA_License_v1#VOR-OA"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["ECCS 1927674"]}],"content-domain":{"domain":["opg.optica.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2020,12,15]]},"abstract":"<jats:p>We implement the cyclic quantum receiver based on the theoretical proposal of Roy Bondurant and demonstrate experimentally below the shot-noise limit (SNL) discrimination of quadrature phase-shift keying signals (PSK). We also experimentally test the receiver generalized for longer communication alphabet lengths and coherent frequency shift keying (CFSK) encoding. Using off-the-shelf components, we obtain state discrimination error rates that are 3 dB and 4.6 dB below the SNLs of ideal classical receivers for quadrature PSK and CFSK encodings, respectively. The receiver unconditionally surpasses the SNL for M=8 PSK and CFSK. This receiver can be used for the simple and robust practical implementation of quantum-enhanced optical communication.</jats:p>","DOI":"10.1364/osac.409200","type":"journal-article","created":{"date-parts":[[2020,11,5]],"date-time":"2020-11-05T06:30:06Z","timestamp":1604557806000},"page":"3324","update-policy":"http://dx.doi.org/10.1364/crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":"Experimental demonstration of the near-quantum optimal receiver","prefix":"10.1364","volume":"3","author":[{"ORCID":"http://orcid.org/0000-0001-6022-6085","authenticated-orcid":true,"given":"M. V.","family":"Jabir","sequence":"first","affiliation":[]},{"given":"I. A.","family":"Burenkov","sequence":"additional","affiliation":[]},{"given":"N. Fajar R.","family":"Annafianto","sequence":"additional","affiliation":[]},{"given":"A.","family":"Battou","sequence":"additional","affiliation":[]},{"given":"S. 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Continuum","original-title":[],"language":"en","link":[{"URL":"https://opg.optica.org/viewmedia.cfm?URI=osac-3-12-3324","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://opg.optica.org/viewmedia.cfm?URI=osac-3-12-3324&seq=0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,7,22]],"date-time":"2022-07-22T19:48:02Z","timestamp":1658519282000},"score":1,"resource":{"primary":{"URL":"https://opg.optica.org/abstract.cfm?URI=osac-3-12-3324"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,19]]},"references-count":29,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2020]]},"published-print":{"date-parts":[[2020]]}},"URL":"http://dx.doi.org/10.1364/OSAC.409200","relation":{},"ISSN":["2578-7519"],"subject":["Electrical and Electronic Engineering","Atomic and Molecular Physics, and Optics","Electronic, Optical and 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We present detailed measurement sequences for magic state distillation protocols which can suppress arbitrary errors on any part of a protocol, assuming the independence of errors across qubits. Provided with input magic states, our protocol operates on a two-dimensional square grid by measurements of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi><mml:mi>Z</mml:mi></mml:math> on horizontal pairs of qubits, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi><mml:mi>X</mml:mi></mml:math> on vertical pairs, and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi></mml:math> on single qubits.</jats:p>","DOI":"10.22331/q-2021-01-20-383","type":"journal-article","created":{"date-parts":[[2021,1,20]],"date-time":"2021-01-20T17:24:30Z","timestamp":1611163470000},"page":"383","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Measurement sequences for magic state distillation","prefix":"10.22331","volume":"5","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Microsoft Quantum, Redmond, Washington, USA"}]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"name":"Microsoft Quantum, Santa Barbara, California, USA"},{"name":"Microsoft Quantum, Redmond, Washington, USA"}]}],"member":"9598","published-online":{"date-parts":[[2021,1,20]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2021-01-20-383/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2021,1,20]],"date-time":"2021-01-20T17:24:49Z","timestamp":1611163489000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2021-01-20-383/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,20]]},"references-count":20,"URL":"http://dx.doi.org/10.22331/q-2021-01-20-383","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2021,1,20]]},"article-number":"383","id":"doi:10.22331/q-2021-01-20-383","_hash":"9e886ba8a94589c03d799deacb490f8db5f258691aae229590d791052d1e0712"},"expire":1702831913980},"doi:10.26421/QIC22.11-12-3":{"value":{"indexed":{"date-parts":[[2022,9,28]],"date-time":"2022-09-28T05:40:12Z","timestamp":1664343612453},"reference-count":0,"publisher":"Rinton Press","issue":"11&12","content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"<jats:p>Quantum error correcting codes protect quantum computation from errors caused by decoherence and other noise. Here we study the problem of designing logical operations for quantum error correcting codes. We present an automated procedure that generates logical operations given known encoding and correcting procedures. Our technique is to use variational circuits for learning both the logical gates and the physical operations implementing them. This procedure can be implemented on near-term quantum computers via quantum process tomography. It enables automatic discovery of logical gates from analytically designed error correcting codes and can be extended to error correcting codes found by numerical optimization. We test the procedure by simulating small quantum codes of four to fifteen qubits showing that our procedure finds most logical gates known in the current literature. Additionally, it generates logical gates not found in the current literature for the [[5,1,2]] code, the [[6,3,2]] code, the [[8,3,2]] code, and the [[10,1,2]] code.</jats:p>","DOI":"10.26421/qic22.11-12-3","type":"journal-article","created":{"date-parts":[[2022,9,27]],"date-time":"2022-09-27T03:32:00Z","timestamp":1664249520000},"page":"947-964","source":"Crossref","is-referenced-by-count":0,"title":"Automated discovery of logical gates for quantum error correction (with Supplementary (153 pages))","prefix":"10.26421","volume":"22","author":[{"given":"Hongxiang","family":"Chen","sequence":"first","affiliation":[]},{"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[]},{"given":"Nikolas P.","family":"Breuckmann","sequence":"additional","affiliation":[]},{"given":"Edward","family":"Grant","sequence":"additional","affiliation":[]}],"member":"10955","published-online":{"date-parts":[[2022,8]]},"container-title":"Quantum Information and Computation","original-title":[],"deposited":{"date-parts":[[2022,9,27]],"date-time":"2022-09-27T03:32:05Z","timestamp":1664249525000},"score":1,"resource":{"primary":{"URL":"https://www.rintonpress.com/journals/doi/QIC22.11-12-3.html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8]]},"references-count":0,"journal-issue":{"issue":"11&12","published-online":{"date-parts":[[2022,8]]}},"URL":"http://dx.doi.org/10.26421/QIC22.11-12-3","relation":{},"ISSN":["1533-7146"],"subject":["Computational Theory and Mathematics","General Physics and Astronomy","Mathematical Physics","Nuclear and High Energy Physics","Statistical and Nonlinear Physics","Theoretical Computer Science"],"container-title-short":"QIC","published":{"date-parts":[[2022,8]]},"reference":[],"id":"doi:10.26421/QIC22.11-12-3","_hash":"c1a0a4c1bebb6ab68dfa6cf34961c5909430e9063835f2b54d1731f7b69b45fd"},"expire":1702831915051},"doi:10.1038/s41534-018-0085-z":{"value":{"indexed":{"date-parts":[[2022,12,21]],"date-time":"2022-12-21T05:48:38Z","timestamp":1671601718358},"reference-count":19,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2018,9,12]],"date-time":"2018-09-12T00:00:00Z","timestamp":1536710400000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2018,9,12]],"date-time":"2018-09-12T00:00:00Z","timestamp":1536710400000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1254119","CCF-1254119"]},{"DOI":"10.13039/100000183","name":"DOD | Army Research Office","doi-asserted-by":"publisher","award":["W911NF-12-1-0541","W911NF-12-1-0541"]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"<jats:title>Abstract</jats:title><jats:p>Reliable qubits are difficult to engineer, but standard fault-tolerance schemes use seven or more physical qubits to encode each logical qubit, with still more qubits required for error correction. The large overhead makes it hard to experiment with fault-tolerance schemes with multiple encoded qubits. Here, we study the 15-qubit Hamming code, which protects seven encoded qubits to distance three. We give fault-tolerant procedures for applying arbitrary Clifford operations on these encoded qubits, using only two extra qubits, 17 in total. In particular, individual encoded qubits within the code block can be targeted. Fault-tolerant universal computation is possible with four extra qubits, 19 in total. The procedures could enable testing more sophisticated protected circuits in small-scale quantum devices. Our main technique is to use gadgets to protect gates against correlated faults. We also take advantage of special code symmetries, and use pieceable fault tolerance.</jats:p>","DOI":"10.1038/s41534-018-0085-z","type":"journal-article","created":{"date-parts":[[2018,9,6]],"date-time":"2018-09-06T07:46:02Z","timestamp":1536219962000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":55,"title":"Fault-tolerant quantum computation with few qubits","prefix":"10.1038","volume":"4","author":[{"ORCID":"http://orcid.org/0000-0002-2570-2426","authenticated-orcid":false,"given":"Rui","family":"Chao","sequence":"first","affiliation":[]},{"given":"Ben W.","family":"Reichardt","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,9,12]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41534-018-0085-z.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-018-0085-z","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-018-0085-z.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T23:02:22Z","timestamp":1671577342000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-018-0085-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,12]]},"references-count":19,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,12]]}},"alternative-id":["85"],"URL":"http://dx.doi.org/10.1038/s41534-018-0085-z","relation":{},"ISSN":["2056-6387"],"subject":["Computational Theory and Mathematics","Computer Networks and Communications","Statistical and Nonlinear Physics","Computer Science (miscellaneous)"],"container-title-short":"npj Quantum Inf","published":{"date-parts":[[2018,9,12]]},"assertion":[{"value":"22 December 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 June 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 July 2018","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 September 2018","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"42","id":"doi:10.1038/s41534-018-0085-z","_hash":"d18c301ea384ae1d8742f39e0e24e389f24f829418a207eb6af5903bab464426"},"expire":1702831916071},"doi:10.1103/PhysRevA.54.4741":{"value":{"indexed":{"date-parts":[[2022,12,8]],"date-time":"2022-12-08T15:28:20Z","timestamp":1670513300426},"reference-count":9,"publisher":"American Physical Society (APS)","issue":"6","license":[{"start":{"date-parts":[[1996,12,1]],"date-time":"1996-12-01T00:00:00Z","timestamp":849398400000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.54.4741","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T02:27:51Z","timestamp":1027736871000},"page":"4741-4751","source":"Crossref","is-referenced-by-count":243,"title":"Simple quantum error-correcting codes","prefix":"10.1103","volume":"54","author":[{"given":"A. 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Rev. A","published":{"date-parts":[[1995,2,1]]},"id":"doi:10.1103/PhysRevA.51.992","_hash":"ccc68444bb3a8ec50bd24967f0e63cb70506583bcbcd8be6b2fb11c51aa411e4"},"expire":1702831921029},"doi:10.22331/q-2017-04-25-4":{"value":{"indexed":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T15:25:22Z","timestamp":1668093922603},"reference-count":56,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2017,4,25]],"date-time":"2017-04-25T00:00:00Z","timestamp":1493078400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>We study the fundamental limits on the reliable storage of quantum information in lattices of qubits by deriving tradeoff bounds for approximate quantum error correcting codes. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing various exact error-correction criteria. Our tradeoff bounds relate the number of physical qubits <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>, the number of encoded qubits <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math>, the code distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>, the accuracy parameter <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B4;</mml:mi></mml:math> that quantifies how well the erasure channel can be reversed, and the locality parameter <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x2113;</mml:mi></mml:math> that specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracy <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B4;</mml:mi></mml:math> that is exponentially small in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x2113;</mml:mi></mml:math>, which is the case for perturbations of local commuting projector codes, our bound reads <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi><mml:msup><mml:mi>d</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mfrac><mml:mn>2</mml:mn><mml:mrow><mml:mi>D</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mo>&amp;#x2264;</mml:mo><mml:mi>O</mml:mi><mml:mrow class=\"MJX-TeXAtom-OPEN\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mi>n</mml:mi><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>D</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mrow class=\"MJX-TeXAtom-CLOSE\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math> for codes on <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>D</mml:mi></mml:math>-dimensional lattices of Euclidean metric. We also find that the code distance of any local approximate code cannot exceed <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mrow class=\"MJX-TeXAtom-OPEN\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mi>&amp;#x2113;</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>D</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mrow class=\"MJX-TeXAtom-CLOSE\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math> if <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B4;</mml:mi><mml:mo>&amp;#x2264;</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>&amp;#x2113;</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. As a corollary of our formulation of correctability in terms of logical operator avoidance, we show that the code distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> and the size <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">&amp;#x007E;</mml:mo></mml:mover></mml:mrow></mml:math> of a minimal region that can support all approximate logical operators satisfies <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">&amp;#x007E;</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>D</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mo>&amp;#x2264;</mml:mo><mml:mi>O</mml:mi><mml:mrow class=\"MJX-TeXAtom-OPEN\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mi>n</mml:mi><mml:msup><mml:mi>&amp;#x2113;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mfrac><mml:mi>D</mml:mi><mml:mrow><mml:mi>D</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mrow class=\"MJX-TeXAtom-CLOSE\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math>, where the logical operators are accurate up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mrow class=\"MJX-TeXAtom-OPEN\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mi>&amp;#x03B4;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow class=\"MJX-TeXAtom-CLOSE\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math> in operator norm. Finally, we prove that for two-dimensional systems if logical operators can be approximated by operators supported on constant-width flexible strings, then the dimension of the code space must be bounded. This supports one of the assumptions of algebraic anyon theories, that there exist only finitely many anyon types.</jats:p>","DOI":"10.22331/q-2017-04-25-4","type":"journal-article","created":{"date-parts":[[2017,4,25]],"date-time":"2017-04-25T08:51:31Z","timestamp":1493110291000},"page":"4","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":20,"title":"Limits on the storage of quantum information in a volume of space","prefix":"10.22331","volume":"1","author":[{"given":"Steven T.","family":"Flammia","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Australia"},{"name":"Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, USA"}]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[{"name":"Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington, USA"},{"name":"Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, USA"}]},{"given":"Michael J.","family":"Kastoryano","sequence":"additional","affiliation":[{"name":"NBIA, Niels Bohr Institute, University of Copenhagen, Denmark"}]},{"given":"Isaac H.","family":"Kim","sequence":"additional","affiliation":[{"name":"IBM T. 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Theory","published":{"date-parts":[[1962,9]]},"id":"doi:10.1109/TIT.1962.1057777","_hash":"54de4d46cbdf30478ce3b68fc251ef505b82283b88a3107150e13dd4fc2f9583"},"expire":1703605015298},"doi:10.1007/978-3-642-29011-4_31":{"value":{"indexed":{"date-parts":[[2022,12,30]],"date-time":"2022-12-30T22:16:49Z","timestamp":1672438609517},"publisher-location":"Berlin, Heidelberg","reference-count":31,"publisher":"Springer Berlin Heidelberg","isbn-type":[{"value":"9783642290107","type":"print"},{"value":"9783642290114","type":"electronic"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2012]]},"DOI":"10.1007/978-3-642-29011-4_31","type":"book-chapter","created":{"date-parts":[[2012,4,5]],"date-time":"2012-04-05T07:05:53Z","timestamp":1333609553000},"page":"520-536","source":"Crossref","is-referenced-by-count":147,"title":"Decoding Random Binary Linear Codes in 2 n/20: How 1 + 1 = 0 Improves Information Set Decoding","prefix":"10.1007","author":[{"given":"Anja","family":"Becker","sequence":"first","affiliation":[]},{"given":"Antoine","family":"Joux","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"May","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Meurer","sequence":"additional","affiliation":[]}],"member":"297","reference":[],"container-title":"Advances in Cryptology – EUROCRYPT 2012","original-title":[],"link":[{"URL":"http://link.springer.com/content/pdf/10.1007/978-3-642-29011-4_31.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,5,4]],"date-time":"2021-05-04T11:18:41Z","timestamp":1620127121000},"score":1,"resource":{"primary":{"URL":"http://link.springer.com/10.1007/978-3-642-29011-4_31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012]]},"ISBN":["9783642290107","9783642290114"],"references-count":31,"URL":"http://dx.doi.org/10.1007/978-3-642-29011-4_31","relation":{},"ISSN":["0302-9743","1611-3349"],"published":{"date-parts":[[2012]]},"id":"doi:10.1007/978-3-642-29011-4_31","_hash":"fc1a6fe095d6193f314bf05e6ccbd42c02c58507f59d3103f5d229a31c595df2"},"expire":1703605016343},"doi:10.1007/JHEP09(2019)021":{"value":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T12:04:33Z","timestamp":1648555473999},"reference-count":79,"publisher":"Springer Science and Business Media LLC","issue":"9","license":[{"start":{"date-parts":[[2019,9,1]],"date-time":"2019-09-01T00:00:00Z","timestamp":1567296000000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"},{"start":{"date-parts":[[2019,9,3]],"date-time":"2019-09-03T00:00:00Z","timestamp":1567468800000},"content-version":"vor","delay-in-days":2,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2019,9]]},"abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>Motivated by the close relationship between quantum error-correction, topological order, the holographic AdS/CFT duality, and tensor networks, we initiate the study of approximate quantum error-detecting codes in matrix product states (MPS). We first show that using open-boundary MPS to define boundary to bulk encoding maps yields at most constant distance error-detecting codes. These are degenerate ground spaces of gapped local Hamiltonians. To get around this no-go result, we consider excited states, i.e., we use the excitation ansatz to construct encoding maps: these yield error-detecting codes with distance Ω(<jats:italic>n</jats:italic>\n            <jats:sup>1−ν</jats:sup> ) for any <jats:italic>ν</jats:italic> ∈ (0<jats:italic>,</jats:italic> 1) and Ω(log <jats:italic>n</jats:italic>) encoded qubits. This shows that gapped systems contain — within isolated energy bands — error-detecting codes spanned by momentum eigenstates. We also consider the gapless Heisenberg-XXX model, whose energy eigenstates can be described via Bethe ansatz tensor networks. We show that it contains — within its low-energy eigenspace — an error-detecting code with the same parameter scaling. All these codes detect arbitrary <jats:italic>d</jats:italic>-local (not necessarily geometrically local) errors even though they are not permutation-invariant. This suggests that a wide range of naturally occurring many-body systems possess intrinsic error-detecting features.</jats:p>","DOI":"10.1007/jhep09(2019)021","type":"journal-article","created":{"date-parts":[[2019,9,5]],"date-time":"2019-09-05T15:02:47Z","timestamp":1567695767000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":"Quantum error-detection at low energies","prefix":"10.1007","volume":"2019","author":[{"given":"Martina","family":"Gschwendtner","sequence":"first","affiliation":[]},{"given":"Robert","family":"König","sequence":"additional","affiliation":[]},{"given":"Burak","family":"Şahinoğlu","sequence":"additional","affiliation":[]},{"given":"Eugene","family":"Tang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,9,3]]},"reference":[],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","link":[{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP09(2019)021.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/article/10.1007/JHEP09(2019)021/fulltext.html","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP09(2019)021.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,3]],"date-time":"2021-09-03T18:13:22Z","timestamp":1630692802000},"score":1,"resource":{"primary":{"URL":"https://link.springer.com/10.1007/JHEP09(2019)021"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9]]},"references-count":79,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2019,9]]}},"alternative-id":["11204"],"URL":"http://dx.doi.org/10.1007/JHEP09(2019)021","relation":{},"ISSN":["1029-8479"],"subject":["Nuclear and High Energy Physics"],"container-title-short":"J. High Energ. Phys.","published":{"date-parts":[[2019,9]]},"assertion":[{"value":"14 June 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 August 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 September 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"21","id":"doi:10.1007/JHEP09(2019)021","_hash":"aedf3a33353963fc29b4c62502b6835ca4216df47ebc6fc530e03d7e0bbf540d"},"expire":1703605017398},"doi:10.1007/JHEP01(2022)170":{"value":{"indexed":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T15:28:08Z","timestamp":1668094088262},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,1,27]],"date-time":"2022-01-27T00:00:00Z","timestamp":1643241600000},"content-version":"vor","delay-in-days":26,"URL":"https://creativecommons.org/licenses/by/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2022,1]]},"abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n                     </jats:title><jats:p>There are two parts to this work: first, we study the error correction properties of the real-space renormalization group (RG). The long-distance operators are the (approximately) correctable operators encoded in the physical algebra of short-distance operators. This is closely related to modeling the holographic map as a quantum error correction code. As opposed to holography, the real-space RG of a many-body quantum system does not have the complementary recovery property. We discuss the role of large <jats:italic>N</jats:italic> and a large gap in the spectrum of operators in the emergence of complementary recovery.</jats:p><jats:p>Second, we study the operator algebra exact quantum error correction for any von Neumann algebra. We show that similar to the finite dimensional case, for any error map in between von Neumann algebras the Petz dual of the error map is a recovery map if the inclusion of the correctable subalgebra of operators has finite index.</jats:p>","DOI":"10.1007/jhep01(2022)170","type":"journal-article","created":{"date-parts":[[2022,1,28]],"date-time":"2022-01-28T03:03:27Z","timestamp":1643339007000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":"Real-space RG, error correction and Petz map","prefix":"10.1007","volume":"2022","author":[{"given":"Keiichiro","family":"Furuya","sequence":"first","affiliation":[]},{"given":"Nima","family":"Lashkari","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-4143-5276","authenticated-orcid":false,"given":"Shoy","family":"Ouseph","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,1,27]]},"reference":[],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","link":[{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP01(2022)170.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/article/10.1007/JHEP01(2022)170/fulltext.html","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP01(2022)170.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,25]],"date-time":"2022-04-25T21:26:33Z","timestamp":1650921993000},"score":1,"resource":{"primary":{"URL":"https://link.springer.com/10.1007/JHEP01(2022)170"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1]]},"references-count":48,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,1]]}},"alternative-id":["17662"],"URL":"http://dx.doi.org/10.1007/JHEP01(2022)170","relation":{},"ISSN":["1029-8479"],"subject":["Nuclear and High Energy Physics"],"container-title-short":"J. High Energ. 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Rev. Lett.","published":{"date-parts":[[2007,3,26]]},"article-number":"130501","id":"doi:10.1103/PhysRevLett.98.130501","_hash":"3bebec79db69e3879d3689444dd6cdc5349cd6dab5b517bf6a6b12027ca77cd7"},"expire":1703605019795},"doi:10.22331/q-2022-09-29-821":{"value":{"indexed":{"date-parts":[[2022,9,30]],"date-time":"2022-09-30T05:07:25Z","timestamp":1664514445154},"reference-count":35,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T00:00:00Z","timestamp":1664409600000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"ARO","award":["W911NF-18-1-0020"]},{"name":"ARO","award":["W911NF-18-1-0212"]},{"name":"ARO MURI","award":["W911NF-16-1-0349"]},{"name":"ARO MURI","award":["W911NF-21-1-0325"]},{"name":"AFOSR MURI","award":["FA9550-19-1-0399"]},{"name":"AFOSR MURI","award":["FA9550-21-1-0209"]},{"DOI":"10.13039/100006602","name":"AFRL","doi-asserted-by":"crossref","award":["FA8649-21-P-0781"]},{"name":"NSF","award":["OMA1936118"]},{"name":"NSF","award":["EEC-1941583"]},{"name":"NSF","award":["OMA-2137642"]},{"name":"NTT Research, and the Packard Foundation","award":["2020-71479"]},{"name":"ERC","award":["134847"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Bosonic qubits encoded in continuous-variable systems provide a promising alternative to two-level qubits for quantum computation and communication. So far, photon loss has been the dominant source of errors in bosonic qubits, but the significant reduction of photon loss in recent bosonic qubit experiments suggests that dephasing errors should also be considered. However, a detailed understanding of the combined photon loss and dephasing channel is lacking. Here, we show that, unlike its constituent parts, the combined loss-dephasing channel is non-degradable, pointing towards a richer structure of this channel. We provide bounds for the capacity of the loss-dephasing channel and use numerical optimization to find optimal single-mode codes for a wide range of error rates.</jats:p>","DOI":"10.22331/q-2022-09-29-821","type":"journal-article","created":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T12:26:49Z","timestamp":1664454409000},"page":"821","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Quantum capacity and codes for the bosonic loss-dephasing channel","prefix":"10.22331","volume":"6","author":[{"given":"Peter","family":"Leviant","sequence":"first","affiliation":[{"name":"Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel"}]},{"given":"Qian","family":"Xu","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA"}]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA"}]},{"given":"Serge","family":"Rosenblum","sequence":"additional","affiliation":[{"name":"Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel"}]}],"member":"9598","published-online":{"date-parts":[[2022,9,29]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-09-29-821/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T12:27:53Z","timestamp":1664454473000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-09-29-821/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,29]]},"references-count":35,"URL":"http://dx.doi.org/10.22331/q-2022-09-29-821","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,9,29]]},"article-number":"821","id":"doi:10.22331/q-2022-09-29-821","_hash":"96bec0481798cafaafa0b8232d56dfde88a97b3fa589719107cd558c8b584d64"},"expire":1703605021755},"doi:10.1103/PhysRevD.106.105007":{"value":{"indexed":{"date-parts":[[2022,11,11]],"date-time":"2022-11-11T05:52:54Z","timestamp":1668145974433},"reference-count":40,"publisher":"American Physical Society (APS)","issue":"10","license":[{"start":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T00:00:00Z","timestamp":1668038400000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["PHY-1911298"]},{"DOI":"10.13039/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE- SC0007884"]},{"name":"QuantiSED Fermilab consortium"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevd.106.105007","type":"journal-article","created":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T15:03:08Z","timestamp":1668092588000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":"Renormalization group and approximate error correction","prefix":"10.1103","volume":"106","author":[{"ORCID":"http://orcid.org/0000-0002-1289-6831","authenticated-orcid":true,"given":"Keiichiro","family":"Furuya","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-3446-5933","authenticated-orcid":true,"given":"Nima","family":"Lashkari","sequence":"additional","affiliation":[]},{"given":"Mudassir","family":"Moosa","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2022,11,10]]},"reference":[],"container-title":"Physical Review D","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevD.106.105007","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevD.106.105007/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T15:03:15Z","timestamp":1668092595000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevD.106.105007"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,10]]},"references-count":40,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2022,11]]}},"URL":"http://dx.doi.org/10.1103/PhysRevD.106.105007","relation":{},"ISSN":["2470-0010","2470-0029"],"container-title-short":"Phys. 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We propose a new encryption with a public key matrix by considering the adding of a random distortion matrix over     F  q m      of full column rank n. We show that IND-CPA security is achievable for our encryption under assumption of the Decisional Rank Syndrome Decoding problem. Furthermore, we also prove some bounds for the number of matrices of a fixed rank with entries over a finite field. Our proposal allows the choice of the error terms with rank up to     r 2    , where r is the error-correcting capability of a code. Our encryption based on Gabidulin codes has public key size of     13 . 68     KB, which is 82 times smaller than the public key size of McEliece Cryptosystem based on Goppa codes. For similar post-quantum security level of     2 140     bits, our encryption scheme has a smaller public key size than the key size suggested by LOI17 Encryption.</jats:p>","DOI":"10.3390/cryptography2040032","type":"journal-article","created":{"date-parts":[[2018,10,16]],"date-time":"2018-10-16T06:52:53Z","timestamp":1539672773000},"page":"32","source":"Crossref","is-referenced-by-count":5,"title":"A New Technique in Rank Metric Code-Based Encryption","prefix":"10.3390","volume":"2","author":[{"given":"Terry","family":"Lau","sequence":"first","affiliation":[]},{"given":"Chik","family":"Tan","sequence":"additional","affiliation":[]}],"member":"1968","published-online":{"date-parts":[[2018,10,15]]},"container-title":"Cryptography","original-title":[],"language":"en","link":[{"URL":"http://www.mdpi.com/2410-387X/2/4/32/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2018,10,16]],"date-time":"2018-10-16T06:59:22Z","timestamp":1539673162000},"score":1,"resource":{"primary":{"URL":"http://www.mdpi.com/2410-387X/2/4/32"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,15]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,12]]}},"alternative-id":["cryptography2040032"],"URL":"http://dx.doi.org/10.3390/cryptography2040032","relation":{},"ISSN":["2410-387X"],"subject":["Applied Mathematics","Computational Theory and Mathematics","Computer Networks and Communications","Computer Science Applications","Software"],"container-title-short":"Cryptography","published":{"date-parts":[[2018,10,15]]},"reference":[],"id":"doi:10.3390/cryptography2040032","_hash":"416a9eb7210550bd1f1160406fb074e984d6f72937194e25a1653b156cc34815"},"expire":1705266811932},"doi:10.1088/1367-2630/16/6/063038":{"value":{"indexed":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T16:36:24Z","timestamp":1671554184791},"reference-count":68,"publisher":"IOP Publishing","issue":"6","license":[{"start":{"date-parts":[[2014,6,17]],"date-time":"2014-06-17T00:00:00Z","timestamp":1402963200000},"content-version":"tdm","delay-in-days":0,"URL":"http://iopscience.iop.org/info/page/text-and-data-mining"},{"start":{"date-parts":[[2014,6,17]],"date-time":"2014-06-17T00:00:00Z","timestamp":1402963200000},"content-version":"vor","delay-in-days":0,"URL":"http://creativecommons.org/licenses/by/3.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1088/1367-2630/16/6/063038","type":"journal-article","created":{"date-parts":[[2014,6,17]],"date-time":"2014-06-17T16:55:28Z","timestamp":1403024128000},"page":"063038","source":"Crossref","is-referenced-by-count":43,"title":"Fast decoders for qudit topological codes","prefix":"10.1088","volume":"16","author":[{"given":"Hussain","family":"Anwar","sequence":"first","affiliation":[]},{"given":"Benjamin J","family":"Brown","sequence":"additional","affiliation":[]},{"given":"Earl T","family":"Campbell","sequence":"additional","affiliation":[]},{"given":"Dan E","family":"Browne","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2014,6,17]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"link":[{"URL":"http://stacks.iop.org/1367-2630/16/i=6/a=063038/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://stacks.iop.org/1367-2630/16/i=6/a=063038?key=crossref.cfc8882fe938407cb31511bc503a0006","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http://stacks.iop.org/1367-2630/16/i=6/a=063038/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"http://stacks.iop.org/1367-2630/16/i=6/a=063038?key=crossref.cfc8882fe938407cb31511bc503a0006","content-type":"text/html","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,8]],"date-time":"2022-04-08T15:46:40Z","timestamp":1649432800000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/16/6/063038"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,6,17]]},"references-count":68,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2014,6,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/16/6/063038","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. 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Quantum measurements to discriminate multiple states at the single-photon level are essential for optimizing information transfer in low-power optical communications and quantum communications, and can enhance the capabilities of many quantum information protocols. Here, we theoretically investigate and experimentally demonstrate the discrimination of multiple coherent states of light with sensitivities surpassing the quantum noise limit (QNL) at the single-photon level under realistic conditions of loss and noise based on strategies implementing globally-optimized adaptive measurements with single photon counting and displacement operations. These discrimination strategies can provide realistic advantages to enhance information transfer at low powers, and are compatible with photon number resolving detection, which provides robustness at high powers, thus allowing for surpassing the QNL at arbitrary input power levels under realistic conditions.</jats:p>","DOI":"10.1038/s41534-017-0042-2","type":"journal-article","created":{"date-parts":[[2017,10,9]],"date-time":"2017-10-09T16:04:29Z","timestamp":1507565069000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":"Multi-state discrimination below the quantum noise limit at the single-photon level","prefix":"10.1038","volume":"3","author":[{"given":"A. R.","family":"Ferdinand","sequence":"first","affiliation":[]},{"given":"M. T.","family":"DiMario","sequence":"additional","affiliation":[]},{"given":"F. 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A recent work by Fawzi, Grospellier and Leverrier (FOCS 2018) building on a result by Gottesman (QIC 2013) has shown that the space overhead can be asymptotically reduced to a constant independent of the circuit provided we only consider circuits with a length bounded by a polynomial in the width. In this work, using a minimal model for quantum fault tolerance, we establish a general lower bound on the space overhead required to achieve fault tolerance. \r\nFor any non-unitary qubit channel 𝒩 and any quantum fault tolerance schemes against i.i.d. noise modeled by 𝒩, we prove a lower bound of max{Q(𝒩)^{-1}n,α_𝒩 log T} on the number of physical qubits, for circuits of length T and width n. Here, Q(𝒩) denotes the quantum capacity of 𝒩 and α_𝒩 > 0 is a constant only depending on the channel 𝒩. In our model, we allow for qubits to be replaced by fresh ones during the execution of the circuit and in the case of unital noise, we allow classical computation to be free and perfect. This improves upon results that assumed classical computations to be also affected by noise, and that sometimes did not allow for fresh qubits to be added. Along the way, we prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude damping noise resolving a conjecture by Ben-Or, Gottesman and Hassidim (2013).","container-title":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","DOI":"10.4230/LIPICS.ITCS.2022.68","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","title":"A Lower Bound on the Space Overhead of Fault-Tolerant Quantum Computation","URL":"https://drops.dagstuhl.de/opus/volltexte/2022/15664/","copyright":"Creative Commons Attribution 4.0 International license","reference":[],"_hash":"8d2072ca718b40d6ed2fb41d2639e0d4f5ab8c9bdacc01ade7c83a8eb6094767"},"expire":1708162723400},"doi:10.22331/q-2022-10-06-828":{"value":{"indexed":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T11:10:18Z","timestamp":1675249818437},"reference-count":202,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,10,6]],"date-time":"2022-10-06T00:00:00Z","timestamp":1665014400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/501100001870","name":"the Foundation for Polish Science","doi-asserted-by":"crossref","award":["MAB/2018/5"]},{"name":"General Research Fund","award":["GRF/16305121"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Quantum error correction is believed to be a necessity for large-scale fault-tolerant quantum computation. In the past two decades, various constructions of quantum error-correcting codes (QECCs) have been developed, leading to many good code families. However, the majority of these codes are not suitable for near-term quantum devices. Here we present VarQEC, a noise-resilient variational quantum algorithm to search for quantum codes with a hardware-efficient encoding circuit. The cost functions are inspired by the most general and fundamental requirements of a QECC, the Knill-Laflamme conditions. Given the target noise channel (or the target code parameters) and the hardware connectivity graph, we optimize a shallow variational quantum circuit to prepare the basis states of an eligible code. In principle, VarQEC can find quantum codes for any error model, whether additive or non-additive, degenerate or non-degenerate, pure or impure. We have verified its effectiveness by (re)discovering some symmetric and asymmetric codes, e.g., <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>n</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math> from 7 to 14. We also found new <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>6</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>7</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> codes that are not equivalent to any stabilizer code, and extensive numerical evidence with VarQEC suggests that a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>7</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> code does not exist. Furthermore, we found many new channel-adaptive codes for error models involving nearest-neighbor correlated errors. Our work sheds new light on the understanding of QECC in general, which may also help to enhance near-term device performance with channel-adaptive error-correcting codes.</jats:p>","DOI":"10.22331/q-2022-10-06-828","type":"journal-article","created":{"date-parts":[[2022,10,6]],"date-time":"2022-10-06T13:21:02Z","timestamp":1665062462000},"page":"828","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":"Quantum variational learning for quantum error-correcting codes","prefix":"10.22331","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0001-5589-7503","authenticated-orcid":false,"given":"Chenfeng","family":"Cao","sequence":"first","affiliation":[{"name":"Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China"}]},{"ORCID":"http://orcid.org/0000-0002-2093-7496","authenticated-orcid":false,"given":"Chao","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China"}]},{"ORCID":"http://orcid.org/0000-0002-9349-1325","authenticated-orcid":false,"given":"Zipeng","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China"}]},{"ORCID":"http://orcid.org/0000-0002-3720-5195","authenticated-orcid":false,"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University of Gdansk, 80-309 Gdansk, Poland"}]},{"ORCID":"http://orcid.org/0000-0003-3989-4948","authenticated-orcid":false,"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[{"name":"Department of 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The Epstein zeta function of <jats:italic>f</jats:italic> is denned to be</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0017089500007047_eqnU1\" /></jats:disp-formula></jats:p><jats:p>Rankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s &gt; 0,</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0017089500007047_eqn1\" /></jats:disp-formula></jats:p><jats:p>We prove a corresponding result for theta functions. 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Leveraging such protection requires testing its robustness against uncontrolled environmental interactions. Using 47 superconducting qubits, we implement the one-dimensional kicked Ising model, which exhibits nonlocal Majorana edge modes (MEMs) with\n            <jats:inline-formula>\n              <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n                <mml:mrow>\n                  <mml:msub>\n                    <mml:mi>ℤ</mml:mi>\n                    <mml:mn>2</mml:mn>\n                  </mml:msub>\n                </mml:mrow>\n              </mml:math>\n            </jats:inline-formula>\n            parity symmetry. We find that any multiqubit Pauli operator overlapping with the MEMs exhibits a uniform late-time decay rate comparable to single-qubit relaxation rates, irrespective of its size or composition. This characteristic allows us to accurately reconstruct the exponentially localized spatial profiles of the MEMs. Furthermore, the MEMs are found to be resilient against certain symmetry-breaking noise owing to a prethermalization mechanism. Our work elucidates the complex interplay between noise and symmetry-protected edge modes in a solid-state environment.\n          </jats:p>","DOI":"10.1126/science.abq5769","type":"journal-article","created":{"date-parts":[[2022,11,17]],"date-time":"2022-11-17T18:58:33Z","timestamp":1668711513000},"page":"785-790","source":"Crossref","is-referenced-by-count":3,"title":"Noise-resilient edge modes on a chain of superconducting qubits","prefix":"10.1126","volume":"378","author":[{"ORCID":"http://orcid.org/0000-0003-0507-0211","authenticated-orcid":true,"given":"X.","family":"Mi","sequence":"first","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"http://orcid.org/0000-0002-9003-5914","authenticated-orcid":true,"given":"M.","family":"Sonner","sequence":"additional","affiliation":[{"name":"Department of Theoretical Physics, University of Geneva, Geneva, Switzerland."}]},{"given":"M. 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Unfortunately, their exchange that can reveal their exotic statistics needs manipulations that are still beyond our experimental capabilities. Here we take an alternative approach. Through the Jordan–Wigner transformation, the Kitaev's chain supporting two Majorana zero modes is mapped to the spin-1/2 chain. We experimentally simulated the spin system and its evolution with a photonic quantum simulator. This allows us to probe the geometric phase, which corresponds to the exchange of two Majorana zero modes positioned at the ends of a three-site chain. Finally, we demonstrate the immunity of quantum information encoded in the Majorana zero modes against local errors through the simulator. Our photonic simulator opens the way for the efficient realization and manipulation of Majorana zero modes in complex architectures.</jats:p>","DOI":"10.1038/ncomms13194","type":"journal-article","created":{"date-parts":[[2016,10,25]],"date-time":"2016-10-25T09:37:45Z","timestamp":1477388265000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":"Simulating the exchange of Majorana zero modes with a photonic system","prefix":"10.1038","volume":"7","author":[{"given":"Jin-Shi","family":"Xu","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-3018-7219","authenticated-orcid":false,"given":"Kai","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Yong-Jian","family":"Han","sequence":"additional","affiliation":[]},{"given":"Chuan-Feng","family":"Li","sequence":"additional","affiliation":[]},{"given":"Jiannis K.","family":"Pachos","sequence":"additional","affiliation":[]},{"given":"Guang-Can","family":"Guo","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2016,10,25]]},"reference":[],"container-title":"Nature 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Call a configuration sharp if there are <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m\">\n  <mml:semantics>\n    <mml:mi>m</mml:mi>\n    <mml:annotation encoding=\"application/x-tex\">m</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> distances between distinct points in it and it is a spherical <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis 2 m minus 1 right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo stretchy=\"false\">(</mml:mo>\n      <mml:mn>2</mml:mn>\n      <mml:mi>m</mml:mi>\n      <mml:mo>−<!-- − --></mml:mo>\n      <mml:mn>1</mml:mn>\n      <mml:mo stretchy=\"false\">)</mml:mo>\n    </mml:mrow>\n    <mml:annotation encoding=\"application/x-tex\">(2m-1)</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula>-design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E 8\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>E</mml:mi>\n      <mml:mn>8</mml:mn>\n    </mml:msub>\n    <mml:annotation encoding=\"application/x-tex\">E_8</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> and Leech lattices. We also prove the same result for the vertices of the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"600\">\n  <mml:semantics>\n    <mml:mn>600</mml:mn>\n    <mml:annotation encoding=\"application/x-tex\">600</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula>-cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact two-point homogeneous spaces, and we conclude with an extension to Euclidean space.</p>","DOI":"10.1090/s0894-0347-06-00546-7","type":"journal-article","created":{"date-parts":[[2006,10,14]],"date-time":"2006-10-14T01:46:17Z","timestamp":1160790377000},"page":"99-148","source":"Crossref","is-referenced-by-count":193,"title":"Universally optimal distribution of points on spheres","prefix":"10.1090","volume":"20","author":[{"given":"Henry","family":"Cohn","sequence":"first","affiliation":[]},{"given":"Abhinav","family":"Kumar","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,9,5]]},"reference":[],"container-title":"Journal of the American Mathematical Society","original-title":[],"language":"en","link":[{"URL":"http://www.ams.org/jams/2007-20-01/S0894-0347-06-00546-7/S0894-0347-06-00546-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://www.ams.org/jams/2007-20-01/S0894-0347-06-00546-7/S0894-0347-06-00546-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,29]],"date-time":"2021-07-29T21:43:35Z","timestamp":1627595015000},"score":1,"resource":{"primary":{"URL":"https://www.ams.org/jams/2007-20-01/S0894-0347-06-00546-7/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,9,5]]},"references-count":62,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["S0894-0347-06-00546-7"],"URL":"http://dx.doi.org/10.1090/S0894-0347-06-00546-7","relation":{},"ISSN":["0894-0347","1088-6834"],"subject":["Applied Mathematics","General Mathematics"],"container-title-short":"J. 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Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.","name":"license_information","label":"License information"},{"value":"2018-11-13","name":"date_received","label":"Date received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2019-01-24","name":"date_accepted","label":"Date accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2019-03-28","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/ab0199","_hash":"78dd9a0055ffd1b60a52c554f811ecc329ca1a08bb6826747c436dcd84f340f9"},"expire":1712098433990},"doi:10.1103/PhysRevA.107.042407":{"value":{"indexed":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T05:13:14Z","timestamp":1680844394312},"reference-count":54,"publisher":"American Physical Society (APS)","issue":"4","license":[{"start":{"date-parts":[[2023,4,6]],"date-time":"2023-04-06T00:00:00Z","timestamp":1680739200000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"funder":[{"name":"Plan France 2030","award":["ANR-22-PETQ-0006"]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physreva.107.042407","type":"journal-article","created":{"date-parts":[[2023,4,6]],"date-time":"2023-04-06T14:03:55Z","timestamp":1680789835000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":"Two-photon driven Kerr quantum oscillator with multiple spectral degeneracies","prefix":"10.1103","volume":"107","author":[{"ORCID":"http://orcid.org/0000-0003-2003-7030","authenticated-orcid":true,"given":"Diego","family":"Ruiz","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-6136-6688","authenticated-orcid":true,"given":"Ronan","family":"Gautier","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-6507-9344","authenticated-orcid":true,"given":"Jérémie","family":"Guillaud","sequence":"additional","affiliation":[]},{"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2023,4,6]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevA.107.042407","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.107.042407/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,6]],"date-time":"2023-04-06T14:04:30Z","timestamp":1680789870000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.107.042407"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,6]]},"references-count":54,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,4]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.107.042407","relation":{},"ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. 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Sloan Foundation","doi-asserted-by":"publisher","award":["BR2013-049"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2018,7,20]]},"abstract":"<jats:title>Fault-tolerant quantum coding</jats:title>\n          <jats:p>\n            Noise and imperfections in a quantum system can result in the presence and propagation of errors through the system. A reliable quantum processor will need to be able to correct for these errors and error syndromes. Rosenblum\n            <jats:italic>et al.</jats:italic>\n            used higher quantum states of a superconducting-based quantum circuit to demonstrate a method for the fault-tolerant measurement of an error-correctable logical qubit. Such fault-tolerant measurements will allow more frequent interrogations of the state of the logical qubit, ultimately leading to the implementation of more quantum operations and more complex entangled quantum circuits.\n          </jats:p>\n          <jats:p>\n            <jats:italic>Science</jats:italic>\n            , this issue p.\n            <jats:related-article xmlns:xlink=\"http://www.w3.org/1999/xlink\" ext-link-type=\"doi\" issue=\"6399\" page=\"266\" related-article-type=\"in-this-issue\" vol=\"361\" xlink:href=\"10.1126/science.aat3996\">266</jats:related-article>\n          </jats:p>","DOI":"10.1126/science.aat3996","type":"journal-article","created":{"date-parts":[[2018,7,19]],"date-time":"2018-07-19T18:05:33Z","timestamp":1532023533000},"page":"266-270","source":"Crossref","is-referenced-by-count":89,"title":"Fault-tolerant detection of a quantum error","prefix":"10.1126","volume":"361","author":[{"given":"S.","family":"Rosenblum","sequence":"first","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]},{"ORCID":"http://orcid.org/0000-0002-8141-1842","authenticated-orcid":true,"given":"P.","family":"Reinhold","sequence":"additional","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]},{"given":"M.","family":"Mirrahimi","sequence":"additional","affiliation":[{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."},{"name":"QUANTIC team, INRIA de Paris, 2 Rue Simone Iff, 75012 Paris, France."}]},{"ORCID":"http://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]},{"ORCID":"http://orcid.org/0000-0002-0272-5481","authenticated-orcid":true,"given":"L.","family":"Frunzio","sequence":"additional","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]},{"ORCID":"http://orcid.org/0000-0002-2469-5068","authenticated-orcid":true,"given":"R. J.","family":"Schoelkopf","sequence":"additional","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]}],"member":"221","reference":[],"container-title":"Science","original-title":[],"language":"en","link":[{"URL":"https://syndication.highwire.org/content/doi/10.1126/science.aat3996","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"https://syndication.highwire.org/content/doi/10.1126/science.aat3996","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,13]],"date-time":"2022-01-13T20:57:29Z","timestamp":1642107449000},"score":1,"resource":{"primary":{"URL":"https://www.science.org/doi/10.1126/science.aat3996"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,20]]},"references-count":23,"journal-issue":{"issue":"6399","published-print":{"date-parts":[[2018,7,20]]}},"alternative-id":["10.1126/science.aat3996"],"URL":"http://dx.doi.org/10.1126/science.aat3996","relation":{},"ISSN":["0036-8075","1095-9203"],"subject":["Multidisciplinary"],"container-title-short":"Science","published":{"date-parts":[[2018,7,20]]},"id":"doi:10.1126/science.aat3996","_hash":"5f11a35a2b98324879304d709db301d9a23e5e9fe357582f5bf9bd71a8cf7244"},"expire":1712098435958},"doi:10.1038/nature13436":{"value":{"indexed":{"date-parts":[[2023,3,28]],"date-time":"2023-03-28T13:51:26Z","timestamp":1680011486778},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"7510","license":[{"start":{"date-parts":[[2014,7,1]],"date-time":"2014-07-01T00:00:00Z","timestamp":1404172800000},"content-version":"tdm","delay-in-days":0,"URL":"http://www.springer.com/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"published-print":{"date-parts":[[2014,7]]},"DOI":"10.1038/nature13436","type":"journal-article","created":{"date-parts":[[2014,7,11]],"date-time":"2014-07-11T14:48:46Z","timestamp":1405090126000},"page":"444-448","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":175,"title":"Tracking photon jumps with repeated quantum non-demolition parity measurements","prefix":"10.1038","volume":"511","author":[{"given":"L.","family":"Sun","sequence":"first","affiliation":[]},{"given":"A.","family":"Petrenko","sequence":"additional","affiliation":[]},{"given":"Z.","family":"Leghtas","sequence":"additional","affiliation":[]},{"given":"B.","family":"Vlastakis","sequence":"additional","affiliation":[]},{"given":"G.","family":"Kirchmair","sequence":"additional","affiliation":[]},{"given":"K. 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Typical measurement paradigms for state discrimination involve a minimum probability of error or unambiguous discrimination with a minimum probability of inconclusive results. Alternatively, an optimal inconclusive measurement, a non-projective measurement, achieves minimal error for a given inconclusive probability. This more general measurement encompasses the standard measurement paradigms for state discrimination and provides a much more powerful tool for quantum information and communication. Here, we experimentally demonstrate the optimal inconclusive measurement for the discrimination of binary coherent states using linear optics and single-photon detection. Our demonstration uses coherent displacement operations based on interference, single-photon detection, and fast feedback to prepare the optimal feedback policy for the optimal non-projective quantum measurement with high fidelity. This generalized measurement allows us to transition among standard measurement paradigms in an optimal way from minimum error to unambiguous measurements for binary coherent states. As a particular case, we use this general measurement to implement the optimal minimum error measurement for phase-coherent states, which is the optimal modulation for communications under the average power constraint. Moreover, we propose a hybrid measurement that leverages the binary optimal inconclusive measurement in conjunction with sequential, unambiguous state elimination to realize higher dimensional inconclusive measurements of coherent states.</jats:p>","DOI":"10.1038/s41534-022-00595-3","type":"journal-article","created":{"date-parts":[[2022,7,18]],"date-time":"2022-07-18T11:03:02Z","timestamp":1658142182000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":"Demonstration of optimal non-projective measurement of binary coherent states with photon counting","prefix":"10.1038","volume":"8","author":[{"given":"M. T.","family":"DiMario","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-2928-310X","authenticated-orcid":false,"given":"F. 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Most of the new packings are nonlattice packings. These new packings increase the previously greatest known numbers of spheres which one sphere may touch, and, except in dimensions 9, 12, 14, 15, they include denser packings than any previously known. The density Δ of the packings in <jats:italic>E<jats:sup>n</jats:sup></jats:italic> for <jats:italic>n</jats:italic> = <jats:italic>2<jats:sup>m</jats:sup></jats:italic> satisfies</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0008414X00053724_eqn1\" /></jats:disp-formula></jats:p><jats:p>In this paper we make systematic use of error-correcting codes to obtain sphere packings in <jats:italic>E<jats:sup>n</jats:sup>,</jats:italic> including several of the densest packings known and several new packings.</jats:p>","DOI":"10.4153/cjm-1971-081-3","type":"journal-article","created":{"date-parts":[[2010,12,7]],"date-time":"2010-12-07T23:24:44Z","timestamp":1291764284000},"page":"718-745","source":"Crossref","is-referenced-by-count":150,"title":"Sphere Packings and Error-Correcting Codes","prefix":"10.4153","volume":"23","author":[{"given":"John","family":"Leech","sequence":"first","affiliation":[]},{"given":"N. 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A famous necessary,\nbut not sufficient, condition for the existence of a topological\nboundary condition is that the chiral central charge\n<jats:inline-formula><jats:alternatives><jats:tex-math>c_-</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msub><mml:mi>c</mml:mi><mml:mo>−</mml:mo></mml:msub></mml:math></jats:alternatives></jats:inline-formula>\nhas to vanish. In this paper, we consider conditions associated with\n``higher\" central charges, which have been introduced recently in\nthe math literature. In terms of these new obstructions, we identify\nnecessary and sufficient conditions for the existence of a topological\nboundary in the case of bosonic, Abelian TQFTs, providing an alternative\nto the identification of a Lagrangian subgroup. Our proof relies on\ngeneral aspects of gauging generalized global symmetries. For\nnon-Abelian TQFTs, we give a geometric way of studying topological\nboundary conditions, and explain certain necessary conditions given\nagain in terms of the higher central charges. Along the way, we find a\ncurious duality in the partition functions of Abelian TQFTs, which begs\nfor an explanation via the 3d-3d correspondence.</jats:p>","DOI":"10.21468/scipostphys.13.3.067","type":"journal-article","created":{"date-parts":[[2022,9,26]],"date-time":"2022-09-26T07:33:00Z","timestamp":1664177580000},"update-policy":"http://dx.doi.org/10.21468/scipost.crossmarkpolicy","source":"Crossref","is-referenced-by-count":7,"title":"Higher central charges and topological boundaries in 2+1-dimensional  TQFTs","prefix":"10.21468","volume":"13","author":[{"given":"Justin","family":"Kaidi","sequence":"first","affiliation":[{"name":"Stony Brook University"}]},{"given":"Zohar","family":"Komargodski","sequence":"additional","affiliation":[{"name":"Stony Brook University"}]},{"given":"Kantaro","family":"Ohmori","sequence":"additional","affiliation":[{"name":"University of Tokyo"},{"name":"Stony Brook University"}]},{"given":"Sahand","family":"Seifnashri","sequence":"additional","affiliation":[{"name":"Stony Brook University"}]},{"given":"Shu-Heng","family":"Shao","sequence":"additional","affiliation":[{"name":"C.N. 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It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a <jats:italic>K</jats:italic>-matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.</jats:p>","DOI":"10.1088/1751-8121/abc6c0","type":"journal-article","created":{"date-parts":[[2020,11,2]],"date-time":"2020-11-02T22:15:47Z","timestamp":1604355347000},"page":"505203","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":"In and around abelian anyon models\n                  <sup>*</sup>","prefix":"10.1088","volume":"53","author":[{"given":"Liang","family":"Wang","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-5253-6400","authenticated-orcid":false,"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2020,11,24]]},"reference":[],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0","content-type":"text/html","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,5]],"date-time":"2022-01-05T10:35:20Z","timestamp":1641378920000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,18]]},"references-count":32,"journal-issue":{"issue":"50","published-online":{"date-parts":[[2020,11,24]]},"published-print":{"date-parts":[[2020,11,18]]}},"URL":"http://dx.doi.org/10.1088/1751-8121/abc6c0","relation":{},"ISSN":["1751-8113","1751-8121"],"subject":["General Physics and Astronomy","Mathematical Physics","Modeling and Simulation","Statistics and Probability","Statistical and Nonlinear Physics"],"container-title-short":"J. 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The notion is a\ncondition on the ground state subspace, rather than on the Hamiltonian,\nand demands that given a collection of ball-like regions, any linear\ntransformation on the ground space be realized by an operator that\navoids the ball-like regions. We derive a bound on the ground state\ndegeneracy <jats:inline-formula><jats:alternatives><jats:tex-math>\\mathcal D</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mstyle mathvariant=\"script\"><mml:mi>𝒟</mml:mi></mml:mstyle></mml:math></jats:alternatives></jats:inline-formula>\nfor systems with homogeneous topological order on an arbitrary closed\nRiemannian manifold of dimension <jats:inline-formula><jats:alternatives><jats:tex-math>d</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>d</mml:mi></mml:math></jats:alternatives></jats:inline-formula>,\nwhich reads [ D c (L/a)^{d-2}.] Here, <jats:inline-formula><jats:alternatives><jats:tex-math>L</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>L</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\nis the diameter of the system, <jats:inline-formula><jats:alternatives><jats:tex-math>a</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>a</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\nis the lattice spacing, and <jats:inline-formula><jats:alternatives><jats:tex-math>c</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>c</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\nis a constant that only depends on the isometry class of the manifold,\nand <jats:inline-formula><jats:alternatives><jats:tex-math>\\mu</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>μ</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\nis a constant that only depends on the density of degrees of freedom. If\n<jats:inline-formula><jats:alternatives><jats:tex-math>d=2</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>,\nthe constant <jats:inline-formula><jats:alternatives><jats:tex-math>c</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>c</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\nis the (demi)genus of the space manifold. This bound is saturated up to\nconstants by known examples.examples.</jats:p>","DOI":"10.21468/scipostphys.10.1.011","type":"journal-article","created":{"date-parts":[[2021,1,20]],"date-time":"2021-01-20T08:06:32Z","timestamp":1611129992000},"update-policy":"http://dx.doi.org/10.21468/scipost.crossmarkpolicy","source":"Crossref","is-referenced-by-count":6,"title":"A degeneracy bound for homogeneous topological order","prefix":"10.21468","volume":"10","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Microsoft"}]}],"member":"8907","published-online":{"date-parts":[[2021,1,20]]},"reference":[],"container-title":"SciPost Physics","original-title":[],"link":[{"URL":"https://scipost.org/10.21468/SciPostPhys.10.1.011/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://scipost.org/10.21468/SciPostPhys.10.1.011/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,1,20]],"date-time":"2021-01-20T08:06:34Z","timestamp":1611129994000},"score":1,"resource":{"primary":{"URL":"https://scipost.org/10.21468/SciPostPhys.10.1.011"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,20]]},"references-count":28,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021]]}},"URL":"http://dx.doi.org/10.21468/SciPostPhys.10.1.011","relation":{"has-review":[{"id-type":"doi","id":"10.21468/SciPost.Report.2200","asserted-by":"object"},{"id-type":"doi","id":"10.21468/SciPost.Report.2127","asserted-by":"object"}]},"ISSN":["2542-4653"],"subject":["General Physics and Astronomy"],"container-title-short":"SciPost Phys.","published":{"date-parts":[[2021,1,20]]},"article-number":"011","id":"doi:10.21468/SciPostPhys.10.1.011","_hash":"f05f4e6399713056c1fb3fb92af95651b64787ab4edcff7f7ea37b36425f6541"},"expire":1712098559773},"doi:10.1103/PhysRevB.107.085134":{"value":{"indexed":{"date-parts":[[2023,2,22]],"date-time":"2023-02-22T05:27:38Z","timestamp":1677043658020},"reference-count":36,"publisher":"American Physical Society (APS)","issue":"8","license":[{"start":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T00:00:00Z","timestamp":1676937600000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"funder":[{"DOI":"10.13039/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevb.107.085134","type":"journal-article","created":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T16:39:53Z","timestamp":1676997593000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":"Disentangling modular Walker-Wang models via fermionic invertible boundaries","prefix":"10.1103","volume":"107","author":[{"ORCID":"http://orcid.org/0000-0002-4881-2003","authenticated-orcid":true,"given":"Andreas","family":"Bauer","sequence":"first","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2023,2,21]]},"reference":[],"container-title":"Physical Review B","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevB.107.085134","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevB.107.085134/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T16:41:01Z","timestamp":1676997661000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevB.107.085134"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,21]]},"references-count":36,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2023,2]]}},"URL":"http://dx.doi.org/10.1103/PhysRevB.107.085134","relation":{},"ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. 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Finally, the opportunities offered for quantum information processing by employing the cavity as a quantum bus are presented.</jats:p>","DOI":"10.1093/acprof:oso/9780199681181.003.0003","type":"book-chapter","created":{"date-parts":[[2014,9,18]],"date-time":"2014-09-18T07:05:22Z","timestamp":1411023922000},"page":"113-256","source":"Crossref","is-referenced-by-count":22,"title":"Circuit QED: superconducting qubits coupled to microwave photons","prefix":"10.1093","author":[{"given":"S. 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Here we propose a cellular automaton decoder for a variation of the color code where the bases of the physical qubits are locally rotated, which we call the XYZ color code. The local transformation means our decoder demonstrates key properties of a two-dimensional fractal code if the noise acting on the system is infinitely biased towards dephasing, namely, no string-like logical operators. As such, in the high-bias limit, our local decoder reproduces the behavior of a partially self-correcting memory. At low error rates, our simulations show that the memory time diverges polynomially with system size without intervention from a global decoder, up to some critical system size that grows as the error rate is lowered. Furthermore, although we find that we cannot reproduce partially self-correcting behavior at finite bias, our numerics demonstrate improved memory times at realistic noise biases. Our results therefore motivate the design of tailored cellular automaton decoders that help to reduce the bandwidth demands of global decoding for realistic noise models.</jats:p>","DOI":"10.22331/q-2023-03-09-940","type":"journal-article","created":{"date-parts":[[2023,3,9]],"date-time":"2023-03-09T12:21:56Z","timestamp":1678364516000},"page":"940","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"A cellular automaton decoder for a noise-bias tailored color code","prefix":"10.22331","volume":"7","author":[{"given":"Jonathan F. San","family":"Miguel","sequence":"first","affiliation":[{"name":"Department of Physics, Stanford University, Stanford, CA 94305"}]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[{"name":"Department of Physics, Stanford University, Stanford, CA 94305"},{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia"}]},{"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, 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(APS)","issue":"2","license":[{"start":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T00:00:00Z","timestamp":1680825600000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/prxquantum.4.020303","type":"journal-article","created":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T15:02:15Z","timestamp":1680879735000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":"Logical Blocks for Fault-Tolerant Topological Quantum Computation","prefix":"10.1103","volume":"4","author":[{"ORCID":"http://orcid.org/0000-0001-6728-5002","authenticated-orcid":true,"given":"Héctor","family":"Bombín","sequence":"first","affiliation":[]},{"given":"Chris","family":"Dawson","sequence":"additional","affiliation":[]},{"given":"Ryan 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The most interesting simulations will require a fault-tolerant quantum computer, and building such a device remains a long-term goal. However, the capabilities of existing noisy quantum processors have steadily improved, sparking an interest in running simulations that, while not necessarily classically intractable, may serve as device benchmarks and help elucidate the challenges to achieving practical applications on near-term devices. Systems of <jats:italic>non</jats:italic>-interacting fermions are ideally suited to serve these purposes. While they display rich physics and generate highly entangled states when simulated on a quantum processor, their classical tractability enables experimental results to be verified even at large system sizes that would typically defy classical simulation. In this work, we use a noisy superconducting quantum processor to prepare Majorana zero modes (MZMs) as eigenstates of the Kitaev chain Hamiltonian, a model of non-interacting fermions. Our work builds on previous experiments with non-interacting fermionic systems. Previous work demonstrated error mitigation techniques applicable to the special case of Slater determinants. Here, we show how to extend these techniques to the case of general fermionic Gaussian states, and demonstrate them by preparing MZMs on systems of up to seven qubits.</jats:p>","DOI":"10.1088/2058-9565/acb796","type":"journal-article","created":{"date-parts":[[2023,1,31]],"date-time":"2023-01-31T22:30:26Z","timestamp":1675204226000},"page":"025010","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":"Simulating Majorana zero modes on a noisy quantum processor","prefix":"10.1088","volume":"8","author":[{"ORCID":"http://orcid.org/0000-0001-6459-6374","authenticated-orcid":true,"given":"Kevin J","family":"Sung","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4077-9744","authenticated-orcid":true,"given":"Marko J","family":"Rančić","sequence":"additional","affiliation":[]},{"given":"Olivia T","family":"Lanes","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-6731-6814","authenticated-orcid":true,"given":"Nicholas T","family":"Bronn","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2023,2,9]]},"reference":[],"container-title":"Quantum Science and Technology","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,9]],"date-time":"2023-02-09T13:27:13Z","timestamp":1675949233000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acb796"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,9]]},"references-count":41,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2,9]]},"published-print":{"date-parts":[[2023,4,1]]}},"URL":"http://dx.doi.org/10.1088/2058-9565/acb796","relation":{},"ISSN":["2058-9565"],"subject":["Electrical and Electronic Engineering","Physics and Astronomy (miscellaneous)","Materials Science (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum Sci. Technol.","published":{"date-parts":[[2023,2,9]]},"assertion":[{"value":"Simulating Majorana zero modes on a noisy quantum processor","name":"article_title","label":"Article Title"},{"value":"Quantum Science and Technology","name":"journal_title","label":"Journal Title"},{"value":"paper","name":"article_type","label":"Article Type"},{"value":"© 2023 IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2022-08-01","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2023-01-31","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2023-02-09","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/2058-9565/acb796","_hash":"49e07071ad2302deba5a24ecebfc6e2ce3f517179d21b5266a165a3db760a087"},"expire":1712590663884},"doi:10.22331/q-2023-03-09-942":{"value":{"indexed":{"date-parts":[[2023,3,10]],"date-time":"2023-03-10T05:51:13Z","timestamp":1678427473305},"reference-count":99,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,3,9]],"date-time":"2023-03-09T00:00:00Z","timestamp":1678320000000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"European Research Council","award":["804247"]},{"name":"European Comission, Horizon 2020","award":["820495"]},{"DOI":"10.13039/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"crossref","award":["449905436"]},{"name":"US A.R.O.","award":["W911NF-21-1-0007"]},{"name":"US A.R.O.","award":["W911NF-16-1-0070"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Active quantum error correction is a central ingredient to achieve robust quantum processors. In this paper we investigate the potential of quantum machine learning for quantum error correction in a quantum memory. Specifically, we demonstrate how quantum neural networks, in the form of quantum autoencoders, can be trained to learn optimal strategies for active detection and correction of errors, including spatially correlated computational errors as well as qubit losses. We highlight that the denoising capabilities of quantum autoencoders are not limited to the protection of specific states but extend to the entire logical codespace. We also show that quantum neural networks can be used to discover new logical encodings that are optimally adapted to the underlying noise. Moreover, we find that, even in the presence of moderate noise in the quantum autoencoders themselves, they may still be successfully used to perform beneficial quantum error correction and thereby extend the lifetime of a logical qubit.</jats:p>","DOI":"10.22331/q-2023-03-09-942","type":"journal-article","created":{"date-parts":[[2023,3,9]],"date-time":"2023-03-09T14:44:56Z","timestamp":1678373096000},"page":"942","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Quantum Error Correction with Quantum Autoencoders","prefix":"10.22331","volume":"7","author":[{"given":"David F.","family":"Locher","sequence":"first","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany"},{"name":"Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52425 Jülich, Germany"}]},{"given":"Lorenzo","family":"Cardarelli","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany"},{"name":"Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52425 Jülich, Germany"}]},{"given":"Markus","family":"Müller","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany"},{"name":"Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52425 Jülich, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2023,3,9]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2023-03-09-942/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,3,9]],"date-time":"2023-03-09T14:45:12Z","timestamp":1678373112000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2023-03-09-942/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,9]]},"references-count":99,"URL":"http://dx.doi.org/10.22331/q-2023-03-09-942","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2023,3,9]]},"article-number":"942","id":"doi:10.22331/q-2023-03-09-942","_hash":"ff4a9445f3b108803f21adbfb99b2a430f90d474edbe28482e3813a5899130f9"},"expire":1712590664967},"doi:10.1146/annurev-conmatphys-031720-030658":{"value":{"indexed":{"date-parts":[[2023,3,27]],"date-time":"2023-03-27T15:24:05Z","timestamp":1679930645482},"reference-count":311,"publisher":"Annual Reviews","issue":"1","license":[{"start":{"date-parts":[[2023,3,10]],"date-time":"2023-03-10T00:00:00Z","timestamp":1678406400000},"content-version":"unspecified","delay-in-days":0,"URL":"http://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2023,3,10]]},"abstract":"<jats:p> Quantum circuits—built from local unitary gates and local measurements—are a new playground for quantum many-body physics and a tractable setting to explore universal collective phenomena far from equilibrium. These models have shed light on longstanding questions about thermalization and chaos, and on the underlying universal dynamics of quantum information and entanglement. In addition, such models generate new sets of questions and give rise to phenomena with no traditional analog, such as dynamical phase transitions in quantum systems that are monitored by an external observer. Quantum circuit dynamics is also topical in view of experimental progress in building digital quantum simulators that allow control of precisely these ingredients. Randomness in the circuit elements allows a high level of theoretical control, with a key theme being mappings between real-time quantum dynamics and effective classical lattice models or dynamical processes. Many of the universal phenomena that can be identified in this tractable setting apply to much wider classes of more structured many-body dynamics. </jats:p>","DOI":"10.1146/annurev-conmatphys-031720-030658","type":"journal-article","created":{"date-parts":[[2022,12,12]],"date-time":"2022-12-12T20:39:21Z","timestamp":1670877561000},"page":"335-379","source":"Crossref","is-referenced-by-count":4,"title":"Random Quantum Circuits","prefix":"10.1146","volume":"14","author":[{"given":"Matthew P.A.","family":"Fisher","sequence":"first","affiliation":[{"name":"Department of Physics, University of California, Santa Barbara, California, USA;,"}]},{"given":"Vedika","family":"Khemani","sequence":"additional","affiliation":[{"name":"Department of Physics, Stanford University, Stanford, California, USA;"}]},{"given":"Adam","family":"Nahum","sequence":"additional","affiliation":[{"name":"Laboratoire de Physique, École Normale Supérieure, CNRS, Université PSL, Sorbonne Université, Université de Paris, Paris, France;"}]},{"given":"Sagar","family":"Vijay","sequence":"additional","affiliation":[{"name":"Department of Physics, University of California, Santa Barbara, California, USA;,"}]}],"member":"22","reference":[],"container-title":"Annual Review of Condensed Matter Physics","original-title":[],"language":"en","link":[{"URL":"https://www.annualreviews.org/doi/pdf/10.1146/annurev-conmatphys-031720-030658","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,10]],"date-time":"2023-03-10T23:37:12Z","timestamp":1678491432000},"score":1,"resource":{"primary":{"URL":"https://www.annualreviews.org/doi/10.1146/annurev-conmatphys-031720-030658"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,10]]},"references-count":311,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,3,10]]}},"alternative-id":["10.1146/annurev-conmatphys-031720-030658"],"URL":"http://dx.doi.org/10.1146/annurev-conmatphys-031720-030658","relation":{},"ISSN":["1947-5454","1947-5462"],"subject":["Condensed Matter Physics","General Materials Science"],"container-title-short":"Annu. 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In most QEC codes<jats:sup>2–8</jats:sup>, a logical qubit is encoded in some discrete variables, for example photon numbers, so that the encoded quantum information can be unambiguously extracted after processing. Over the past decade, repetitive QEC has been demonstrated with various discrete-variable-encoded scenarios<jats:sup>9–17</jats:sup>. However, extending the lifetimes of thus-encoded logical qubits beyond the best available physical qubit still remains elusive, which represents a break-even point for judging the practical usefulness of QEC. Here we demonstrate a QEC procedure in a circuit quantum electrodynamics architecture<jats:sup>18</jats:sup>, where the logical qubit is binomially encoded in photon-number states of a microwave cavity<jats:sup>8</jats:sup>, dispersively coupled to an auxiliary superconducting qubit. By applying a pulse featuring a tailored frequency comb to the auxiliary qubit, we can repetitively extract the error syndrome with high fidelity and perform error correction with feedback control accordingly, thereby exceeding the break-even point by about 16% lifetime enhancement. 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Optimizing the resource and time overheads needed to implement QEC is one of the most pressing challenges. Here, we introduce a new topological quantum error-correcting code, the three-dimensional subsystem toric code (3D STC). The 3D STC can be realized with geometrically-local parity checks of weight at most three on the cubic lattice with open boundary conditions. We prove that one round of parity-check measurements suffices to perform reliable QEC with the 3D STC even in the presence of measurement errors. We also propose an efficient single-shot QEC decoding strategy for the 3D STC and numerically estimate the resulting storage threshold against independent bit-flip, phase-flip and measurement errors to be<jats:italic>p</jats:italic><jats:sub>STC</jats:sub> ≈ 1.045%. Such a high threshold together with local parity-check measurements make the 3D STC particularly appealing for realizing fault-tolerant quantum computing.</jats:p>","DOI":"10.1038/s41467-022-33923-4","type":"journal-article","created":{"date-parts":[[2022,10,21]],"date-time":"2022-10-21T14:08:47Z","timestamp":1666361327000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":"Single-shot quantum error correction with the three-dimensional subsystem toric code","prefix":"10.1038","volume":"13","author":[{"ORCID":"http://orcid.org/0000-0001-8213-8190","authenticated-orcid":false,"given":"Aleksander","family":"Kubica","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-6711-5924","authenticated-orcid":false,"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,10,21]]},"reference":[],"container-title":"Nature 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Published by Elsevier Inc.","name":"copyright","label":"Copyright"}],"article-number":"169286","id":"doi:10.1016/j.aop.2023.169286","_hash":"96192b1ddfeb64b839f7308081c632c75fd6b63585a655f159b86ca7f39e2225"},"expire":1715612576802},"doi:10.1038/s41467-023-37725-0":{"value":{"indexed":{"date-parts":[[2023,5,11]],"date-time":"2023-05-11T17:32:26Z","timestamp":1683826346465},"reference-count":53,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T00:00:00Z","timestamp":1682035200000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T00:00:00Z","timestamp":1682035200000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"<jats:title>Abstract</jats:title><jats:p>Electrons are indivisible elementary particles, yet paradoxically a collection of them can act as a fraction of a single electron, exhibiting exotic and useful properties. One such collective excitation, known as a topological Majorana mode, is naturally stable against perturbations, such as unwanted local noise, and can thereby robustly store quantum information. As such, Majorana modes serve as the basic primitive of topological quantum computing, providing resilience to errors. However, their demonstration on quantum hardware has remained elusive. Here, we demonstrate a verifiable identification and braiding of topological Majorana modes using a superconducting quantum processor as a quantum simulator. By simulating fermions on a one-dimensional lattice subject to a periodic drive, we confirm the existence of Majorana modes localized at the edges, and distinguish them from other trivial modes. To simulate a basic logical operation of topological quantum computing known as braiding, we propose a non-adiabatic technique, whose implementation reveals correct braiding statistics in our experiments. This work could further be used to study topological models of matter using circuit-based simulations, and shows that long-sought quantum phenomena can be realized by anyone in cloud-run quantum simulations, whereby accelerating fundamental discoveries in quantum science and technology.</jats:p>","DOI":"10.1038/s41467-023-37725-0","type":"journal-article","created":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T08:02:54Z","timestamp":1682064174000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":"Observing and braiding topological Majorana modes on programmable quantum 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They break the fermion-boson dichotomy and obey non-Abelian braiding statistics: their interchanges yield unitary operations, rather than merely a phase factor, in a space spanned by topologically degenerate wavefunctions. They are the building blocks of topological quantum computing. However, experimental observation of non-Abelian anyons and their characterizing braiding statistics is notoriously challenging and has remained elusive hitherto, in spite of various theoretical proposals. Here, we report an experimental quantum digital simulation of projective non-Abelian anyons and their braiding statistics with up to 68 programmable superconducting qubits arranged on a two-dimensional lattice. By implementing the ground states of the toric-code model with twists through quantum circuits, we demonstrate that twists exchange electric and magnetic charges and behave as a particular type of non-Abelian anyons—the Ising anyons. 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Such a degeneracy is known as topological degeneracy and can be usually seen under the periodic boundary condition regardless of the choice of the system sizes L1 and L2 in each direction. In this work, we introduce a family of extensions of the Kitaev toric code to N level spins (N ≥ 2). The model realizes topologically ordered phases or symmetry-protected topological phases depending on the parameters in the model. The most remarkable feature of topologically ordered phases is that the ground state may be unique, depending on L1 and L2, despite that the translation symmetry of the model remains unbroken. Nonetheless, the topological entanglement entropy takes the nontrivial value. We argue that this behavior originates from the nontrivial action of translations permuting anyon species.</jats:p>","DOI":"10.1063/5.0134010","type":"journal-article","created":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T10:19:06Z","timestamp":1684318746000},"update-policy":"http://dx.doi.org/10.1063/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":"Ground state degeneracy on torus in a family of ZN toric code","prefix":"10.1063","volume":"64","author":[{"ORCID":"http://orcid.org/0000-0002-8112-021X","authenticated-orcid":false,"given":"Haruki","family":"Watanabe","sequence":"first","affiliation":[{"name":"Department of Applied Physics, The University of Tokyo 1 , Tokyo, Japan"}]},{"given":"Meng","family":"Cheng","sequence":"additional","affiliation":[{"name":"Department of Physics, Yale University 2 , New Haven, Connecticut 06520, USA"}]},{"ORCID":"http://orcid.org/0000-0002-3381-6642","authenticated-orcid":false,"given":"Yohei","family":"Fuji","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, The University of Tokyo 1 , Tokyo, Japan"}]}],"member":"317","published-online":{"date-parts":[[2023,5,17]]},"reference":[],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","link":[{"URL":"https://pubs.aip.org/aip/jmp/article-pdf/doi/10.1063/5.0134010/17614211/051901_1_5.0134010.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://pubs.aip.org/aip/jmp/article-pdf/doi/10.1063/5.0134010/17614211/051901_1_5.0134010.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T10:19:16Z","timestamp":1684318756000},"score":1,"resource":{"primary":{"URL":"https://pubs.aip.org/jmp/article/64/5/051901/2891377/Ground-state-degeneracy-on-torus-in-a-family-of-ZN"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,1]]},"references-count":54,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,5,1]]}},"URL":"http://dx.doi.org/10.1063/5.0134010","relation":{},"ISSN":["0022-2488","1089-7658"],"subject":["Mathematical Physics","Statistical and Nonlinear Physics"],"published-other":{"date-parts":[[2023,5,1]]},"published":{"date-parts":[[2023,5,1]]},"id":"doi:10.1063/5.0134010","_hash":"1fdf11bd915d05d4602acb584d34e5fa06289c8f81e3bb4563e0be2857d4a1ac"},"expire":1718831603910},"doi:10.1038/s41467-023-38247-5":{"value":{"indexed":{"date-parts":[[2023,5,19]],"date-time":"2023-05-19T05:13:29Z","timestamp":1684473209864},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T00:00:00Z","timestamp":1684368000000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T00:00:00Z","timestamp":1684368000000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/100011039","name":"ODNI | Intelligence Advanced Research Projects Activity","doi-asserted-by":"publisher","award":["W911NF-16-1-0114","youngseok.kim1@ibm.com","W911NF-16-1-0114","W911NF-16-1-0114","W911NF-16-1-0114","W911NF-16-1-0114","W911NF-16-1-0114","W911NF-16-1-0114","W911NF-16-1-0114"]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum error correction offers a promising path for performing high fidelity quantum computations. Although fully fault-tolerant executions of algorithms remain unrealized, recent improvements in control electronics and quantum hardware enable increasingly advanced demonstrations of the necessary operations for error correction. Here, we perform quantum error correction on superconducting qubits connected in a heavy-hexagon lattice. We encode a logical qubit with distance three and perform several rounds of fault-tolerant syndrome measurements that allow for the correction of any single fault in the circuitry. Using real-time feedback, we reset syndrome and flag qubits conditionally after each syndrome extraction cycle. We report decoder dependent logical error, with average logical error per syndrome measurement in Z(X)-basis of ~0.040 (~0.088) and ~0.037 (~0.087) for matching and maximum likelihood decoders, respectively, on leakage post-selected data.</jats:p>","DOI":"10.1038/s41467-023-38247-5","type":"journal-article","created":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T18:31:30Z","timestamp":1684434690000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":"Demonstrating multi-round subsystem quantum error correction using matching and maximum likelihood decoders","prefix":"10.1038","volume":"14","author":[{"ORCID":"http://orcid.org/0000-0002-0772-2853","authenticated-orcid":false,"given":"Neereja","family":"Sundaresan","sequence":"first","affiliation":[]},{"given":"Theodore J.","family":"Yoder","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-8486-9162","authenticated-orcid":false,"given":"Youngseok","family":"Kim","sequence":"additional","affiliation":[]},{"given":"Muyuan","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-7217-7137","authenticated-orcid":false,"given":"Edward H.","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Grace","family":"Harper","sequence":"additional","affiliation":[]},{"given":"Ted","family":"Thorbeck","sequence":"additional","affiliation":[]},{"given":"Andrew W.","family":"Cross","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-7800-0399","authenticated-orcid":false,"given":"Antonio D.","family":"Córcoles","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-8871-4638","authenticated-orcid":false,"given":"Maika","family":"Takita","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,5,18]]},"reference":[],"container-title":"Nature Communications","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41467-023-38247-5.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-023-38247-5","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-023-38247-5.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T18:31:40Z","timestamp":1684434700000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41467-023-38247-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,18]]},"references-count":46,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,12]]}},"alternative-id":["38247"],"URL":"http://dx.doi.org/10.1038/s41467-023-38247-5","relation":{},"ISSN":["2041-1723"],"subject":["General Physics and Astronomy","General Biochemistry, Genetics and Molecular Biology","General Chemistry","Multidisciplinary"],"container-title-short":"Nat Commun","published":{"date-parts":[[2023,5,18]]},"assertion":[{"value":"15 July 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 April 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 May 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"2852","id":"doi:10.1038/s41467-023-38247-5","_hash":"4e93b21633c5eaec23e9a0fcf05249ed1d5eb4daf7b1920b58098588cec04b8c"},"expire":1718831605081},"doi:10.22331/q-2023-06-07-1037":{"value":{"indexed":{"date-parts":[[2023,6,8]],"date-time":"2023-06-08T04:30:07Z","timestamp":1686198607392},"reference-count":46,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,6,7]],"date-time":"2023-06-07T00:00:00Z","timestamp":1686096000000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"EU Quantum Technology Flagship","award":["AQTION/820495"]},{"name":"BMBF","award":["MUNIQC-ATOMS"]},{"name":"European Research Council","award":["ERC/682726"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Belief propagation (BP) is well-known as a low complexity decoding algorithm with a strong performance for important classes of quantum error correcting codes, e.g. notably for the quantum low-density parity check (LDPC) code class of random expander codes. However, it is also well-known that the performance of BP breaks down when facing topological codes such as the surface code, where naive BP fails entirely to reach a below-threshold regime, i.e. the regime where error correction becomes useful. Previous works have shown, that this can be remedied by resorting to post-processing decoders outside the framework of BP. In this work, we present a generalized belief propagation method with an outer re-initialization loop that successfully decodes surface codes, i.e. opposed to naive BP it recovers the sub-threshold regime known from decoders tailored to the surface code and from statistical-mechanical mappings. We report a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">17%</mml:mtext></mml:mrow></mml:math> under independent bit-and phase-flip data noise (to be compared to the ideal threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">20.6%</mml:mtext></mml:mrow></mml:math>) and a threshold value of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">14%</mml:mtext></mml:mrow></mml:math> under depolarizing data noise (compared to the ideal threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">18.9%</mml:mtext></mml:mrow></mml:math>), which are on par with thresholds achieved by non-BP post-processing methods.</jats:p>","DOI":"10.22331/q-2023-06-07-1037","type":"journal-article","created":{"date-parts":[[2023,6,7]],"date-time":"2023-06-07T10:53:02Z","timestamp":1686135182000},"page":"1037","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Generalized Belief Propagation Algorithms for Decoding of Surface Codes","prefix":"10.22331","volume":"7","author":[{"given":"Josias","family":"Old","sequence":"first","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, Aachen, Germany"},{"name":"Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich, Jülich, Germany"}]},{"given":"Manuel","family":"Rispler","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, Aachen, Germany"},{"name":"Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich, Jülich, Germany"},{"name":"QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands"}]}],"member":"9598","published-online":{"date-parts":[[2023,6,7]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2023-06-07-1037/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,6,7]],"date-time":"2023-06-07T10:53:16Z","timestamp":1686135196000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2023-06-07-1037/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,7]]},"references-count":46,"URL":"http://dx.doi.org/10.22331/q-2023-06-07-1037","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2023,6,7]]},"article-number":"1037","id":"doi:10.22331/q-2023-06-07-1037","_hash":"888e955c0ce0e381d34fe03109132bc44c43bf500d7ee3074b63f68063f941e6"},"expire":1718831606005},"doi:10.1088/1367-2630/13/4/043016":{"value":{"indexed":{"date-parts":[[2023,6,19]],"date-time":"2023-06-19T08:30:42Z","timestamp":1687163442615},"reference-count":0,"publisher":"IOP Publishing","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1088/1367-2630/13/4/043016","type":"journal-article","created":{"date-parts":[[2011,4,15]],"date-time":"2011-04-15T03:34:39Z","timestamp":1302838479000},"page":"043016","source":"Crossref","is-referenced-by-count":139,"title":"Interacting quantum observables: categorical algebra and diagrammatics","prefix":"10.1088","volume":"13","author":[{"given":"Bob","family":"Coecke","sequence":"first","affiliation":[]},{"given":"Ross","family":"Duncan","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2011,4,14]]},"container-title":"New Journal of Physics","original-title":[],"deposited":{"date-parts":[[2020,4,11]],"date-time":"2020-04-11T15:15:12Z","timestamp":1586618112000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/13/4/043016"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,4,14]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2011,4,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/13/4/043016","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. Phys.","published":{"date-parts":[[2011,4,14]]},"reference":[],"id":"doi:10.1088/1367-2630/13/4/043016","_hash":"5152c192586b62cb6171ddc27668f1174492e7f6a3437dae777abee234e74cb5"},"expire":1718831606972},"doi:10.22331/q-2019-04-30-135":{"value":{"indexed":{"date-parts":[[2023,6,18]],"date-time":"2023-06-18T10:50:45Z","timestamp":1687085445826},"reference-count":32,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T00:00:00Z","timestamp":1556582400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"<jats:p>We present magic state factory constructions for producing<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>states and<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>states. For the<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>factory we apply the surface code lattice surgery construction techniques described in \\cite{fowler2018} to the fault-tolerant Toffoli \\cite{jones2013, eastin2013distilling}. The resulting factory has a footprint of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>12</mml:mn><mml:mi>d</mml:mi><mml:mo>×</mml:mo><mml:mn>6</mml:mn><mml:mi>d</mml:mi></mml:math>(where<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>is the code distance) and produces one<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>every<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>5.5</mml:mn><mml:mi>d</mml:mi></mml:math>surface code cycles. Our<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>state factory uses the<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>factory's output and a catalyst<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>state to exactly transform one<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>state into two<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>states. It has a footprint<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>25</mml:mn><mml:mi mathvariant=\"normal\">%</mml:mi></mml:math>smaller than the factory in \\cite{fowler2018} but outputs<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>states twice as quickly. We show how to generalize the catalyzed transformation to arbitrary phase angles, and note that the case<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>22.5</mml:mn><mml:mo>∘</mml:mo></mml:msup></mml:math>produces a particularly efficient circuit for producing<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msqrt><mml:mi>T</mml:mi></mml:msqrt><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>states. Compared to using the<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>12</mml:mn><mml:mi>d</mml:mi><mml:mo>×</mml:mo><mml:mn>8</mml:mn><mml:mi>d</mml:mi><mml:mo>×</mml:mo><mml:mn>6.5</mml:mn><mml:mi>d</mml:mi></mml:math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>factory of \\cite{fowler2018}, our<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>factory can quintuple the speed of algorithms that are dominated by the cost of applying Toffoli gates, including Shor's algorithm \\cite{shor1994} and the chemistry algorithm of Babbush et al. \\cite{babbush2018}. Assuming a physical gate error rate of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mn>10</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>, our CCZ factory can produce<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>∼</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>10</mml:mn></mml:mrow></mml:msup></mml:math>states on average before an error occurs. This is sufficient for classically intractable instantiations of the chemistry algorithm, but for more demanding algorithms such as Shor's algorithm the mean number of states until failure can be increased to<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>∼</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>12</mml:mn></mml:mrow></mml:msup></mml:math>by increasing the factory footprint<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>∼</mml:mo><mml:mn>20</mml:mn><mml:mi mathvariant=\"normal\">%</mml:mi></mml:math>.</jats:p>","DOI":"10.22331/q-2019-04-30-135","type":"journal-article","created":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T03:12:22Z","timestamp":1556593942000},"page":"135","source":"Crossref","is-referenced-by-count":46,"title":"Efficient magic state factories with a catalyzed<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>to<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>transformation","prefix":"10.22331","volume":"3","author":[{"given":"Craig","family":"Gidney","sequence":"first","affiliation":[{"name":"Google Inc., Santa Barbara, California 93117, USA"}]},{"given":"Austin G.","family":"Fowler","sequence":"additional","affiliation":[{"name":"Google Inc., Santa Barbara, California 93117, USA"}]}],"member":"9598","published-online":{"date-parts":[[2019,4,30]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2019-04-30-135/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2020,12,9]],"date-time":"2020-12-09T03:13:27Z","timestamp":1607483607000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2019-04-30-135/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,30]]},"references-count":32,"URL":"http://dx.doi.org/10.22331/q-2019-04-30-135","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2019,4,30]]},"article-number":"135","id":"doi:10.22331/q-2019-04-30-135","_hash":"6d4e746ea6d62d0218955310e6ec6696b5cfec48535d9cef7b22ffac8ceaec7d"},"expire":1718831608007},"doi:10.22331/q-2020-01-09-218":{"value":{"indexed":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T15:35:26Z","timestamp":1680881726443},"reference-count":47,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2020,1,9]],"date-time":"2020-01-09T00:00:00Z","timestamp":1578528000000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"<jats:p>A leading choice of error correction for scalable quantum computing is the surface code with lattice surgery. The basic lattice surgery operations, the merging and splitting of logical qubits, act non-unitarily on the logical states and are not easily captured by standard circuit notation. This raises the question of how best to design, verify, and optimise protocols that use lattice surgery, in particular in architectures with complex resource management issues. In this paper we demonstrate that the operations of the ZX calculus --- a form of quantum diagrammatic reasoning based on bialgebras --- match exactly the operations of lattice surgery. Red and green ``spider'' nodes match rough and smooth merges and splits, and follow the axioms of a dagger special associative Frobenius algebra. Some lattice surgery operations require non-trivial correction operations, which are captured natively in the use of the ZX calculus in the form of ensembles of diagrams. 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Rev. A","published":{"date-parts":[[2002,5,3]]},"article-number":"052316","id":"doi:10.1103/PhysRevA.65.052316","_hash":"78a4826461d12bff41f0e39b2abb20a8b820ee1525fcebcb235d1eaa91a92035"},"expire":1718831616974},"doi:10.21468/SciPostPhysLectNotes.72":{"value":{"indexed":{"date-parts":[[2023,6,17]],"date-time":"2023-06-17T04:31:58Z","timestamp":1686976318323},"reference-count":79,"publisher":"Stichting SciPost","license":[{"start":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T00:00:00Z","timestamp":1686873600000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"content-domain":{"domain":["scipost.org"],"crossmark-restriction":false},"abstract":"<jats:p>These are the lecture notes from the 2019 Les Houches Summer School on “Quantum Information Machines”. After a brief introduction to quantum error correction and bosonic codes, we focus on the case of cat qubits stabilized by a nonlinear multi-photon driven dissipation process. We argue that such a system can be seen as a self-correcting qubit where bit-flip errors are robustly and exponentially suppressed. Next, we provide some experimental directions to engineer such a multi-photon driven dissipation process with superconducting circuits. Finally, we analyze various logical gates that can be implemented without re-introducing bit-flip errors. This set of bias-preserving gates pave the way towards a hardware-efficient and fault-tolerant quantum processor.</jats:p>","DOI":"10.21468/scipostphyslectnotes.72","type":"journal-article","created":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T09:24:40Z","timestamp":1686907480000},"update-policy":"http://dx.doi.org/10.21468/scipost.crossmarkpolicy","source":"Crossref","is-referenced-by-count":0,"title":"Quantum computation with cat qubits","prefix":"10.21468","author":[{"given":"Jérémie","family":"Guillaud","sequence":"first","affiliation":[{"name":"Alice and Bob"}]},{"given":"Joachim","family":"Cohen","sequence":"additional","affiliation":[{"name":"Université de Sherbrooke"}]},{"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[{"name":"Physics Laboratory of the École Normale Supérieure"}]}],"member":"8907","published-online":{"date-parts":[[2023,6,16]]},"reference":[],"container-title":"SciPost Physics Lecture Notes","original-title":[],"link":[{"URL":"https://scipost.org/10.21468/SciPostPhysLectNotes.72/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://scipost.org/10.21468/SciPostPhysLectNotes.72/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T09:24:43Z","timestamp":1686907483000},"score":1,"resource":{"primary":{"URL":"https://scipost.org/10.21468/SciPostPhysLectNotes.72"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,16]]},"references-count":79,"URL":"http://dx.doi.org/10.21468/SciPostPhysLectNotes.72","relation":{},"ISSN":["2590-1990"],"subject":["Condensed Matter Physics","Nuclear and High Energy Physics","Atomic and Molecular Physics, and Optics","Statistical and Nonlinear Physics"],"container-title-short":"SciPost Phys. 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Notes","published":{"date-parts":[[2023,6,16]]},"article-number":"72","id":"doi:10.21468/SciPostPhysLectNotes.72","_hash":"a4da1b77cfcb34ee2c896010e9640dc4b58ec48493df27cfe3ab55c716d31255"},"expire":1718831617964},"doi:10.22331/q-2023-06-05-1032":{"value":{"indexed":{"date-parts":[[2023,6,6]],"date-time":"2023-06-06T04:39:56Z","timestamp":1686026396136},"reference-count":48,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T00:00:00Z","timestamp":1685923200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"crossref","award":["ANR-22-PETQ-0006"]},{"DOI":"10.13039/501100000780","name":"European Union","doi-asserted-by":"crossref","award":["H2020 CiViQ"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Quantum computers often manipulate physical qubits encoded on two-level quantum systems. Bosonic qubit codes depart from this idea by encoding information in a well-chosen subspace of an infinite-dimensional Fock space. This larger physical space provides a natural protection against experimental imperfections and allows bosonic codes to circumvent no-go results that apply to states constrained by a 2-dimensional Hilbert space. A bosonic qubit is usually defined in a single bosonic mode but it makes sense to look for multimode versions that could exhibit better performance.In this work, building on the observation that the cat code lives in the span of coherent states indexed by a finite subgroup of the complex numbers, we consider a two-mode generalisation living in the span of 24 coherent states indexed by the binary tetrahedral group <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>T</mml:mi></mml:math> of the quaternions. The resulting <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>T</mml:mi></mml:math>-qutrit naturally inherits the algebraic properties of the group <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>T</mml:mi></mml:math> and appears to be quite robust in the low-loss regime. We initiate its study and identify stabilisers as well as some logical operators for this bosonic code.</jats:p>","DOI":"10.22331/q-2023-06-05-1032","type":"journal-article","created":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T13:34:27Z","timestamp":1685972067000},"page":"1032","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"The <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>T</mml:mi></mml:math>-qutrit, a two-mode bosonic qutrit","prefix":"10.22331","volume":"7","author":[{"given":"Aurélie","family":"Denys","sequence":"first","affiliation":[{"name":"Inria Paris, France"}]},{"ORCID":"http://orcid.org/0000-0002-6707-1458","authenticated-orcid":false,"given":"Anthony","family":"Leverrier","sequence":"additional","affiliation":[{"name":"Inria Paris, France"}]}],"member":"9598","published-online":{"date-parts":[[2023,6,5]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2023-06-05-1032/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T13:34:34Z","timestamp":1685972074000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2023-06-05-1032/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,5]]},"references-count":48,"URL":"http://dx.doi.org/10.22331/q-2023-06-05-1032","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2023,6,5]]},"article-number":"1032","id":"doi:10.22331/q-2023-06-05-1032","_hash":"0f6247e3c14e9281652b61c4aadfb9562e70296f32cf88ca17c69955e167efc9"},"expire":1718831618916},"doi:10.22331/q-2023-05-31-1025":{"value":{"indexed":{"date-parts":[[2023,6,1]],"date-time":"2023-06-01T10:41:18Z","timestamp":1685616078944},"reference-count":59,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,5,31]],"date-time":"2023-05-31T00:00:00Z","timestamp":1685491200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"EPSRC Quantum Communications Hub","award":["EP/T001011/1"]},{"name":"NSF Center for Quantum Networks","award":["EEC-1941583"]},{"name":"PNRR MUR Project","award":["PE0000023-NQSTI"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Coherent states of the quantum electromagnetic field, the quantum description of ideal laser light, are prime candidates as information carriers for optical communications. A large body of literature exists on their quantum-limited estimation and discrimination. However, very little is known about the practical realizations of receivers for unambiguous state discrimination (USD) of coherent states. Here we fill this gap and outline a theory of USD with receivers that are allowed to employ: passive multimode linear optics, phase-space displacements, auxiliary vacuum modes, and on-off photon detection. Our results indicate that, in some regimes, these currently-available optical components are typically sufficient to achieve near-optimal unambiguous discrimination of multiple, multimode coherent states.</jats:p>","DOI":"10.22331/q-2023-05-31-1025","type":"journal-article","created":{"date-parts":[[2023,5,31]],"date-time":"2023-05-31T15:09:03Z","timestamp":1685545743000},"page":"1025","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Linear optics and photodetection achieve near-optimal unambiguous coherent state discrimination","prefix":"10.22331","volume":"7","author":[{"ORCID":"http://orcid.org/0000-0002-6167-8224","authenticated-orcid":false,"given":"Jasminder S.","family":"Sidhu","sequence":"first","affiliation":[{"name":"SUPA Department of Physics, The University of Strathclyde, Glasgow, G4 0NG, UK"}]},{"ORCID":"http://orcid.org/0000-0002-3528-7473","authenticated-orcid":false,"given":"Michael S.","family":"Bullock","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, The University of Arizona, Tucson, Arizona 85721, USA"}]},{"ORCID":"http://orcid.org/0000-0002-2581-4380","authenticated-orcid":false,"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, The University of Arizona, Tucson, Arizona 85721, USA"},{"name":"College of Optical Sciences, The University of Arizona, Tucson, Arizona 85721, USA"}]},{"ORCID":"http://orcid.org/0000-0002-5227-4009","authenticated-orcid":false,"given":"Cosmo","family":"Lupo","sequence":"additional","affiliation":[{"name":"Dipartimento Interateneo di Fisica, Politecnico & Università di Bari, 70126 Bari, Italy"},{"name":"INFN, Sezione di Bari, 70126 Bari, Italy"}]}],"member":"9598","published-online":{"date-parts":[[2023,5,31]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2023-05-31-1025/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,5,31]],"date-time":"2023-05-31T15:09:18Z","timestamp":1685545758000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2023-05-31-1025/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,31]]},"references-count":59,"URL":"http://dx.doi.org/10.22331/q-2023-05-31-1025","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2023,5,31]]},"article-number":"1025","id":"doi:10.22331/q-2023-05-31-1025","_hash":"4e82dae06e0390386ba0aee2fe5463a966ea23a4814ec79875ae1b3bdf5c8f53"},"expire":1718831619915},"doi:10.1103/PhysRevLett.92.027902":{"value":{"indexed":{"date-parts":[[2023,6,25]],"date-time":"2023-06-25T02:18:15Z","timestamp":1687659495766},"reference-count":19,"publisher":"American Physical Society (APS)","issue":"2","license":[{"start":{"date-parts":[[2004,1,15]],"date-time":"2004-01-15T00:00:00Z","timestamp":1074124800000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physrevlett.92.027902","type":"journal-article","created":{"date-parts":[[2004,1,16]],"date-time":"2004-01-16T03:55:00Z","timestamp":1074225300000},"source":"Crossref","is-referenced-by-count":247,"title":"Classical Capacity of the Lossy Bosonic Channel: The Exact Solution","prefix":"10.1103","volume":"92","author":[{"given":"V.","family":"Giovannetti","sequence":"first","affiliation":[]},{"given":"S.","family":"Guha","sequence":"additional","affiliation":[]},{"given":"S.","family":"Lloyd","sequence":"additional","affiliation":[]},{"given":"L.","family":"Maccone","sequence":"additional","affiliation":[]},{"given":"J. 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Reconstructions exist for any microstate, but no reconstruction works for all microstates. We refine this dichotomy, demonstrating that the same boundary operator can often be used for large subspaces of black hole microstates, corresponding to a constant fraction <jats:italic>α</jats:italic> of the black hole entropy. In the Schrödinger picture, the boundary subregion encodes the <jats:italic>α</jats:italic>-bits (a concept from quantum information) of a bulk region containing the black hole and bounded by extremal surfaces. These results have important consequences for the structure of AdS/CFT and for quantum information. Firstly, they imply that the bulk reconstruction is necessarily only approximate and allow us to place non-perturbative lower bounds on the error when doing so. Second, they provide a simple and tractable limit in which the entanglement wedge is state dependent, but in a highly controlled way. Although the state dependence of operators comes from ordinary quantum error correction, there are clear connections to the Papadodimas-Raju proposal for understanding operators behind black hole horizons. In tensor network toy models of AdS/CFT, we see how state dependence arises from the bulk operator being ‘pushed’ through the black hole itself. Finally, we show that black holes provide the first ‘explicit’ examples of capacity-achieving <jats:italic>α</jats:italic>-bit codes. Unintuitively, Hawking radiation always reveals the <jats:italic>α</jats:italic>-bits of a black hole as soon as possible. In an appendix, we apply a result from the quantum information literature to prove that entanglement wedge reconstruction can be made exact to all orders in 1<jats:italic>/N</jats:italic>.</jats:p>","DOI":"10.1007/jhep12(2019)007","type":"journal-article","created":{"date-parts":[[2019,12,5]],"date-time":"2019-12-05T17:16:52Z","timestamp":1575566212000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":54,"title":"Learning the Alpha-bits of black holes","prefix":"10.1007","volume":"2019","author":[{"given":"Patrick","family":"Hayden","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-8627-5237","authenticated-orcid":false,"given":"Geoffrey","family":"Penington","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,12,2]]},"reference":[],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","link":[{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP12(2019)007.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/article/10.1007/JHEP12(2019)007/fulltext.html","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP12(2019)007.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,3]],"date-time":"2021-09-03T19:54:45Z","timestamp":1630698885000},"score":1,"resource":{"primary":{"URL":"https://link.springer.com/10.1007/JHEP12(2019)007"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12]]},"references-count":58,"journal-issue":{"issue":"12","published-print":{"date-parts":[[2019,12]]}},"alternative-id":["11902"],"URL":"http://dx.doi.org/10.1007/JHEP12(2019)007","relation":{},"ISSN":["1029-8479"],"subject":["Nuclear and High Energy Physics"],"container-title-short":"J. 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Developing the subject 'from the ground up' it covers classical results as well as major advances of the past decade. Beginning with an extensive overview of classical information theory suitable for the non-expert, the author then turns his attention to quantum mechanics for quantum information theory, and the important protocols of teleportation, super-dense coding and entanglement distribution. He develops all of the tools necessary for understanding important results in quantum information theory, including capacity theorems for classical, entanglement-assisted, private and quantum communication. The book also covers important recent developments such as superadditivity of private, coherent and Holevo information, and the superactivation of quantum capacity. This book will be warmly welcomed by the upcoming generation of quantum information theorists and the already established community of classical information theorists.</jats:p>","DOI":"10.1017/cbo9781139525343","type":"monograph","created":{"date-parts":[[2013,5,5]],"date-time":"2013-05-05T04:17:06Z","timestamp":1367727426000},"source":"Crossref","is-referenced-by-count":604,"title":"Quantum Information Theory","prefix":"10.1017","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2013,5,5]]},"container-title":[],"original-title":[],"deposited":{"date-parts":[[2022,6,13]],"date-time":"2022-06-13T18:21:44Z","timestamp":1655144504000},"score":1,"resource":{"primary":{"URL":"https://www.cambridge.org/core/product/identifier/9781139525343/type/book"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,4,18]]},"ISBN":["9781107034259","9781139525343"],"references-count":0,"URL":"http://dx.doi.org/10.1017/CBO9781139525343","relation":{},"published":{"date-parts":[[2013,4,18]]},"reference":[],"id":"doi:10.1017/CBO9781139525343","_hash":"803666f83242193b9f3f627c62c7b7294df2c8ce32ed7f9f688da8fbe6c8f689"},"expire":1721576483936},"doi:10.22331/q-2023-07-06-1048":{"value":{"indexed":{"date-parts":[[2023,7,7]],"date-time":"2023-07-07T04:25:15Z","timestamp":1688703915460},"reference-count":65,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,7,6]],"date-time":"2023-07-06T00:00:00Z","timestamp":1688601600000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"National Science and Technology Council, Taiwan","award":["NSTC 112-2119-M-002-014"]},{"name":"National Science and Technology Council, Taiwan","award":["NSTC 111- 2119-M-002-006-MY3"]},{"name":"National Science and Technology Council, Taiwan","award":["NSTC 110-2627-M002-002"]},{"name":"National Science and Technology Council, Taiwan","award":["NSTC 111-2627-M-002-001"]},{"name":"Center for Advanced Computing and Imaging in Biomedicine","award":["NTU-112L900702"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>In this work, we compute the number of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mi>d</mml:mi></mml:msub></mml:math> stabilizer codes made up of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>-dimensional qudits, for arbitrary positive integers <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>. In a seminal work by Gross \\cite{Gross2006} the number of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mi>d</mml:mi></mml:msub></mml:math> stabilizer codes was computed for the case when <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> is a prime (or the power of a prime, i.e., <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mi>m</mml:mi></mml:msup></mml:math>, but when the qudits are Galois-qudits). The proof in \\cite{Gross2006} is inapplicable to the non-prime case. For our proof, we introduce a group structure to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mi>d</mml:mi></mml:msub></mml:math> codes, and use this in conjunction with the Chinese remainder theorem to count the number of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mi>d</mml:mi></mml:msub></mml:math> codes. Our work overlaps with \\cite{Gross2006} when <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> is a prime and in this case our results match exactly, but the results differ for the more generic case. Despite that, the overall order of magnitude of the number of stabilizer codes scales agnostic of whether the dimension is prime or non-prime. This is surprising since the method employed to count the number of stabilizer states (or more generally stabilizer codes) depends on whether <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> is prime or not. The cardinality of stabilizer states, which was so far known only for the prime-dimensional case (and the Galois qudit prime-power dimensional case) plays an important role as a quantifier in many topics in quantum computing. Salient among these are the resource theory of magic, design theory, de Finetti theorem for stabilizer states, the study and optimisation of the classical simulability of Clifford circuits, the study of quantum contextuality of small-dimensional systems and the study of Wigner-functions. Our work makes available this quantifier for the generic case, and thus is an important step needed to place results for quantum computing with non-prime dimensional quantum systems on the same pedestal as prime-dimensional systems.</jats:p>","DOI":"10.22331/q-2023-07-06-1048","type":"journal-article","created":{"date-parts":[[2023,7,6]],"date-time":"2023-07-06T14:16:26Z","timestamp":1688652986000},"page":"1048","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Counting stabiliser codes for arbitrary dimension","prefix":"10.22331","volume":"7","author":[{"ORCID":"http://orcid.org/0000-0001-5986-248X","authenticated-orcid":false,"given":"Tanmay","family":"Singal","sequence":"first","affiliation":[{"name":"Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5/7, 87-100 Toruń, 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A","published":{"date-parts":[[2023,8,3]]},"article-number":"022401","id":"doi:10.1103/PhysRevA.108.022401","_hash":"5c80135598e943d1e2727429fceb1b59730f49d84c257cb15ff6e97903cc8933"},"expire":1728583703489},"doi:10.1103/PhysRevA.80.052312":{"value":{"indexed":{"date-parts":[[2023,10,10]],"date-time":"2023-10-10T09:53:57Z","timestamp":1696931637113},"reference-count":16,"publisher":"American Physical Society (APS)","issue":"5","license":[{"start":{"date-parts":[[2009,11,11]],"date-time":"2009-11-11T00:00:00Z","timestamp":1257897600000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.80.052312","type":"journal-article","created":{"date-parts":[[2009,11,12]],"date-time":"2009-11-12T14:50:40Z","timestamp":1258037440000},"source":"Crossref","is-referenced-by-count":280,"title":"High-threshold universal quantum computation on the surface code","prefix":"10.1103","volume":"80","author":[{"given":"Austin G.","family":"Fowler","sequence":"first","affiliation":[]},{"given":"Ashley M.","family":"Stephens","sequence":"additional","affiliation":[]},{"given":"Peter","family":"Groszkowski","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2009,11,11]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.80.052312","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.80.052312/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,19]],"date-time":"2017-06-19T01:42:05Z","timestamp":1497836525000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.80.052312"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,11,11]]},"references-count":16,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2009,11]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.80.052312","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. Rev. A","published":{"date-parts":[[2009,11,11]]},"article-number":"052312","id":"doi:10.1103/PhysRevA.80.052312","_hash":"06a65d9770741ab5c89f6a9ed3ed10ddd0711e0bef2b18bae617b44ba29573c6"},"expire":1728583704551},"doi:10.1088/2058-9565/ace64d":{"value":{"indexed":{"date-parts":[[2023,8,22]],"date-time":"2023-08-22T15:04:31Z","timestamp":1692716671192},"reference-count":72,"publisher":"IOP Publishing","issue":"4","license":[{"start":{"date-parts":[[2023,7,27]],"date-time":"2023-07-27T00:00:00Z","timestamp":1690416000000},"content-version":"vor","delay-in-days":0,"URL":"http://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2023,7,27]],"date-time":"2023-07-27T00:00:00Z","timestamp":1690416000000},"content-version":"tdm","delay-in-days":0,"URL":"https://iopscience.iop.org/info/page/text-and-data-mining"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2023,10,1]]},"abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Implementing algorithms on a fault-tolerant quantum computer will require fast decoding throughput and latency times to prevent an exponential increase in buffer times between the applications of gates. In this work we begin by quantifying these requirements. We then introduce the construction of local neural network (NN) decoders using three-dimensional convolutions. These local decoders are adapted to circuit-level noise and can be applied to surface code volumes of arbitrary size. Their application removes errors arising from a certain number of faults, which serves to substantially reduce the syndrome density. Remaining errors can then be corrected by a global decoder, such as Blossom or union find, with their implementation significantly accelerated due to the reduced syndrome density. However, in the circuit-level setting, the corrections applied by the local decoder introduce many vertical pairs of highlighted vertices. To obtain a low syndrome density in the presence of vertical pairs, we consider a strategy of performing a syndrome collapse which removes many vertical pairs and reduces the size of the decoding graph used by the global decoder. We also consider a strategy of performing a vertical cleanup, which consists of removing all local vertical pairs prior to implementing the global decoder. By applying our local NN decoder and the vertical cleanup strategy to a <jats:italic>d</jats:italic> = 17 surface code volume, we show a <jats:inline-formula>\n                     <jats:tex-math><?CDATA $10^6 \\times$?></jats:tex-math>\n                     <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n                        <mml:msup>\n                           <mml:mn>10</mml:mn>\n                           <mml:mn>6</mml:mn>\n                        </mml:msup>\n                        <mml:mo>×</mml:mo>\n                     </mml:math>\n                     <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"qstace64dieqn1.gif\" xlink:type=\"simple\" />\n                  </jats:inline-formula> speedup of the minimum-weight perfect matching decoder. Lastly, we estimate the cost of implementing our local decoders on field programmable gate arrays.</jats:p>","DOI":"10.1088/2058-9565/ace64d","type":"journal-article","created":{"date-parts":[[2023,7,11]],"date-time":"2023-07-11T22:26:47Z","timestamp":1689114407000},"page":"045011","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":"Techniques for combining fast local decoders with global decoders under circuit-level noise","prefix":"10.1088","volume":"8","author":[{"ORCID":"http://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"given":"Luis","family":"Goncalves","sequence":"additional","affiliation":[]},{"given":"Prasahnt","family":"Sivarajah","sequence":"additional","affiliation":[]},{"given":"Eric","family":"Peterson","sequence":"additional","affiliation":[]},{"given":"Sebastian","family":"Grimberg","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2023,7,27]]},"reference":[],"container-title":"Quantum Science and Technology","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,27]],"date-time":"2023-07-27T08:17:12Z","timestamp":1690445832000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ace64d"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,27]]},"references-count":72,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,7,27]]},"published-print":{"date-parts":[[2023,10,1]]}},"URL":"http://dx.doi.org/10.1088/2058-9565/ace64d","relation":{},"ISSN":["2058-9565"],"subject":["Electrical and Electronic Engineering","Physics and Astronomy (miscellaneous)","Materials Science (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum Sci. Technol.","published":{"date-parts":[[2023,7,27]]},"assertion":[{"value":"Techniques for combining fast local decoders with global decoders under circuit-level noise","name":"article_title","label":"Article Title"},{"value":"Quantum Science and Technology","name":"journal_title","label":"Journal Title"},{"value":"paper","name":"article_type","label":"Article Type"},{"value":"© 2023 The Author(s). Published by IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2022-09-27","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2023-07-11","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2023-07-27","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/2058-9565/ace64d","_hash":"203cf122aba29aca7b6f09d001376da37ba1c19f8c74f37840759ce8b0ad858e"},"expire":1728583705548},"doi:10.22331/q-2023-09-26-1122":{"value":{"indexed":{"date-parts":[[2023,9,27]],"date-time":"2023-09-27T15:14:09Z","timestamp":1695827649819},"reference-count":40,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,9,26]],"date-time":"2023-09-26T00:00:00Z","timestamp":1695686400000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"Simons Collaboration on Ultra-Quantum Matter","award":["651438"]},{"name":"Institute for Quantum Information and Matter, an NSF Physics Frontiers Center","award":["PHY-1733907"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Recently, a class of fractal surface codes (FSCs), has been constructed on fractal lattices with Hausdorff dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mi>&amp;#x03F5;</mml:mi></mml:math>, which admits a fault-tolerant non-Clifford CCZ gate \\cite{zhu2021topological}. We investigate the performance of such FSCs as fault-tolerant quantum memories. We prove that there exist decoding strategies with non-zero thresholds for bit-flip and phase-flip errors in the FSCs with Hausdorff dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mi>&amp;#x03F5;</mml:mi></mml:math>. For the bit-flip errors, we adapt the sweep decoder, developed for string-like syndromes in the regular 3D surface code, to the FSCs by designing suitable modifications on the boundaries of the holes in the fractal lattice. Our adaptation of the sweep decoder for the FSCs maintains its self-correcting and single-shot nature. For the phase-flip errors, we employ the minimum-weight-perfect-matching (MWPM) decoder for the point-like syndromes. We report a sustainable fault-tolerant threshold (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x223C;</mml:mo><mml:mn>1.7</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>) under phenomenological noise for the sweep decoder and the code capacity threshold (lower bounded by <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2.95</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>) for the MWPM decoder for a particular FSC with Hausdorff dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>D</mml:mi><mml:mi>H</mml:mi></mml:msub><mml:mo>&amp;#x2248;</mml:mo><mml:mn>2.966</mml:mn></mml:math>. The latter can be mapped to a lower bound of the critical point of a  confinement-Higgs transition on the fractal lattice, which is tunable via the Hausdorff dimension.</jats:p>","DOI":"10.22331/q-2023-09-26-1122","type":"journal-article","created":{"date-parts":[[2023,9,26]],"date-time":"2023-09-26T14:03:22Z","timestamp":1695737002000},"page":"1122","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Quantum error correction with fractal topological codes","prefix":"10.22331","volume":"7","author":[{"given":"Arpit","family":"Dua","sequence":"first","affiliation":[{"name":"Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125 USA"}]},{"given":"Tomas","family":"Jochym-O&apos;Connor","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598 USA"},{"name":"IBM Almaden Research Center, San Jose, CA 95120 USA"}]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598 USA"},{"name":"IBM Almaden Research Center, San Jose, CA 95120 USA"}]}],"member":"9598","published-online":{"date-parts":[[2023,9,26]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2023-09-26-1122/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,9,26]],"date-time":"2023-09-26T14:03:38Z","timestamp":1695737018000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2023-09-26-1122/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,26]]},"references-count":40,"URL":"http://dx.doi.org/10.22331/q-2023-09-26-1122","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2023,9,26]]},"article-number":"1122","id":"doi:10.22331/q-2023-09-26-1122","_hash":"7381ca4773759af3ae52a37cdb05b677625605b33de8d7b9f130e5ff6ca614e6"},"expire":1728583706562},"doi:10.22331/q-2023-08-29-1093":{"value":{"indexed":{"date-parts":[[2023,10,10]],"date-time":"2023-10-10T22:48:51Z","timestamp":1696978131549},"reference-count":34,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,8,29]],"date-time":"2023-08-29T00:00:00Z","timestamp":1693267200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"name":"JST Moonshot R&D Grant","award":["JPMJMS2061"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"monospace\">f</mml:mi><mml:mi mathvariant=\"monospace\">l</mml:mi><mml:mi mathvariant=\"monospace\">i</mml:mi><mml:mi mathvariant=\"monospace\">p</mml:mi></mml:mrow></mml:math> is an extremely simple and maximally local classical decoder which has been used to great effect in certain classes of classical codes. When applied to quantum codes there exist constant-weight errors (such as half of a stabiliser) which are uncorrectable for this decoder, so previous studies have considered modified versions of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"monospace\">f</mml:mi><mml:mi mathvariant=\"monospace\">l</mml:mi><mml:mi mathvariant=\"monospace\">i</mml:mi><mml:mi mathvariant=\"monospace\">p</mml:mi></mml:mrow></mml:math>, sometimes in conjunction with other decoders. We argue that this may not always be necessary, and present numerical evidence for the existence of a threshold for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"monospace\">f</mml:mi><mml:mi mathvariant=\"monospace\">l</mml:mi><mml:mi mathvariant=\"monospace\">i</mml:mi><mml:mi mathvariant=\"monospace\">p</mml:mi></mml:mrow></mml:math> when applied to the looplike syndromes of a three-dimensional toric code on a cubic lattice. This result can be attributed to the fact that the lowest-weight uncorrectable errors for this decoder are closer (in terms of Hamming distance) to correctable errors than to other uncorrectable errors, and so they are likely to become correctable in future code cycles after transformation by additional noise. Introducing randomness into the decoder can allow it to correct these \"uncorrectable\" errors with finite probability, and for a decoding strategy that uses a combination of belief propagation and probabilistic <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"monospace\">f</mml:mi><mml:mi mathvariant=\"monospace\">l</mml:mi><mml:mi mathvariant=\"monospace\">i</mml:mi><mml:mi mathvariant=\"monospace\">p</mml:mi></mml:mrow></mml:math> we observe a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x223C;</mml:mo><mml:mn>5.5</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> under phenomenological noise. This is comparable to the best known threshold for this code (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x223C;</mml:mo><mml:mn>7.1</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>) which was achieved using belief propagation and ordered statistics decoding [Higgott and Breuckmann, 2022], a strategy with a runtime of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math> as opposed to the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math> when parallelised) runtime of our local decoder. We expect that this strategy could be generalised to work well in other low-density parity check codes, and hope that these results will prompt investigation of other previously overlooked decoders.</jats:p>","DOI":"10.22331/q-2023-08-29-1093","type":"journal-article","created":{"date-parts":[[2023,8,29]],"date-time":"2023-08-29T11:18:41Z","timestamp":1693307921000},"page":"1093","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":"Local Probabilistic Decoding of a Quantum Code","prefix":"10.22331","volume":"7","author":[{"given":"T. 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For a stabilizer code of distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> that can correct up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x230A;</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>d</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x230B;</mml:mo></mml:math> errors, the traditional Shor scheme handles ancilla preparation and measurement faults by performing syndrome measurements until the syndromes are repeated <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math> times in a row; in the worst-case scenario, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:math> rounds of measurements are required. In this work, we improve the Shor FTEC scheme using an adaptive syndrome measurement technique. The syndrome for error correction is determined based on information from the differences of syndromes obtained from consecutive rounds. Our protocols that satisfy the strong and the weak FTEC conditions require no more than <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:math> rounds and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:mo>&amp;#x2212;</mml:mo><mml:mn>2</mml:mn></mml:math> rounds, respectively, and are applicable to any stabilizer code. Our simulations of FTEC protocols with the adaptive schemes on hexagonal color codes of small distances verify that our protocols preserve the code distance, can increase the pseudothreshold, and can decrease the average number of rounds compared to the traditional Shor scheme. We also find that for the code of distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>, our FTEC protocols with the adaptive schemes require no more than <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> rounds on average.</jats:p>","DOI":"10.22331/q-2023-08-08-1075","type":"journal-article","created":{"date-parts":[[2023,8,8]],"date-time":"2023-08-08T13:47:26Z","timestamp":1691502446000},"page":"1075","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":"Adaptive syndrome measurements for Shor-style error correction","prefix":"10.22331","volume":"7","author":[{"ORCID":"http://orcid.org/0000-0002-2865-0705","authenticated-orcid":false,"given":"Theerapat","family":"Tansuwannont","sequence":"first","affiliation":[{"name":"Duke Quantum Center, Duke University, Durham, NC 27701, USA"},{"name":"Department of Electrical and Computer 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Unfortunately, there are no field-theoretic\narguments about why QEC holds in known holographic systems. The purpose\nof this paper is to fill this gap by studying the error correcting\nproperties of the fermionic sector of various large\n<jats:inline-formula><jats:alternatives><jats:tex-math>N</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>N</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\ntheories. 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Steane is one of the best known quantum codes. The Steane code has a desirable property that most basic operations can be performed easily in a fault-tolerant manner. A major obstacle to fault-tolerant quantum computation with the Steane code is fault-tolerant preparation of encoded states, which requires large computational resources. Here we propose efficient state preparation methods for zero and magic states encoded with the Steane code, where the zero state is one of the computational basis states and the magic state allows us to achieve universality in fault-tolerant quantum computation. The methods minimize resource overheads for the fault-tolerant state preparation and therefore reduce necessary resources for quantum computation with the Steane code. Thus, the present results will open a new possibility for efficient fault-tolerant quantum computation.</jats:p>","DOI":"10.1038/srep19578","type":"journal-article","created":{"date-parts":[[2016,1,27]],"date-time":"2016-01-27T10:12:05Z","timestamp":1453889525000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":"Minimizing resource overheads for fault-tolerant preparation of encoded states of the Steane code","prefix":"10.1038","volume":"6","author":[{"given":"Hayato","family":"Goto","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2016,1,27]]},"reference":[],"container-title":"Scientific 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Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> at inverse temperature <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B2;</mml:mi><mml:mo>&amp;#x2265;</mml:mo><mml:mn>0</mml:mn></mml:math> starting from a purification of the thermal state of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math>. The complexity of the quantum algorithm, given by the number of uses of certain unitaries, is <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi><mml:mo stretchy=\"false\">&amp;#x007E;</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>&amp;#x03B2;</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">&amp;#x0394;</mml:mi><mml:mspace width=\"negativethinmathspace\" /><mml:mi>A</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:msub><mml:mi>w</mml:mi><mml:mi>l</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math>, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x0394;</mml:mi><mml:mspace width=\"negativethinmathspace\" /><mml:mi>A</mml:mi></mml:math> is the free-energy difference between <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>w</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:math> is a work cutoff that depends on the properties of the work distribution and the approximation error <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03F5;</mml:mi><mml:mo>&amp;#x003E;</mml:mo><mml:mn>0</mml:mn></mml:math>. If the non-equilibrium process is trivial, this complexity is exponential in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B2;</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:mi>V</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo></mml:math>, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:mi>V</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo></mml:math> is the spectral norm of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>V</mml:mi></mml:math>. This represents a significant improvement of prior quantum algorithms that have complexity exponential in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B2;</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo></mml:math> in the regime where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:mi>V</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:mo>&amp;#x226A;</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo></mml:math>. The dependence of the complexity in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03F5;</mml:mi></mml:math> varies according to the structure of the quantum systems. It can be exponential in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi></mml:math> in general, but we show it to be sublinear in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi></mml:math> if <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> commute, or polynomial in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi></mml:math> if <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> are local spin systems. The possibility of applying a unitary that drives the system out of equilibrium allows one to increase the value of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>w</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:math> and improve the complexity even further. To this end, we analyze the complexity for preparing the thermal state of the transverse field Ising model using different non-equilibrium unitary processes and see significant complexity improvements.</jats:p>","DOI":"10.22331/q-2022-10-06-825","type":"journal-article","created":{"date-parts":[[2022,10,6]],"date-time":"2022-10-06T11:19:57Z","timestamp":1665055197000},"page":"825","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":"Quantum algorithms from fluctuation theorems: Thermal-state preparation","prefix":"10.22331","volume":"6","author":[{"given":"Zoe","family":"Holmes","sequence":"first","affiliation":[{"name":"Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]},{"given":"Gopikrishnan","family":"Muraleedharan","sequence":"additional","affiliation":[{"name":"Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]},{"given":"Rolando D.","family":"Somma","sequence":"additional","affiliation":[{"name":"Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]},{"given":"Yigit","family":"Subasi","sequence":"additional","affiliation":[{"name":"Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]},{"given":"Burak","family":"Şahinoğlu","sequence":"additional","affiliation":[{"name":"Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]}],"member":"9598","published-online":{"date-parts":[[2022,10,6]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2022-10-06-825/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,10,6]],"date-time":"2022-10-06T11:20:19Z","timestamp":1665055219000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-10-06-825/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,6]]},"references-count":96,"URL":"http://dx.doi.org/10.22331/q-2022-10-06-825","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2022,10,6]]},"article-number":"825","id":"doi:10.22331/q-2022-10-06-825","_hash":"8c137ec792e740d77f5204f58ab2c048e2f9881f2d0c146dd439ced70dc7e9a8"},"expire":1731787379037},"doi:10.22331/q-2019-07-12-163":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T22:43:02Z","timestamp":1700606582477},"reference-count":51,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2019,7,12]],"date-time":"2019-07-12T00:00:00Z","timestamp":1562889600000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"<jats:p>We present the problem of approximating the time-evolution operator<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>e</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup></mml:math>to error<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>ϵ</mml:mi></mml:math>, where the Hamiltonian<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:mi>G</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>⊗</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>G</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo><mml:mo>⊗</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math>is the projection of a unitary oracle<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math>onto the state<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>G</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>created by another unitary oracle. Our algorithm solves this with a query complexity<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math>to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math>is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math>in an invariant<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mtext>SU</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math>subspace. A large class of operator functions of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math>can then be computed with optimal query complexity, of which<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>e</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup></mml:math>is a special case.</jats:p>","DOI":"10.22331/q-2019-07-12-163","type":"journal-article","created":{"date-parts":[[2019,7,12]],"date-time":"2019-07-12T02:43:01Z","timestamp":1562899381000},"page":"163","source":"Crossref","is-referenced-by-count":307,"title":"Hamiltonian Simulation by Qubitization","prefix":"10.22331","volume":"3","author":[{"ORCID":"http://orcid.org/0000-0002-6934-1052","authenticated-orcid":false,"given":"Guang Hao","family":"Low","sequence":"first","affiliation":[{"name":"Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA"}]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, Department of Physics, Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA"}]}],"member":"9598","published-online":{"date-parts":[[2019,7,12]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2019-07-12-163/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,9,23]],"date-time":"2022-09-23T18:50:42Z","timestamp":1663959042000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2019-07-12-163/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,12]]},"references-count":51,"URL":"http://dx.doi.org/10.22331/q-2019-07-12-163","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2019,7,12]]},"article-number":"163","id":"doi:10.22331/q-2019-07-12-163","_hash":"094c3bb32994e43fa8bfbb74dc13e33fcc80889296333c4a667d6c73ef7ba5f0"},"expire":1731787379991},"doi:10.1103/PhysRevLett.114.090502":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T22:38:03Z","timestamp":1700606283548},"reference-count":19,"publisher":"American Physical Society (APS)","issue":"9","license":[{"start":{"date-parts":[[2015,3,3]],"date-time":"2015-03-03T00:00:00Z","timestamp":1425340800000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"},{"start":{"date-parts":[[2016,3,2]],"date-time":"2016-03-02T00:00:00Z","timestamp":1456876800000},"content-version":"am","delay-in-days":365,"URL":"http://link.aps.org/licenses/aps-default-accepted-manuscript-license"}],"funder":[{"DOI":"10.13039/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["FT100100761"]},{"DOI":"10.13039/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-12-1-0057"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physrevlett.114.090502","type":"journal-article","created":{"date-parts":[[2015,3,3]],"date-time":"2015-03-03T22:08:25Z","timestamp":1425420505000},"source":"Crossref","is-referenced-by-count":337,"title":"Simulating Hamiltonian Dynamics with a Truncated Taylor Series","prefix":"10.1103","volume":"114","author":[{"given":"Dominic W.","family":"Berry","sequence":"first","affiliation":[]},{"given":"Andrew M.","family":"Childs","sequence":"additional","affiliation":[]},{"given":"Richard","family":"Cleve","sequence":"additional","affiliation":[]},{"given":"Robin","family":"Kothari","sequence":"additional","affiliation":[]},{"given":"Rolando D.","family":"Somma","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2015,3,3]]},"reference":[],"container-title":"Physical Review Letters","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/accepted/10.1103/PhysRevLett.114.090502","content-type":"application/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"http://link.aps.org/article/10.1103/PhysRevLett.114.090502","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.114.090502/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,5,1]],"date-time":"2022-05-01T10:01:39Z","timestamp":1651399299000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevLett.114.090502"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,3,3]]},"references-count":19,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2015,3]]}},"URL":"http://dx.doi.org/10.1103/PhysRevLett.114.090502","relation":{},"ISSN":["0031-9007","1079-7114"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. Rev. Lett.","published":{"date-parts":[[2015,3,3]]},"article-number":"090502","id":"doi:10.1103/PhysRevLett.114.090502","_hash":"33a49cb0c2e8c003335717d45f8b1997bfad2af0f8d9ee5124c3791fe62c09ba"},"expire":1731787380990},"doi:10.22331/q-2023-10-12-1137":{"value":{"indexed":{"date-parts":[[2023,10,13]],"date-time":"2023-10-13T09:18:09Z","timestamp":1697188689578},"reference-count":90,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,10,12]],"date-time":"2023-10-12T00:00:00Z","timestamp":1697068800000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/100000893","name":"Simons Foundation","doi-asserted-by":"crossref","award":["651438, AD"]},{"DOI":"10.13039/100000893","name":"Simons Foundation","doi-asserted-by":"crossref","award":["651444, WS"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>We construct Pauli topological subsystem codes characterized by arbitrary two-dimensional Abelian anyon theories–this includes anyon theories with degenerate braiding relations and those without a gapped boundary to the vacuum. Our work both extends the classification of two-dimensional Pauli topological subsystem codes to systems of composite-dimensional qudits and establishes that the classification is at least as rich as that of Abelian anyon theories. We exemplify the construction with topological subsystem codes defined on four-dimensional qudits based on the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msubsup><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"double-struck\">Z</mml:mi></mml:mrow><mml:mn>4</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup></mml:math> anyon theory with degenerate braiding relations and the chiral semion theory–both of which cannot be captured by topological stabilizer codes. The construction proceeds by \"gauging out\" certain anyon types of a topological stabilizer code. This amounts to defining a gauge group generated by the stabilizer group of the topological stabilizer code and a set of anyonic string operators for the anyon types that are gauged out. The resulting topological subsystem code is characterized by an anyon theory containing a proper subset of the anyons of the topological stabilizer code. We thereby show that every Abelian anyon theory is a subtheory of a stack of toric codes and a certain family of twisted quantum doubles that generalize the double semion anyon theory. We further prove a number of general statements about the logical operators of translation invariant topological subsystem codes and define their associated anyon theories in terms of higher-form symmetries.</jats:p>","DOI":"10.22331/q-2023-10-12-1137","type":"journal-article","created":{"date-parts":[[2023,10,12]],"date-time":"2023-10-12T15:25:17Z","timestamp":1697124317000},"page":"1137","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Pauli topological subsystem codes from Abelian anyon theories","prefix":"10.22331","volume":"7","author":[{"given":"Tyler D.","family":"Ellison","sequence":"first","affiliation":[{"name":"Department of Physics, Yale University, New Haven, CT 06511, USA"}]},{"given":"Yu-An","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Physics, Condensed Matter Theory Center, Joint Quantum Institute, and Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742, USA"}]},{"given":"Arpit","family":"Dua","sequence":"additional","affiliation":[{"name":"Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA"}]},{"given":"Wilbur","family":"Shirley","sequence":"additional","affiliation":[{"name":"School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA"}]},{"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[{"name":"Walter Burke Institute for Theoretical Physics and Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA"},{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA"}]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia"}]}],"member":"9598","published-online":{"date-parts":[[2023,10,12]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2023-10-12-1137/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,10,12]],"date-time":"2023-10-12T15:25:23Z","timestamp":1697124323000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2023-10-12-1137/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,12]]},"references-count":90,"URL":"http://dx.doi.org/10.22331/q-2023-10-12-1137","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2023,10,12]]},"article-number":"1137","id":"doi:10.22331/q-2023-10-12-1137","_hash":"39d0dcccf46f7311848b3f4ae88e1992e17c9056c809046e8177af4599ef32a5"},"expire":1731787382087},"doi:10.1103/PhysRevA.83.022307":{"value":{"indexed":{"date-parts":[[2022,9,2]],"date-time":"2022-09-02T05:31:29Z","timestamp":1662096689543},"reference-count":37,"publisher":"American Physical Society (APS)","issue":"2","license":[{"start":{"date-parts":[[2011,2,10]],"date-time":"2011-02-10T00:00:00Z","timestamp":1297296000000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"},{"start":{"date-parts":[[2012,2,10]],"date-time":"2012-02-10T00:00:00Z","timestamp":1328832000000},"content-version":"am","delay-in-days":365,"URL":"http://link.aps.org/licenses/aps-default-accepted-manuscript-license"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.83.022307","type":"journal-article","created":{"date-parts":[[2011,2,11]],"date-time":"2011-02-11T16:24:31Z","timestamp":1297441471000},"source":"Crossref","is-referenced-by-count":3,"title":"Automated searching for quantum subsystem codes","prefix":"10.1103","volume":"83","author":[{"given":"Gregory M.","family":"Crosswhite","sequence":"first","affiliation":[]},{"given":"Dave","family":"Bacon","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2011,2,10]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/accepted/10.1103/PhysRevA.83.022307","content-type":"application/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"http://link.aps.org/article/10.1103/PhysRevA.83.022307","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.83.022307/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,4,6]],"date-time":"2017-04-06T16:29:26Z","timestamp":1491496166000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.83.022307"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,2,10]]},"references-count":37,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2011,2]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.83.022307","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. Rev. A","published":{"date-parts":[[2011,2,10]]},"article-number":"022307","id":"doi:10.1103/PhysRevA.83.022307","_hash":"62eb08e3c34a72f3f3d70354673d14aea601eab1049974b0722e5f6d535982f1"},"expire":1731787383017},"doi:10.1098/rspa.2007.0028":{"value":{"indexed":{"date-parts":[[2023,8,23]],"date-time":"2023-08-23T11:47:32Z","timestamp":1692791252581},"reference-count":16,"publisher":"The Royal Society","issue":"2087","license":[{"start":{"date-parts":[[2007,8,21]],"date-time":"2007-08-21T00:00:00Z","timestamp":1187654400000},"content-version":"tdm","delay-in-days":0,"URL":"https://royalsociety.org/journals/ethics-policies/data-sharing-mining/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2007,11,8]]},"abstract":"<jats:p>\n            Subsystem codes are a generalization of noiseless subsystems, decoherence-free subspaces and stabilizer codes. We generalize the quantum Singleton bound to\n            <jats:inline-formula>\n              <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"2887equ1.gif\" />\n            </jats:inline-formula>\n            <jats:sub>\n              <jats:italic>q</jats:italic>\n            </jats:sub>\n            -linear subsystem codes. It follows that no subsystem code over a prime field can beat the quantum Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the quantum Hamming bound. A number of open problems concern the comparison in the performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements than an optimal\n            <jats:inline-formula>\n              <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"2887equ2.gif\" />\n            </jats:inline-formula>\n            <jats:sub>\n              <jats:italic>q</jats:italic>\n            </jats:sub>\n            -linear maximum distance separable stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding.\n          </jats:p>","DOI":"10.1098/rspa.2007.0028","type":"journal-article","created":{"date-parts":[[2007,8,21]],"date-time":"2007-08-21T21:46:03Z","timestamp":1187732763000},"page":"2887-2905","source":"Crossref","is-referenced-by-count":14,"title":"On subsystem codes beating the quantum Hamming or Singleton bound","prefix":"10.1098","volume":"463","author":[{"given":"Andreas","family":"Klappenecker","sequence":"first","affiliation":[{"name":"Texas A&amp;M University, College StationTexas, TX 77843, USA"}]},{"given":"Pradeep Kiran","family":"Sarvepalli","sequence":"additional","affiliation":[{"name":"Texas A&amp;M University, College StationTexas, TX 77843, USA"}]}],"member":"175","published-online":{"date-parts":[[2007,8,21]]},"reference":[],"container-title":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","link":[{"URL":"https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2007.0028","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://royalsocietypublishing.org/doi/full-xml/10.1098/rspa.2007.0028","content-type":"application/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2007.0028","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,21]],"date-time":"2021-02-21T07:40:09Z","timestamp":1613893209000},"score":1,"resource":{"primary":{"URL":"https://royalsocietypublishing.org/doi/10.1098/rspa.2007.0028"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,8,21]]},"references-count":16,"journal-issue":{"issue":"2087","published-print":{"date-parts":[[2007,11,8]]}},"alternative-id":["10.1098/rspa.2007.0028"],"URL":"http://dx.doi.org/10.1098/rspa.2007.0028","relation":{},"ISSN":["1364-5021","1471-2946"],"subject":["General Physics and Astronomy","General Engineering","General Mathematics"],"container-title-short":"Proc. R. Soc. A.","published":{"date-parts":[[2007,8,21]]},"id":"doi:10.1098/rspa.2007.0028","_hash":"ad0867f37944e6ff2f622163ab824be756bbcc013d6b2ce3e8d69584afc52bd1"},"expire":1731787384069},"doi:10.1103/PhysRevA.99.052333":{"value":{"indexed":{"date-parts":[[2023,9,3]],"date-time":"2023-09-03T20:09:42Z","timestamp":1693771782466},"reference-count":32,"publisher":"American Physical Society (APS)","issue":"5","license":[{"start":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T00:00:00Z","timestamp":1558483200000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physreva.99.052333","type":"journal-article","created":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T13:53:58Z","timestamp":1558533238000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":8,"title":"Optimal quantum subsystem codes in two dimensions","prefix":"10.1103","volume":"99","author":[{"given":"Theodore J.","family":"Yoder","sequence":"first","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2019,5,22]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevA.99.052333","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.99.052333/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T13:54:01Z","timestamp":1558533241000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.99.052333"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,22]]},"references-count":32,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2019,5]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.99.052333","relation":{},"ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","published":{"date-parts":[[2019,5,22]]},"article-number":"052333","id":"doi:10.1103/PhysRevA.99.052333","_hash":"8a73062c4de3527f9b81a47d2f0f25188707e2d6c7319234687e214f48f2bb1f"},"expire":1731787384991},"doi:10.1038/s41467-017-01418-2":{"value":{"indexed":{"date-parts":[[2023,4,4]],"date-time":"2023-04-04T12:09:21Z","timestamp":1680610161565},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2017,11,6]],"date-time":"2017-11-06T00:00:00Z","timestamp":1509926400000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2017,11,6]],"date-time":"2017-11-06T00:00:00Z","timestamp":1509926400000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"<jats:title>Abstract</jats:title><jats:p>Topological error correction codes are promising candidates to protect quantum computations from the deteriorating effects of noise. While some codes provide high noise thresholds suitable for robust quantum memories, others allow straightforward gate implementation needed for data processing. To exploit the particular advantages of different topological codes for fault-tolerant quantum computation, it is necessary to be able to switch between them. Here we propose a practical solution, subsystem lattice surgery, which requires only two-body nearest-neighbor interactions in a fixed layout in addition to the indispensable error correction. This method can be used for the fault-tolerant transfer of quantum information between arbitrary topological subsystem codes in two dimensions and beyond. In particular, it can be employed to create a simple interface, a quantum bus, between noise resilient surface code memories and flexible color code processors.</jats:p>","DOI":"10.1038/s41467-017-01418-2","type":"journal-article","created":{"date-parts":[[2017,10,31]],"date-time":"2017-10-31T15:11:20Z","timestamp":1509462680000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":"Fault-tolerant interface between quantum memories and quantum processors","prefix":"10.1038","volume":"8","author":[{"ORCID":"http://orcid.org/0000-0001-7815-7006","authenticated-orcid":false,"given":"Hendrik","family":"Poulsen Nautrup","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-1950-8640","authenticated-orcid":false,"given":"Nicolai","family":"Friis","sequence":"additional","affiliation":[]},{"given":"Hans J.","family":"Briegel","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,11,6]]},"reference":[],"container-title":"Nature Communications","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41467-017-01418-2.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-017-01418-2","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-017-01418-2.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,22]],"date-time":"2022-12-22T22:16:28Z","timestamp":1671747388000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41467-017-01418-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,11,6]]},"references-count":47,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,12]]}},"alternative-id":["1418"],"URL":"http://dx.doi.org/10.1038/s41467-017-01418-2","relation":{},"ISSN":["2041-1723"],"subject":["General Physics and Astronomy","General Biochemistry, Genetics and Molecular Biology","General Chemistry","Multidisciplinary"],"container-title-short":"Nat Commun","published":{"date-parts":[[2017,11,6]]},"assertion":[{"value":"20 March 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 September 2017","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 November 2017","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing financial interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"1321","id":"doi:10.1038/s41467-017-01418-2","_hash":"f4bad35157b8593c125d2709bc3b41629074aac9f59c0c717c60768ad5fceea3"},"expire":1731787386004},"doi:10.22331/q-2023-10-25-1156":{"value":{"indexed":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T16:17:23Z","timestamp":1698337043223},"reference-count":16,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T00:00:00Z","timestamp":1698192000000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>In this paper, I present a way to compile the surface code into two-body parity measurements (\"pair measurements\"), where the pair measurements run along the edges of a Cairo pentagonal tiling. The resulting circuit improves on prior work by Chao et al. by using fewer pair measurements per four-body stabilizer measurement (5 instead of 6) and fewer time steps per round of stabilizer measurement (6 instead of 10). Using Monte Carlo sampling, I show that these improvements increase the threshold of the surface code when compiling into pair measurements from <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>0.2</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>0.4</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>, and also that they improve the teraquop footprint at a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.1</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> physical gate error rate from <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>6000</mml:mn></mml:math> qubits to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>3000</mml:mn></mml:math> qubits. However, I also show that the teraquop footprint of Chao et al's construction improves more quickly than mine as physical error rate decreases, and is likely better below a physical gate error rate of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>0.03</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> (due to bidirectional hook errors in my construction). I also compare to the planar honeycomb code, showing that although this work does noticeably reduce the gap between the surface code and the honeycomb code (when compiling into pair measurements), the honeycomb code is still more efficient (threshold <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>0.8</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>, teraquop footprint at <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.1</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>1000</mml:mn></mml:math>).</jats:p>","DOI":"10.22331/q-2023-10-25-1156","type":"journal-article","created":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T15:06:17Z","timestamp":1698246377000},"page":"1156","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"A Pair Measurement Surface Code on Pentagons","prefix":"10.22331","volume":"7","author":[{"given":"Craig","family":"Gidney","sequence":"first","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]}],"member":"9598","published-online":{"date-parts":[[2023,10,25]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2023-10-25-1156/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T15:06:22Z","timestamp":1698246382000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2023-10-25-1156/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,25]]},"references-count":16,"URL":"http://dx.doi.org/10.22331/q-2023-10-25-1156","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2023,10,25]]},"article-number":"1156","id":"doi:10.22331/q-2023-10-25-1156","_hash":"c8f0a0499172ead884eb2b46221cb66a6ae4986273824dcccf2cff52d9d03bf5"},"expire":1731787386996},"doi:10.1103/PhysRevLett.109.260401":{"value":{"indexed":{"date-parts":[[2023,2,2]],"date-time":"2023-02-02T17:07:55Z","timestamp":1675357675164},"reference-count":14,"publisher":"American Physical Society (APS)","issue":"26","license":[{"start":{"date-parts":[[2012,12,26]],"date-time":"2012-12-26T00:00:00Z","timestamp":1356480000000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"},{"start":{"date-parts":[[2013,12,26]],"date-time":"2013-12-26T00:00:00Z","timestamp":1388016000000},"content-version":"am","delay-in-days":365,"URL":"http://link.aps.org/licenses/aps-default-accepted-manuscript-license"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher"},{"DOI":"10.13039/100000183","name":"Army Research Office","doi-asserted-by":"publisher"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physrevlett.109.260401","type":"journal-article","created":{"date-parts":[[2012,12,26]],"date-time":"2012-12-26T21:29:24Z","timestamp":1356557364000},"source":"Crossref","is-referenced-by-count":15,"title":"Gapless Hamiltonians for the Toric Code Using the Projected Entangled Pair State Formalism","prefix":"10.1103","volume":"109","author":[{"given":"Carlos","family":"Fernández-González","sequence":"first","affiliation":[]},{"given":"Norbert","family":"Schuch","sequence":"additional","affiliation":[]},{"given":"Michael M.","family":"Wolf","sequence":"additional","affiliation":[]},{"given":"J. Ignacio","family":"Cirac","sequence":"additional","affiliation":[]},{"given":"David","family":"Pérez-García","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2012,12,26]]},"reference":[],"container-title":"Physical Review Letters","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/accepted/10.1103/PhysRevLett.109.260401","content-type":"application/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"http://link.aps.org/article/10.1103/PhysRevLett.109.260401","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.109.260401/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,2,4]],"date-time":"2022-02-04T17:03:27Z","timestamp":1643994207000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevLett.109.260401"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,12,26]]},"references-count":14,"journal-issue":{"issue":"26","published-print":{"date-parts":[[2012,12]]}},"URL":"http://dx.doi.org/10.1103/PhysRevLett.109.260401","relation":{},"ISSN":["0031-9007","1079-7114"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. Rev. Lett.","published":{"date-parts":[[2012,12,26]]},"article-number":"260401","id":"doi:10.1103/PhysRevLett.109.260401","_hash":"44165e4ec2731dd0e7fa2ae97524161d179b9506959d354a60bae069c597ead8"},"expire":1731787387996},"doi:10.22331/q-2023-11-07-1172":{"value":{"indexed":{"date-parts":[[2023,11,8]],"date-time":"2023-11-08T00:49:10Z","timestamp":1699404550842},"reference-count":75,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,11,7]],"date-time":"2023-11-07T00:00:00Z","timestamp":1699315200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>The typical time-independent view of quantum error correction (QEC) codes hides significant freedom in the decomposition into circuits that are executable on hardware. Using the concept of detecting regions, we design time-dynamic QEC circuits directly instead of designing static QEC codes to decompose into circuits. In particular, we improve on the standard circuit constructions for the surface code, presenting new circuits that can embed on a hexagonal grid instead of a square grid, that can use ISWAP gates instead of CNOT or CZ gates, that can exchange qubit data and measure roles, and that move logical patches around the physical qubit grid while executing. All these constructions use no additional entangling gate layers and display essentially the same logical performance, having teraquop footprints within 25% of the standard surface code circuit. We expect these circuits to be of great interest to quantum hardware engineers, because they achieve essentially the same logical performance as standard surface code circuits while relaxing demands on hardware.</jats:p>","DOI":"10.22331/q-2023-11-07-1172","type":"journal-article","created":{"date-parts":[[2023,11,7]],"date-time":"2023-11-07T14:44:57Z","timestamp":1699368297000},"page":"1172","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Relaxing Hardware Requirements for Surface Code Circuits using Time-dynamics","prefix":"10.22331","volume":"7","author":[{"given":"Matt","family":"McEwen","sequence":"first","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Dave","family":"Bacon","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Seattle, Washington 98103, USA"}]},{"given":"Craig","family":"Gidney","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]}],"member":"9598","published-online":{"date-parts":[[2023,11,7]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2023-11-07-1172/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,11,7]],"date-time":"2023-11-07T14:45:17Z","timestamp":1699368317000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2023-11-07-1172/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,7]]},"references-count":75,"URL":"http://dx.doi.org/10.22331/q-2023-11-07-1172","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2023,11,7]]},"article-number":"1172","id":"doi:10.22331/q-2023-11-07-1172","_hash":"b0be23f270a65abf8f521a64987863921f722f44b9884aba895eba4ed0af769a"},"expire":1731787388998},"doi:10.22331/q-2023-11-14-1183":{"value":{"indexed":{"date-parts":[[2023,11,15]],"date-time":"2023-11-15T00:17:15Z","timestamp":1700007435383},"reference-count":41,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,11,14]],"date-time":"2023-11-14T00:00:00Z","timestamp":1699920000000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"crossref","award":["EP/T001062/1"]},{"DOI":"10.13039/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"crossref","award":["EP/W032635/1"]},{"DOI":"10.13039/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"crossref","award":["EP/Y004655/1"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>Fault-tolerant quantum computing requires classical hardware to perform the decoding necessary for error correction. The Union–Find decoder is one of the best candidates for this. It has remarkably organic characteristics, involving the growth and merger of data structures through nearest-neighbour steps; this naturally suggests the possibility of its realisation using a lattice of simple processors with nearest-neighbour links. In this way the computational load can be distributed with near-ideal parallelism. Here we show for the first time that this strict (rather than partial) locality is practical, with a worst-case runtime <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>d</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math> and mean runtime subquadratic in the surface code distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>. A novel parity-calculation scheme is employed which can simplify previously proposed architectures, and our approach is optimised for circuit-level noise. We compare our local realisation with one augmented by long-range links; while the latter is of course faster, we note that local asynchronous logic could negate the difference.</jats:p>","DOI":"10.22331/q-2023-11-14-1183","type":"journal-article","created":{"date-parts":[[2023,11,14]],"date-time":"2023-11-14T13:34:48Z","timestamp":1699968888000},"page":"1183","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Actis: A Strictly Local Union–Find Decoder","prefix":"10.22331","volume":"7","author":[{"ORCID":"http://orcid.org/0000-0001-6187-7402","authenticated-orcid":false,"given":"Tim","family":"Chan","sequence":"first","affiliation":[{"name":"Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom"}]},{"ORCID":"http://orcid.org/0000-0002-7766-5348","authenticated-orcid":false,"given":"Simon C.","family":"Benjamin","sequence":"additional","affiliation":[{"name":"Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom"},{"name":"Quantum Motion, 9 Sterling Way, London N7 9HJ, United Kingdom"}]}],"member":"9598","published-online":{"date-parts":[[2023,11,14]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2023-11-14-1183/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,11,14]],"date-time":"2023-11-14T13:34:59Z","timestamp":1699968899000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2023-11-14-1183/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,14]]},"references-count":41,"URL":"http://dx.doi.org/10.22331/q-2023-11-14-1183","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2023,11,14]]},"article-number":"1183","id":"doi:10.22331/q-2023-11-14-1183","_hash":"05d6abed0f2066d32fdd16ea94f3e440334bb763aca76c3269a85ab5b5622b21"},"expire":1731787390090},"doi:10.1103/PhysRevB.75.075103":{"value":{"indexed":{"date-parts":[[2023,10,12]],"date-time":"2023-10-12T16:30:26Z","timestamp":1697128226893},"reference-count":46,"publisher":"American Physical Society (APS)","issue":"7","license":[{"start":{"date-parts":[[2007,2,7]],"date-time":"2007-02-07T00:00:00Z","timestamp":1170806400000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physrevb.75.075103","type":"journal-article","created":{"date-parts":[[2007,2,8]],"date-time":"2007-02-08T04:07:54Z","timestamp":1170907674000},"source":"Crossref","is-referenced-by-count":98,"title":"Exact topological quantum order in<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math>and beyond: Branyons and brane-net condensates","prefix":"10.1103","volume":"75","author":[{"given":"H.","family":"Bombin","sequence":"first","affiliation":[]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2007,2,7]]},"reference":[],"container-title":"Physical Review B","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevB.75.075103","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevB.75.075103/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,24]],"date-time":"2019-04-24T12:34:15Z","timestamp":1556109255000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevB.75.075103"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,2,7]]},"references-count":46,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2007,2]]}},"URL":"http://dx.doi.org/10.1103/PhysRevB.75.075103","relation":{},"ISSN":["1098-0121","1550-235X"],"subject":["Condensed Matter Physics","Electronic, Optical and Magnetic Materials"],"container-title-short":"Phys. 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B","published":{"date-parts":[[2007,2,7]]},"article-number":"075103","id":"doi:10.1103/PhysRevB.75.075103","_hash":"a8749850cf082a42786e7b6dd1aa7d8beb6fa57c518e77f5db7b487ee9e95f14"},"expire":1731787391086},"doi:10.1109/TIT.2020.3015683":{"value":{"indexed":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T23:19:06Z","timestamp":1669418346908},"reference-count":65,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","license":[{"start":{"date-parts":[[2020,10,1]],"date-time":"2020-10-01T00:00:00Z","timestamp":1601510400000},"content-version":"vor","delay-in-days":0,"URL":"https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html"},{"start":{"date-parts":[[2020,10,1]],"date-time":"2020-10-01T00:00:00Z","timestamp":1601510400000},"content-version":"am","delay-in-days":0,"URL":"https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html"},{"start":{"date-parts":[[2020,10,1]],"date-time":"2020-10-01T00:00:00Z","timestamp":1601510400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https://doi.org/10.15223/policy-029"},{"start":{"date-parts":[[2020,10,1]],"date-time":"2020-10-01T00:00:00Z","timestamp":1601510400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https://doi.org/10.15223/policy-037"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1718494","1908730"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2020,10]]},"DOI":"10.1109/tit.2020.3015683","type":"journal-article","created":{"date-parts":[[2020,8,11]],"date-time":"2020-08-11T22:30:11Z","timestamp":1597185011000},"page":"6104-6120","source":"Crossref","is-referenced-by-count":4,"title":"Kerdock Codes Determine Unitary 2-Designs","prefix":"10.1109","volume":"66","author":[{"given":"Trung","family":"Can","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-2369-3159","authenticated-orcid":false,"given":"Narayanan","family":"Rengaswamy","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-2084-9717","authenticated-orcid":false,"given":"Robert","family":"Calderbank","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-5521-4397","authenticated-orcid":false,"given":"Henry D.","family":"Pfister","sequence":"additional","affiliation":[]}],"member":"263","reference":[],"container-title":"IEEE Transactions on Information Theory","original-title":[],"link":[{"URL":"https://ieeexplore.ieee.org/ielam/18/9201197/9164978-aam.pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"http://xplorestaging.ieee.org/ielx7/18/9201197/09164978.pdf?arnumber=9164978","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T17:26:14Z","timestamp":1651080374000},"score":1,"resource":{"primary":{"URL":"https://ieeexplore.ieee.org/document/9164978/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10]]},"references-count":65,"journal-issue":{"issue":"10"},"URL":"http://dx.doi.org/10.1109/TIT.2020.3015683","relation":{},"ISSN":["0018-9448","1557-9654"],"subject":["Library and Information Sciences","Computer Science Applications","Information Systems"],"container-title-short":"IEEE Trans. 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Quantum Optimisation and Machine Learning"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2023,10,1]]},"abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>We introduce a high-level graphical framework for designing and analysing quantum error correcting codes, centred on what we term the coherent parity check (CPC). The graphical formulation is based on the diagrammatic tools of the <jats:italic>ZX</jats:italic>-calculus of quantum observables. The resulting framework leads to a construction for stabilizer codes that allows us to design and verify a broad range of quantum codes based on classical ones, and that gives a means of discovering large classes of codes using both analytical and numerical methods. We focus in particular on the smaller codes that will be the first used by near-term devices. We show how CSS codes form a subset of CPC codes and, more generally, how to compute stabilizers for a CPC code. As an explicit example of this framework, we give a method for turning almost any pair of classical <jats:inline-formula>\n                     <jats:tex-math><?CDATA $[n,k,3]$?></jats:tex-math>\n                     <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n                        <mml:mo stretchy=\"false\">[</mml:mo>\n                        <mml:mi>n</mml:mi>\n                        <mml:mo>,</mml:mo>\n                        <mml:mi>k</mml:mi>\n                        <mml:mo>,</mml:mo>\n                        <mml:mn>3</mml:mn>\n                        <mml:mo stretchy=\"false\">]</mml:mo>\n                     </mml:math>\n                     <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"qstacf157ieqn1.gif\" xlink:type=\"simple\" />\n                  </jats:inline-formula> codes into a <jats:inline-formula>\n                     <jats:tex-math><?CDATA $[[2n - k + 2, k, 3]]$?></jats:tex-math>\n                     <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n                        <mml:mo stretchy=\"false\">[</mml:mo>\n                        <mml:mo stretchy=\"false\">[</mml:mo>\n                        <mml:mn>2</mml:mn>\n                        <mml:mi>n</mml:mi>\n                        <mml:mo>−</mml:mo>\n                        <mml:mi>k</mml:mi>\n                        <mml:mo>+</mml:mo>\n                        <mml:mn>2</mml:mn>\n                        <mml:mo>,</mml:mo>\n                        <mml:mi>k</mml:mi>\n                        <mml:mo>,</mml:mo>\n                        <mml:mn>3</mml:mn>\n                        <mml:mo stretchy=\"false\">]</mml:mo>\n                        <mml:mo stretchy=\"false\">]</mml:mo>\n                     </mml:math>\n                     <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"qstacf157ieqn2.gif\" xlink:type=\"simple\" />\n                  </jats:inline-formula> CPC code. Further, we give a simple technique for machine search which yields thousands of potential codes, and demonstrate its operation for distance 3 and 5 codes. Finally, we use the graphical tools to demonstrate how Clifford computation can be performed within CPC codes. As our framework gives a new tool for constructing small- to medium-sized codes with relatively high code rates, it provides a new source for codes that could be suitable for emerging devices, while its <jats:italic>ZX</jats:italic>-calculus foundations enable natural integration of error correction with graphical compiler toolchains. It also provides a powerful framework for reasoning about all stabilizer quantum error correction codes of any size.</jats:p>","DOI":"10.1088/2058-9565/acf157","type":"journal-article","created":{"date-parts":[[2023,8,17]],"date-time":"2023-08-17T22:29:20Z","timestamp":1692311360000},"page":"045028","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":"Graphical structures for design and verification of quantum error correction","prefix":"10.1088","volume":"8","author":[{"ORCID":"http://orcid.org/0000-0002-1293-0761","authenticated-orcid":true,"given":"Nicholas","family":"Chancellor","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-6090-9684","authenticated-orcid":true,"given":"Aleks","family":"Kissinger","sequence":"additional","affiliation":[]},{"given":"Stefan","family":"Zohren","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-9202-1156","authenticated-orcid":true,"given":"Joschka","family":"Roffe","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4965-0584","authenticated-orcid":true,"given":"Dominic","family":"Horsman","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2023,9,22]]},"reference":[],"container-title":"Quantum Science and Technology","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acf157","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acf157/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acf157","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acf157/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acf157/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acf157/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acf157/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acf157/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,22]],"date-time":"2023-09-22T07:29:29Z","timestamp":1695367769000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/acf157"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,22]]},"references-count":81,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,9,22]]},"published-print":{"date-parts":[[2023,10,1]]}},"URL":"http://dx.doi.org/10.1088/2058-9565/acf157","relation":{},"ISSN":["2058-9565"],"subject":["Electrical and Electronic Engineering","Physics and Astronomy (miscellaneous)","Materials Science (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum Sci. Technol.","published":{"date-parts":[[2023,9,22]]},"assertion":[{"value":"Graphical structures for design and verification of quantum error correction","name":"article_title","label":"Article Title"},{"value":"Quantum Science and Technology","name":"journal_title","label":"Journal Title"},{"value":"paper","name":"article_type","label":"Article Type"},{"value":"© 2023 The Author(s). Published by IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2022-11-01","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2023-08-17","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2023-09-22","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/2058-9565/acf157","_hash":"9a0b2dcdae776aac95ae00e64a9afb6e9d45317ed07c3675834a03939a8e675f"},"expire":1731787394082},"doi:10.22331/q-2023-10-24-1153":{"value":{"indexed":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T16:15:00Z","timestamp":1698250500999},"reference-count":56,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,10,24]],"date-time":"2023-10-24T00:00:00Z","timestamp":1698105600000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"<jats:p>The promise of high-rate low-density parity check (LDPC) codes to substantially reduce the overhead of fault-tolerant quantum computation depends on constructing efficient, fault-tolerant implementations of logical gates on such codes. Transversal gates are the simplest type of fault-tolerant gate, but the potential of transversal gates on LDPC codes has hitherto been largely neglected. We investigate the transversal gates that can be implemented in hypergraph product codes, a class of LDPC codes. Our analysis is aided by the construction of a symplectic canonical basis for the logical operators of hypergraph product codes, a result that may be of independent interest. We show that in these codes transversal gates can implement Hadamard (up to logical SWAP gates) and control-Z on all logical qubits. Moreover, we show that sequences of transversal operations, interleaved with error correction, allow implementation of entangling gates between arbitrary pairs of logical qubits in the same code block. We thereby demonstrate that transversal gates can be used as the basis for universal quantum computing on LDPC codes, when supplemented with state injection.</jats:p>","DOI":"10.22331/q-2023-10-24-1153","type":"journal-article","created":{"date-parts":[[2023,10,24]],"date-time":"2023-10-24T16:28:16Z","timestamp":1698164896000},"page":"1153","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Partitioning qubits in hypergraph product codes to implement logical gates","prefix":"10.22331","volume":"7","author":[{"given":"Armanda O.","family":"Quintavalle","sequence":"first","affiliation":[{"name":"Department of Physics & Astronomy, University of Sheffield, Sheffield, S3 7RH, United Kingdom"},{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"}]},{"given":"Paul","family":"Webster","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, 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Progress towards the NLTS conjecture was made by Eldar and Harrow (Foundations of Computer Science 2017), who proved a closely related theorem called No Low-Error Trivial States (NLETS). In this paper, we give a much simpler proof of the NLETS theorem and use the same technique to establish superpolynomial circuit size lower bounds for noisy ground states of local Hamiltonians (assuming QCMA != QMA), resolving an open question of Eldar and Harrow. We discuss the new light our results cast on the relationship between NLTS and NLETS.\nFinally, our techniques imply the existence of approximate quantum low-weight check (qLWC) codes with linear rate, linear distance, and constant weight checks. These codes are similar to quantum LDPC codes except (1) each particle may participate in a large number of checks, and (2) errors only need to be corrected up to fidelity 1 - 1/poly(n). This stands in contrast to the best-known stabilizer LDPC codes due to Freedman, Meyer, and Luo which achieve a distance of O(sqrt{n log n}).\nThe principal technique used in our results is to leverage the Feynman-Kitaev clock construction to approximately embed a subspace of states defined by a circuit as the ground space of a local Hamiltonian.","container-title":"Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH, Wadern/Saarbruecken, Germany","DOI":"10.4230/LIPICS.ICALP.2018.91","publisher":"Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH, Wadern/Saarbruecken, Germany","title":"Approximate Low-Weight Check Codes and Circuit Lower Bounds for Noisy Ground States","URL":"http://drops.dagstuhl.de/opus/volltexte/2018/9095/","reference":[],"_hash":"36bdcf0cfd03b96f24f45cab6f999614f99a6cdc3e8c1ed3a18199c79a5110da"},"expire":1731787400861},"doi:10.1103/PhysRevA.66.032304":{"value":{"indexed":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T21:43:37Z","timestamp":1694468617914},"reference-count":7,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2002,9,13]],"date-time":"2002-09-13T00:00:00Z","timestamp":1031875200000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.66.032304","type":"journal-article","created":{"date-parts":[[2002,9,13]],"date-time":"2002-09-13T22:41:55Z","timestamp":1031956915000},"source":"Crossref","is-referenced-by-count":39,"title":"Exact performance of concatenated quantum codes","prefix":"10.1103","volume":"66","author":[{"given":"Benjamin","family":"Rahn","sequence":"first","affiliation":[]},{"given":"Andrew C.","family":"Doherty","sequence":"additional","affiliation":[]},{"given":"Hideo","family":"Mabuchi","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2002,9,13]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.66.032304","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.66.032304/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,15]],"date-time":"2017-06-15T07:41:33Z","timestamp":1497512493000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.66.032304"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,9,13]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2002,9]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.66.032304","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. 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We obtain their weight distribution and present additional bounds that arise from Rains' shadow inequalities. Our main result can be seen as a generalization of bounds that are known for the two special cases of stabilizer QMDS codes and absolutely maximally entangled states, and confirms the quantum MDS conjecture in the special case of distance-three codes. As the existence of QMDS codes is linked to that of highly entangled subspaces (in which every vector has uniform <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>r</mml:mi></mml:math>-body marginals) of maximal dimension, our methods directly carry over to address questions in multipartite entanglement.</jats:p>","DOI":"10.22331/q-2020-06-18-284","type":"journal-article","created":{"date-parts":[[2020,6,18]],"date-time":"2020-06-18T14:08:48Z","timestamp":1592489328000},"page":"284","source":"Crossref","is-referenced-by-count":25,"title":"Quantum Codes of Maximal Distance and Highly Entangled Subspaces","prefix":"10.22331","volume":"4","author":[{"ORCID":"http://orcid.org/0000-0002-3856-4018","authenticated-orcid":false,"given":"Felix","family":"Huber","sequence":"first","affiliation":[{"name":"ICFO - The Institute of Photonic Sciences, 08860 Castelldefels (Barcelona), Spain"},{"name":"Institut für Theoretische Physik, Universität zu Köln, 50937 Köln, Germany"},{"name":"Naturwissenschaftlich-Technische Fakultät, Universität Siegen, 57068 Siegen, Germany"}]},{"ORCID":"http://orcid.org/0000-0002-3720-5195","authenticated-orcid":false,"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University of Gdansk, 80-308 Gdańsk, Poland"},{"name":"Max Planck Institute for the Science of Light, 91058 Erlangen, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2020,6,18]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2020-06-18-284/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2020,6,18]],"date-time":"2020-06-18T14:08:51Z","timestamp":1592489331000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2020-06-18-284/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,18]]},"references-count":72,"URL":"http://dx.doi.org/10.22331/q-2020-06-18-284","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2020,6,18]]},"article-number":"284","id":"doi:10.22331/q-2020-06-18-284","_hash":"19beab8638844b5171e6dd99a11342d8b7cac850f4e8719889d7b987c5f3bcb1"},"expire":1731787403088},"doi:10.1103/PhysRevA.102.012607":{"value":{"indexed":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T17:00:51Z","timestamp":1698253251236},"reference-count":53,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[2020,7,6]],"date-time":"2020-07-06T00:00:00Z","timestamp":1593993600000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"funder":[{"DOI":"10.13039/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-14-CE26-0029"]},{"DOI":"10.13039/100012946","name":"Association Sorbonne Université","doi-asserted-by":"publisher"},{"name":"Labex SEAM","award":["ANR-10-LABX-0096","ANR-18-IDEX-0001"]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physreva.102.012607","type":"journal-article","created":{"date-parts":[[2020,7,6]],"date-time":"2020-07-06T16:23:59Z","timestamp":1594052639000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":26,"title":"Generation of a time-frequency grid state with integrated biphoton frequency combs","prefix":"10.1103","volume":"102","author":[{"given":"N.","family":"Fabre","sequence":"first","affiliation":[]},{"given":"G.","family":"Maltese","sequence":"additional","affiliation":[]},{"given":"F.","family":"Appas","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-0447-5380","authenticated-orcid":true,"given":"S.","family":"Felicetti","sequence":"additional","affiliation":[]},{"given":"A.","family":"Ketterer","sequence":"additional","affiliation":[]},{"given":"A.","family":"Keller","sequence":"additional","affiliation":[]},{"given":"T.","family":"Coudreau","sequence":"additional","affiliation":[]},{"given":"F.","family":"Baboux","sequence":"additional","affiliation":[]},{"given":"M. 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To this end, we introduce the circuit-Quantum Electrodynamics (QED) dual-rail qubit in which our physical qubit is encoded in the single-photon subspace,\n            <jats:inline-formula>\n              <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\">\n                <mml:mrow>\n                  <mml:mo stretchy=\"false\">{</mml:mo>\n                  <mml:mo>|</mml:mo>\n                  <mml:mn>01</mml:mn>\n                  <mml:mo>〉</mml:mo>\n                  <mml:mo>,</mml:mo>\n                  <mml:mo>|</mml:mo>\n                  <mml:mn>10</mml:mn>\n                  <mml:mo>〉</mml:mo>\n                  <mml:mo stretchy=\"false\">}</mml:mo>\n                </mml:mrow>\n              </mml:math>\n            </jats:inline-formula>\n            , of two superconducting microwave cavities. The dominant photon loss errors can be detected and converted into erasure errors, which are in general much easier to correct. In contrast to linear optics, a circuit-QED implementation of the dual-rail code offers unique capabilities. Using just one additional transmon ancilla per dual-rail qubit, we describe how to perform a gate-based set of universal operations that includes state preparation, logical readout, and parametrizable single and two-qubit gates. Moreover, first-order hardware errors in the cavities and the transmon can be detected and converted to erasure errors in all operations, leaving background Pauli errors that are orders of magnitude smaller. Hence, the dual-rail cavity qubit exhibits a favorable hierarchy of error rates and is expected to perform well below the relevant QEC thresholds with today’s coherence times.\n          </jats:p>","DOI":"10.1073/pnas.2221736120","type":"journal-article","created":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T17:42:22Z","timestamp":1696614142000},"update-policy":"http://dx.doi.org/10.1073/pnas.cm10313","source":"Crossref","is-referenced-by-count":0,"title":"Dual-rail encoding with superconducting cavities","prefix":"10.1073","volume":"120","author":[{"ORCID":"http://orcid.org/0009-0009-3086-3513","authenticated-orcid":false,"given":"James D.","family":"Teoh","sequence":"first","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"ORCID":"http://orcid.org/0000-0001-6457-6971","authenticated-orcid":false,"given":"Patrick","family":"Winkel","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"given":"Harshvardhan K.","family":"Babla","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"ORCID":"http://orcid.org/0000-0003-1248-7568","authenticated-orcid":false,"given":"Benjamin J.","family":"Chapman","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"ORCID":"http://orcid.org/0000-0002-6711-4585","authenticated-orcid":false,"given":"Jahan","family":"Claes","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"ORCID":"http://orcid.org/0000-0003-1111-4460","authenticated-orcid":false,"given":"Stijn J.","family":"de Graaf","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"given":"John W. 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Berlekamp, \\emph{Algebraic Coding Theory}, McGraw-Hill, 1968"},"id":"preset:Berlekamp","_hash":"d49c294e3b88574a6373189934cafbddb057b78d8f85f09423d4133ebcedeb97"},"expire":1700686113993},"preset:splag":{"value":{"_ready_formatted":{"flm":"10.1007/978-1-4757-6568-7"},"id":"preset:splag","_hash":"592116cfe3f41d1d02dbdbcd0624c1d2e35eaf4c0dab417117719c76db91c920"},"expire":1700686114040},"preset:HPRainsSloane":{"value":{"_ready_formatted":{"flm":"E. M. Rains and N. J. A. Sloane, “Self-dual codes,” in Handbook of Coding Theory, eds. V. S. Pless and W. C. Huffman. Amsterdam: Elsevier, 1998, pp. 177–294."},"id":"preset:HPRainsSloane","_hash":"80664b888f5fd618980aadcbecddaaf0b9caffd9083b0197ec1ab0bb93448b77"},"expire":1700686114086},"preset:HKSbasics":{"value":{"_ready_formatted":{"flm":"W. C. Huffman, J.-L. Kim, and P. 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MacKay. \\emph{Information Theory, Inference and Learning Algorithms}. Cambridge university press, 2003."},"id":"preset:MacKay","_hash":"2c4d23d3a87dd312d64e5a96a40d2b0580457dc416972b644b6318426fffe45c"},"expire":1700686114500},"preset:HKSpolar":{"value":{"_ready_formatted":{"flm":"N. Presman, Simon Litsyn, \"Polar codes.\" Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) \\href{https://doi.org/10.1201/9781315147901}{DOI}"},"id":"preset:HKSpolar","_hash":"97687b96086f5a2c3b4252ac5883544789e0ffdf83f21fb9669fc1bde8d7ceb4"},"expire":1700686114547},"preset:Hill":{"value":{"_ready_formatted":{"flm":"R. Hill. \\emph{A First Course In Coding Theory}. Oxford University Press, 1988."},"id":"preset:Hill","_hash":"9cc60c1f3d98ee23143d8d36bcc39f0c42916712a9e59478e8ed8d917d659044"},"expire":1700686114594},"manual:{V. A. Kotelnikov, “The theory of optimum noise immunity,” PhD Thesis, Molotov Energy Institute, Moscow, Jan. 1947}":{"value":{"_ready_formatted":{"flm":"{V. A. Kotelnikov, “The theory of optimum noise immunity,” PhD Thesis, Molotov Energy Institute, Moscow, Jan. 1947}"},"id":"manual:{V. A. Kotelnikov, “The theory of optimum noise immunity,” PhD Thesis, Molotov Energy Institute, Moscow, Jan. 1947}","_hash":"7e85a0a05b41f411b5615fe53e7c9c08d765807745f01fdab75a8babc7f08225"},"expire":1700686058769}}
\ No newline at end of file
diff --git a/eczoodb/eczoodb.js b/eczoodb/eczoodb.js
index ab768bc..d914e26 100644
--- a/eczoodb/eczoodb.js
+++ b/eczoodb/eczoodb.js
@@ -435,6 +435,8 @@ export class EcZooDb extends ZooDb
     async validate()
     {
         // Check that there are no cycles in parent-child relationships.
+        
+        // eslint-disable-next-line no-unused-vars
         let all_parent_child_sorted_codes = this.code_parent_child_sort(
             Object.values(this.objects.code)
         );
diff --git a/site/site_data/eczoodb.js b/site/site_data/eczoodb.js
index a55b727..7f0b8c7 100644
--- a/site/site_data/eczoodb.js
+++ b/site/site_data/eczoodb.js
@@ -1,6 +1,27 @@
 
 let cached_eczoodb = null;
 
+
+function get_error_string(err)
+{
+    let errstr = null;
+    try {
+        if (err && err.__class__ != null) {
+            if (err.args) {
+                errstr = `${err.__class__.__name__}: ${err.args}`;
+            } else {
+                errstr = `${err.__class__.__name__} (no information)`;
+            }
+        } else {
+            errstr = ''+err;
+        }
+    } catch (tostrerr) {
+        errstr = Object.toString(err);
+    }
+    return errstr;
+}
+
+
 module.exports = async (configData) => {
 
     const eczoo_config = configData.eczoo_config;
@@ -35,8 +56,8 @@ module.exports = async (configData) => {
             cached_eczoodb.install_zoo_loader_handler(loader_handler);
 
         } catch (err) {
+            console.error(`ERROR INITIALIZING ZOO: ${get_error_string(err)}`);
             console.error(err);
-            console.error(`ERROR INITIALIZING ZOO: ${err}`);
             throw err;
         }
     }
@@ -46,7 +67,7 @@ module.exports = async (configData) => {
         await cached_eczoodb.load();
 
     } catch (err) {
-        console.error(`ERROR LOADING ZOO [eczoodb.js]:`);
+        console.error(`ERROR LOADING ZOO [site/site_data/eczoodb.js]: ${get_error_string(err)}`);
         console.error(err);
         throw err;
     }
diff --git a/yarn.lock b/yarn.lock
index e519ef8..03a1d7a 100644
--- a/yarn.lock
+++ b/yarn.lock
@@ -1896,7 +1896,7 @@ __metadata:
 
 "@phfaist/zoodb@https://github.com/phfaist/zoodb.git#main":
   version: 0.1.1
-  resolution: "@phfaist/zoodb@https://github.com/phfaist/zoodb.git#commit=4b73b10a1116e4343020146203e25b88fe14288b"
+  resolution: "@phfaist/zoodb@https://github.com/phfaist/zoodb.git#commit=7095e22bff35dda56a223b0d18afcd266caad7c0"
   dependencies:
     "@apidevtools/json-schema-ref-parser": ^10.1.0
     base32-encode: ^2.0.0
@@ -1918,7 +1918,7 @@ __metadata:
     stream-parser: ^0.3.1
     tippy.js: ^6.3.7
     url: ^0.11.0
-  checksum: 17dc7df482ff9d19ed35a9f4b6dc7fd6cdb98677c8bda89aa229d6d3c3872da8c08e59fe17927465f61876fa65c0fce748cce45bb9b255775529759dbd71199d
+  checksum: 75169962b87489405f11f8dcd5994252dc574d26f2d4a0e105d4acd8db716c6e643a85e7bc44f5a4e681392d4f1c07907e6ffd48b89435d269a12ca95e0d3c0f
   languageName: node
   linkType: hard