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The data and ancilla qubits of the small stellated dodecahedron code can be located on the edges respectively vertices of a small stellated dodecahedron, making this code suitable for three-dimensional connectivity. This code encodes eight logical qubits into 30 physical qubits (plus 22 ancilla qubits for parity check measurements) in contrast with one logical qubit into nine physical qubits (plus eight ancilla qubits) for the surface code. We develop fault-tolerant parity check circuits and a decoder for this code, allowing us to numerically assess the circuit-based pseudo-threshold.\n This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.","DOI":"10.1098/rsta.2017.0323","type":"journal-article","created":{"date-parts":[[2018,5,28]],"date-time":"2018-05-28T18:10:15Z","timestamp":1527531015000},"page":"20170323","update-policy":"http://dx.doi.org/10.1098/crossmark-policy","source":"Crossref","is-referenced-by-count":10,"title":"The small stellated dodecahedron code and friends","prefix":"10.1098","volume":"376","author":[{"given":"J.","family":"Conrad","sequence":"first","affiliation":[{"name":"JARA Institute for Quantum Information, RWTH Aachen University, Aachen 52056, Germany"}]},{"ORCID":"http://orcid.org/0000-0003-3239-5783","authenticated-orcid":false,"given":"C.","family":"Chamberland","sequence":"additional","affiliation":[{"name":"Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1"}]},{"given":"N. P.","family":"Breuckmann","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, London WC1E 6BT, UK"}]},{"given":"B. M.","family":"Terhal","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands"},{"name":"Institute for Theoretical Nanoelectronics, Forschungszentrum Juelich, 52425 Juelich, Germany"}]}],"member":"175","published-online":{"date-parts":[[2018,5,28]]},"reference":[],"container-title":"Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","link":[{"URL":"https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2017.0323","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://royalsocietypublishing.org/doi/full-xml/10.1098/rsta.2017.0323","content-type":"application/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2017.0323","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,18]],"date-time":"2021-02-18T19:24:22Z","timestamp":1613676262000},"score":1,"resource":{"primary":{"URL":"https://royalsocietypublishing.org/doi/10.1098/rsta.2017.0323"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,28]]},"references-count":31,"journal-issue":{"issue":"2123","published-print":{"date-parts":[[2018,7,13]]}},"alternative-id":["10.1098/rsta.2017.0323"],"URL":"http://dx.doi.org/10.1098/rsta.2017.0323","relation":{},"ISSN":["1364-503X","1471-2962"],"subject":["General Physics and Astronomy","General Engineering","General Mathematics"],"container-title-short":"Phil. Trans. R. Soc. A.","published":{"date-parts":[[2018,5,28]]},"assertion":[{"value":"2018-03-16","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2018-05-28","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"id":"doi:10.1098/rsta.2017.0323","_hash":"73cbc6485aefaf5f6472aad3ca88131193cdb59e95203c044203e15eed1eb417"},"expire":1706743963457},"doi:10.7907/5NDZ-W890":{"value":{"type":"thesis","id":"doi:10.7907/5NDZ-W890","categories":["Physics","anyon interferometry"],"language":"en","author":[{"family":"Bonderson","given":"Parsa Hassan"}],"issued":{"date-parts":[[2007]]},"abstract":"This thesis is primarily a study of the measurement theory of non-Abelian anyons through interference experiments. We give an introduction to the theory of anyon models, providing all the formalism necessary to apply standard quantum measurement theory to such systems. This formalism is then applied to give a detailed analysis of a Mach-Zehnder interferometer for arbitrary anyon models. In this treatment, we find that the collapse behavior exhibited by a target anyon in a superposition of states is determined by the monodromy of the probe anyons with the target. Such measurements may also be used to gain knowledge that would help to properly identify the anyon model describing an unknown system. The techniques used and results obtained from this model interferometer have general applicability, and we use them to also describe the interferometry measurements in a two point-contact interferometer proposed for non-Abelian fractional quantum Hall states. Additionally, we give the complete description of a number of important examples of anyon models, as well as their corresponding quantities that are relevant for interferometry. Finally, we give a partial classification of anyon models with small numbers of particle types.","DOI":"10.7907/5NDZ-W890","publisher":"California Institute of Technology","title":"Non-Abelian Anyons and Interferometry","URL":"https://resolver.caltech.edu/CaltechETD:etd-06042007-101617","copyright":"No commercial reproduction, distribution, display or performance rights in this work are provided.","version":"Final","reference":[],"_hash":"945aff280ddabba6091440ae1d06301a9fffd6edf79c4bf91387ffd4f654cc13"},"expire":1708162717058},"doi:10.1103/PhysRevA.55.900":{"value":{"indexed":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T09:11:40Z","timestamp":1676970700992},"reference-count":21,"publisher":"American Physical Society (APS)","issue":"2","license":[{"start":{"date-parts":[[1997,2,1]],"date-time":"1997-02-01T00:00:00Z","timestamp":854755200000},"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.55.900","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T02:27:51Z","timestamp":1027736871000},"page":"900-911","source":"Crossref","is-referenced-by-count":922,"title":"Theory of quantum error-correcting codes","prefix":"10.1103","volume":"55","author":[{"given":"Emanuel","family":"Knill","sequence":"first","affiliation":[]},{"given":"Raymond","family":"Laflamme","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[1997,2,1]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.55.900","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.55.900/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,15]],"date-time":"2017-06-15T08:47:24Z","timestamp":1497516444000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.55.900"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,2,1]]},"references-count":21,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1997,2]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.55.900","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. 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H.","family":"van Lint","sequence":"first","affiliation":[]}],"member":"297","container-title":"Graduate Texts in Mathematics","original-title":[],"link":[{"URL":"http://link.springer.com/content/pdf/10.1007/978-3-642-58575-3.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://link.springer.com/content/pdf/10.1007/978-3-642-58575-3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,5]],"date-time":"2019-04-05T20:56:40Z","timestamp":1554497800000},"score":1,"resource":{"primary":{"URL":"http://link.springer.com/10.1007/978-3-642-58575-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999]]},"ISBN":["9783642636530","9783642585753"],"references-count":0,"URL":"http://dx.doi.org/10.1007/978-3-642-58575-3","relation":{},"ISSN":["0072-5285"],"published":{"date-parts":[[1999]]},"reference":[],"id":"doi:10.1007/978-3-642-58575-3","_hash":"a34d5dfc32764bef033dc2832190a365b970bf834fe269fcd323bbac5cb7ca8a"},"expire":1708162719652},"doi:10.1103/PhysRevA.55.R839":{"value":{"indexed":{"date-parts":[[2023,2,15]],"date-time":"2023-02-15T22:19:16Z","timestamp":1676499556014},"reference-count":5,"publisher":"American Physical Society (APS)","issue":"2","license":[{"start":{"date-parts":[[1997,2,1]],"date-time":"1997-02-01T00:00:00Z","timestamp":854755200000},"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.55.r839","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T02:27:51Z","timestamp":1027736871000},"page":"R839-R841","source":"Crossref","is-referenced-by-count":29,"title":"Correcting quantum errors in higher spin systems","prefix":"10.1103","volume":"55","author":[{"given":"H. 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Rev. B","published":{"date-parts":[[2005,7,6]]},"article-number":"035307","id":"doi:10.1103/PhysRevB.72.035307","_hash":"affdb1fbf700e80df47bcad501b785c878e1af2311278835b8d8f1da47a4372f"},"expire":1708162722665},"doi:10.4230/LIPIcs.ITCS.2022.68":{"value":{"type":"article-journal","id":"doi:10.4230/LIPIcs.ITCS.2022.68","categories":["Fault-tolerant quantum computation","quantum error correction","Theory of computation → Quantum computation theory","Hardware → Quantum error correction and fault tolerance"],"language":"en","author":[{"family":"Fawzi","given":"Omar"},{"family":"Müller-Hermes","given":"Alexander"},{"family":"Shayeghi","given":"Ala"}],"contributor":[{"family":"Braverman","given":"Mark"}],"issued":{"date-parts":[[2022]]},"abstract":"The threshold theorem is a fundamental result in the theory of fault-tolerant quantum computation stating that arbitrarily long quantum computations can be performed with a polylogarithmic overhead provided the noise level is below a constant level. 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":1708162723432},"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":"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., ((n,2n&#x2212;6,3))2 for n from 7 to 14. We also found new ((6,2,3))2 and ((7,2,3))2 codes that are not equivalent to any stabilizer code, and extensive numerical evidence with VarQEC suggests that a ((7,3,3))2 code does not exist. Furthermore, we found many new channel-adaptive codes for error models involving nearest-neighbor correlated errors. 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We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are \n\n \n m\n m\n \n\n distances between distinct points in it and it is a spherical \n\n \n \n (\n 2\n m\n −\n 1\n )\n \n (2m-1)\n \n\n-design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the \n\n \n \n E\n 8\n \n E_8\n \n\n and Leech lattices. We also prove the same result for the vertices of the \n\n \n 600\n 600\n \n\n-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.

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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":1712098434027},"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":"Fault-tolerant quantum coding\n \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 et al.\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 \n \n Science\n , this issue p.\n 266\n ","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. 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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.","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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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 K-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.","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 *","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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It starts with a demonstration of how,in these electrodynamic systems, both the superconducting gap and the long-range Coulomb interactions contribute to the existence of collective modesthat have extremely low dissipationand that can be quantized. The chapter also gives in-depth coverage of quantum harmonic oscillators in all their glory from various perspectives. Superconducting qubits are then introduced, starting with the fundamental theory of the Josephson effect and moving on to a presentation of the variety of existing superconducting qubits. Particular attention is given to the dispersive readout of qubits through their interaction with a cavity into which flying modes can be scattered. 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M.","family":"Girvin","sequence":"first","affiliation":[]}],"member":"286","reference":[],"container-title":"Quantum Machines: Measurement and Control of Engineered Quantum Systems","original-title":[],"link":[{"URL":"https://academic.oup.com/book/chapter-pdf/45138561/acprof-9780199681181-chapter-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,2]],"date-time":"2022-08-02T11:03:58Z","timestamp":1659438238000},"score":1,"resource":{"primary":{"URL":"https://academic.oup.com/book/43704/chapter/367194888"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,6,12]]},"references-count":248,"URL":"http://dx.doi.org/10.1093/acprof:oso/9780199681181.003.0003","relation":{},"published":{"date-parts":[[2014,6,12]]},"id":"doi:10.1093/acprof:oso/9780199681181.003.0003","_hash":"c4ec4f0aba8ac42fefe8c973386406a05af65d2cea1b0c0aca354613e96566fd"},"expire":1712590642941},"doi:10.1017/CBO9781139034807.009":{"value":{"indexed":{"date-parts":[[2022,6,12]],"date-time":"2022-06-12T01:10:33Z","timestamp":1654996233263},"edition-number":"1","reference-count":0,"publisher":"Cambridge University Press","license":[{"start":{"date-parts":[[2013,9,5]],"date-time":"2013-09-05T00:00:00Z","timestamp":1378339200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://www.cambridge.org/core/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2013,9,12]]},"DOI":"10.1017/cbo9781139034807.009","type":"book-chapter","created":{"date-parts":[[2013,9,5]],"date-time":"2013-09-05T05:03:20Z","timestamp":1378357400000},"page":"181-200","source":"Crossref","is-referenced-by-count":1,"title":"Entanglement-assisted quantum error-correcting codes","prefix":"10.1017","author":[{"given":"Todd A.","family":"Brun","sequence":"first","affiliation":[]},{"given":"Min-Hsiu","family":"Hsieh","sequence":"additional","affiliation":[]}],"member":"56","container-title":"Quantum Error Correction","original-title":[],"link":[{"URL":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F5EFA6A75DF07CEAD178295DDC20C995","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,12]],"date-time":"2022-06-12T00:52:14Z","timestamp":1654995134000},"score":1,"resource":{"primary":{"URL":"https://www.cambridge.org/core/product/identifier/CBO9781139034807A065/type/book_part"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,9,12]]},"references-count":0,"URL":"http://dx.doi.org/10.1017/CBO9781139034807.009","relation":{},"published":{"date-parts":[[2013,9,12]]},"reference":[],"id":"doi:10.1017/CBO9781139034807.009","_hash":"8ff42c42e2a0d5104ef4f960bf29dde5622147e7d206c45f0e62c0ed4bf5cd55"},"expire":1712590643928},"doi:10.1109/12.286310":{"value":{"indexed":{"date-parts":[[2023,4,12]],"date-time":"2023-04-12T20:53:16Z","timestamp":1681332796473},"reference-count":16,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","license":[{"start":{"date-parts":[[1994,6,1]],"date-time":"1994-06-01T00:00:00Z","timestamp":770428800000},"content-version":"vor","delay-in-days":0,"URL":"https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[1994,6]]},"DOI":"10.1109/12.286310","type":"journal-article","created":{"date-parts":[[2002,8,24]],"date-time":"2002-08-24T20:26:37Z","timestamp":1030220797000},"page":"759-764","source":"Crossref","is-referenced-by-count":44,"title":"Design of CAECC - cellular automata based error correcting code","prefix":"10.1109","volume":"43","author":[{"given":"D.R.","family":"Chowdhury","sequence":"first","affiliation":[]},{"given":"S.","family":"Basu","sequence":"additional","affiliation":[]},{"given":"I.S.","family":"Gupta","sequence":"additional","affiliation":[]},{"given":"P.P.","family":"Chaudhuri","sequence":"additional","affiliation":[]}],"member":"263","reference":[],"container-title":"IEEE Transactions on Computers","original-title":[],"link":[{"URL":"http://xplorestaging.ieee.org/ielx1/12/7122/00286310.pdf?arnumber=286310","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,12]],"date-time":"2022-01-12T16:23:15Z","timestamp":1642004595000},"score":1,"resource":{"primary":{"URL":"http://ieeexplore.ieee.org/document/286310/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,6]]},"references-count":16,"journal-issue":{"issue":"6"},"URL":"http://dx.doi.org/10.1109/12.286310","relation":{},"ISSN":["0018-9340"],"subject":["Computational Theory and Mathematics","Hardware and Architecture","Theoretical Computer Science","Software"],"container-title-short":"IEEE Trans. Comput.","published":{"date-parts":[[1994,6]]},"id":"doi:10.1109/12.286310","_hash":"8a4d5b681fe50ad1567a603016ed8df730c09112fcab739cf7a5c4ad1c556140"},"expire":1712590644841},"doi:10.1103/PhysRevApplied.19.034050":{"value":{"indexed":{"date-parts":[[2023,3,16]],"date-time":"2023-03-16T04:40:00Z","timestamp":1678941600763},"reference-count":49,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2023,3,15]],"date-time":"2023-03-15T00:00:00Z","timestamp":1678838400000},"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":["W911NF-21-1-0007"]},{"DOI":"10.13039/100010582","name":"National Computational Infrastructure","doi-asserted-by":"publisher"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevapplied.19.034050","type":"journal-article","created":{"date-parts":[[2023,3,15]],"date-time":"2023-03-15T14:58:57Z","timestamp":1678892337000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":"Local Predecoder to Reduce the Bandwidth and Latency of Quantum Error Correction","prefix":"10.1103","volume":"19","author":[{"ORCID":"http://orcid.org/0000-0002-4865-7015","authenticated-orcid":true,"given":"Samuel C.","family":"Smith","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4387-670X","authenticated-orcid":true,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2023,3,15]]},"reference":[],"container-title":"Physical Review Applied","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevApplied.19.034050","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevApplied.19.034050/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,15]],"date-time":"2023-03-15T14:59:03Z","timestamp":1678892343000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevApplied.19.034050"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,15]]},"references-count":49,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,3]]}},"URL":"http://dx.doi.org/10.1103/PhysRevApplied.19.034050","relation":{},"ISSN":["2331-7019"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. 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Applied","published":{"date-parts":[[2023,3,15]]},"article-number":"034050","id":"doi:10.1103/PhysRevApplied.19.034050","_hash":"47028a04808b75dd61a2521b939fd7619da446aecae797da4c58369fed34f989"},"expire":1712590660919},"doi:10.22331/q-2023-03-09-940":{"value":{"indexed":{"date-parts":[[2023,3,10]],"date-time":"2023-03-10T05:54:06Z","timestamp":1678427646331},"reference-count":80,"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":"National Science Foundation","award":["DGE-1656518"]},{"name":"Centre of Excellence in Engineered Quantum Systems","award":["CE170100009"]},{"name":"ARO","award":["W911NF-21-1-0007"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"Self-correcting quantum memories demonstrate robust properties that can be exploited to improve active quantum error-correction protocols. 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.","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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Quantum","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PRXQuantum.4.020303","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PRXQuantum.4.020303/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T15:02:17Z","timestamp":1680879737000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PRXQuantum.4.020303"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,7]]},"references-count":85,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,4]]}},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.020303","relation":{},"ISSN":["2691-3399"],"subject":["General Physics and Astronomy","Mathematical Physics","Applied Mathematics","Electronic, Optical and Magnetic Materials","Electrical and Electronic Engineering","General Computer Science"],"container-title-short":"PRX Quantum","published":{"date-parts":[[2023,4,7]]},"article-number":"020303","id":"doi:10.1103/PRXQuantum.4.020303","_hash":"44c98d9952119717dd84cab38233bcd2540d31749938934be2b3ad9b768fe933"},"expire":1712590662945},"doi:10.1088/2058-9565/acb796":{"value":{"indexed":{"date-parts":[[2023,3,30]],"date-time":"2023-03-30T12:28:53Z","timestamp":1680179333147},"reference-count":41,"publisher":"IOP Publishing","issue":"2","license":[{"start":{"date-parts":[[2023,2,9]],"date-time":"2023-02-09T00:00:00Z","timestamp":1675900800000},"content-version":"vor","delay-in-days":0,"URL":"https://iopscience.iop.org/page/copyright"},{"start":{"date-parts":[[2023,2,9]],"date-time":"2023-02-09T00:00:00Z","timestamp":1675900800000},"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":[[2023,4,1]]},"abstract":"Abstract\n The simulation of systems of interacting fermions is one of the most anticipated applications of quantum computers. 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 non-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.","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":1712590663926},"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":"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.","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":1712590665009},"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":" 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. 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In most QEC codes2–8, 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 scenarios9–17. 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 architecture18, where the logical qubit is binomially encoded in photon-number states of a microwave cavity8, 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. Our work illustrates the potential of hardware-efficient discrete-variable encodings for fault-tolerant quantum computation19.","DOI":"10.1038/s41586-023-05784-4","type":"journal-article","created":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T17:03:38Z","timestamp":1679504618000},"page":"56-60","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":"Beating the break-even point with a discrete-variable-encoded logical 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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 bepSTC ≈ 1.045%. Such a high threshold together with local parity-check measurements make the 3D STC particularly appealing for realizing fault-tolerant quantum computing.","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":1715612576849},"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":"AbstractElectrons 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.","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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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.","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":1718831603958},"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":"AbstractQuantum 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.","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":1718831605129},"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":"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 17% under independent bit-and phase-flip data noise (to be compared to the ideal threshold of 20.6%) and a threshold value of 14% under depolarizing data noise (compared to the ideal threshold of 18.9%), which are on par with thresholds achieved by non-BP post-processing methods.","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":1718831606053},"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":1718831607020},"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":"We present magic state factory constructions for producing|CCZ⟩states and|T⟩states. For the|CCZ⟩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 of12d×6d(wheredis the code distance) and produces one|CCZ⟩every5.5dsurface code cycles. Our|T⟩state factory uses the|CCZ⟩factory's output and a catalyst|T⟩state to exactly transform one|CCZ⟩state into two|T⟩states. It has a footprint25%smaller than the factory in \\cite{fowler2018} but outputs|T⟩states twice as quickly. We show how to generalize the catalyzed transformation to arbitrary phase angles, and note that the caseθ=22.5∘produces a particularly efficient circuit for producing|T⟩states. Compared to using the12d×8d×6.5d|T⟩factory of \\cite{fowler2018}, our|CCZ⟩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 of10−3, our CCZ factory can produce∼1010states 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∼1012by increasing the factory footprint∼20%.","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|CCZ⟩to2|T⟩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":1718831608055},"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":"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. We give a first taste of the power of the calculus as a language for lattice surgery by considering two operations (T gates and producing a CNOT) and show how ZX diagram re-write rules give lattice surgery procedures for these operations that are novel, efficient, and highly configurable.","DOI":"10.22331/q-2020-01-09-218","type":"journal-article","created":{"date-parts":[[2020,1,9]],"date-time":"2020-01-09T19:54:16Z","timestamp":1578599656000},"page":"218","source":"Crossref","is-referenced-by-count":13,"title":"The ZX calculus is a language for surface code lattice surgery","prefix":"10.22331","volume":"4","author":[{"ORCID":"http://orcid.org/0000-0001-9549-5146","authenticated-orcid":false,"given":"Niel","family":"de Beaudrap","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Oxford, Parks Road, Oxford, OX1 3QD"}]},{"ORCID":"http://orcid.org/0000-0003-4965-0584","authenticated-orcid":false,"given":"Dominic","family":"Horsman","sequence":"additional","affiliation":[{"name":"Department of Physics, Durham University, South Road, Durham, DH1 1LE Department of Computer Science, University of Oxford, Parks Road, Oxford, OX1 3QD"}]}],"member":"9598","published-online":{"date-parts":[[2020,1,9]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2020-01-09-218/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2020,6,5]],"date-time":"2020-06-05T18:19:50Z","timestamp":1591381190000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2020-01-09-218/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,9]]},"references-count":47,"URL":"http://dx.doi.org/10.22331/q-2020-01-09-218","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2020,1,9]]},"article-number":"218","id":"doi:10.22331/q-2020-01-09-218","_hash":"5dd2287e91568ba6d4ffc89ce0afc2402586ba6ca11c443b1f830231880b4dfc"},"expire":1718831608955},"doi:10.1088/1751-8113/42/9/095302":{"value":{"indexed":{"date-parts":[[2023,6,13]],"date-time":"2023-06-13T17:51:13Z","timestamp":1686678673142},"reference-count":18,"publisher":"IOP Publishing","issue":"9","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2009,3,6]]},"DOI":"10.1088/1751-8113/42/9/095302","type":"journal-article","created":{"date-parts":[[2009,2,5]],"date-time":"2009-02-05T04:15:17Z","timestamp":1233807317000},"page":"095302","source":"Crossref","is-referenced-by-count":79,"title":"Quantum measurements and gates by code deformation","prefix":"10.1088","volume":"42","author":[{"given":"H","family":"Bombin","sequence":"first","affiliation":[]},{"given":"M A","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2009,2,4]]},"reference":[],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"link":[{"URL":"http://stacks.iop.org/1751-8121/42/i=9/a=095302/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,1]],"date-time":"2021-10-01T03:26:38Z","timestamp":1633058798000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1751-8113/42/9/095302"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,2,4]]},"references-count":18,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2009,3,6]]}},"alternative-id":["S1751-8113(09)86826-1"],"URL":"http://dx.doi.org/10.1088/1751-8113/42/9/095302","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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Theor.","published":{"date-parts":[[2009,2,4]]},"id":"doi:10.1088/1751-8113/42/9/095302","_hash":"125ec259e206ce78c5591e149b311124aece02b018574439ddb2f8a66d9daa2f"},"expire":1718831609949},"doi:10.21468/SciPostPhysLectNotes.70":{"value":{"indexed":{"date-parts":[[2023,6,13]],"date-time":"2023-06-13T04:28:45Z","timestamp":1686630525727},"reference-count":132,"publisher":"Stichting SciPost","license":[{"start":{"date-parts":[[2023,6,12]],"date-time":"2023-06-12T00:00:00Z","timestamp":1686528000000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["W911NF-18-1-0212"]},{"DOI":"10.13039/100005326","name":"Yale University","doi-asserted-by":"publisher"}],"content-domain":{"domain":["scipost.org"],"crossmark-restriction":false},"abstract":"These lecture notes from the 2019 Les Houches Summer School on “Quantum Information Machines” are intended to provide an introduction to classical and quantum error correction with bits and qubits, and with continuous variable systems (harmonic oscillators). The focus on the latter will be on practical examples that can be realized today or in the near future with a modular architecture based on superconducting electrical circuits and microwave photons. The goal and vision is “hardware-efficient” quantum error correction that does not require exponentially large hardware overhead in order to achieve practical and useful levels of fault tolerance and circuit depth.","DOI":"10.21468/scipostphyslectnotes.70","type":"journal-article","created":{"date-parts":[[2023,6,12]],"date-time":"2023-06-12T14:52:35Z","timestamp":1686581555000},"update-policy":"http://dx.doi.org/10.21468/scipost.crossmarkpolicy","source":"Crossref","is-referenced-by-count":0,"title":"Introduction to quantum error correction and fault tolerance","prefix":"10.21468","author":[{"given":"Steven M.","family":"Girvin","sequence":"first","affiliation":[{"name":"Yale University"}]}],"member":"8907","published-online":{"date-parts":[[2023,6,12]]},"reference":[],"container-title":"SciPost Physics Lecture Notes","original-title":[],"link":[{"URL":"https://scipost.org/10.21468/SciPostPhysLectNotes.70/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://scipost.org/10.21468/SciPostPhysLectNotes.70/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,12]],"date-time":"2023-06-12T14:52:48Z","timestamp":1686581568000},"score":1,"resource":{"primary":{"URL":"https://scipost.org/10.21468/SciPostPhysLectNotes.70"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,12]]},"references-count":132,"URL":"http://dx.doi.org/10.21468/SciPostPhysLectNotes.70","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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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 2T of the quaternions. The resulting 2T-qutrit naturally inherits the algebraic properties of the group 2T 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.","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 2T-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":1718831618964},"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":"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.","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":1718831619963},"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 α of the black hole entropy. In the Schrödinger picture, the boundary subregion encodes the α-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 α-bit codes. Unintuitively, Hawking radiation always reveals the α-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/N.","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.","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":1721576483984},"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":"In this work, we compute the number of [[n,k]]d stabilizer codes made up of d-dimensional qudits, for arbitrary positive integers d. In a seminal work by Gross \\cite{Gross2006} the number of [[n,k]]d stabilizer codes was computed for the case when d is a prime (or the power of a prime, i.e., d=pm, 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 [[n,k]]d codes, and use this in conjunction with the Chinese remainder theorem to count the number of [[n,k]]d codes. Our work overlaps with \\cite{Gross2006} when d 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 d 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.","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 d = 17 surface code volume, we show a \n \n \n \n 10\n 6\n \n ×\n \n \n speedup of the minimum-weight perfect matching decoder. Lastly, we estimate the cost of implementing our local decoders on field programmable gate arrays.","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":1728583705597},"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":"Recently, a class of fractal surface codes (FSCs), has been constructed on fractal lattices with Hausdorff dimension 2+&#x03F5;, 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 2+&#x03F5;. 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 (&#x223C;1.7&#x0025;) under phenomenological noise for the sweep decoder and the code capacity threshold (lower bounded by 2.95&#x0025;) for the MWPM decoder for a particular FSC with Hausdorff dimension DH&#x2248;2.966. 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.","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'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":1728583706611},"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":"flip 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 flip, 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 flip 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 flip we observe a threshold of &#x223C;5.5&#x0025; under phenomenological noise. This is comparable to the best known threshold for this code (&#x223C;7.1&#x0025;) which was achieved using belief propagation and ordered statistics decoding [Higgott and Breuckmann, 2022], a strategy with a runtime of O(n3) as opposed to the O(n) (O(1) 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.","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. R.","family":"Scruby","sequence":"first","affiliation":[{"name":"Okinawa Institute of Science and Technology, Okinawa, 904-0495, Japan"}]},{"given":"K.","family":"Nemoto","sequence":"additional","affiliation":[{"name":"Okinawa Institute of Science and Technology, Okinawa, 904-0495, Japan"}]}],"member":"9598","published-online":{"date-parts":[[2023,8,29]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2023-08-29-1093/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,8,29]],"date-time":"2023-08-29T11:18:57Z","timestamp":1693307937000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2023-08-29-1093/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,29]]},"references-count":34,"URL":"http://dx.doi.org/10.22331/q-2023-08-29-1093","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2023,8,29]]},"article-number":"1093","id":"doi:10.22331/q-2023-08-29-1093","_hash":"6ea5e213eafec4ed7fe8e9dcbe5e817b287878b74d155e0f7ef28c9e9d3028b9"},"expire":1728583707704},"doi:10.22331/q-2023-08-08-1075":{"value":{"indexed":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T17:11:44Z","timestamp":1697044304821},"reference-count":40,"publisher":"Verein 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":"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 d that can correct up to t=&#x230A;(d&#x2212;1)/2&#x230B; errors, the traditional Shor scheme handles ancilla preparation and measurement faults by performing syndrome measurements until the syndromes are repeated t+1 times in a row; in the worst-case scenario, (t+1)2 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 (t+3)2/4&#x2212;1 rounds and (t+3)2/4&#x2212;2 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. 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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 price 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 detailed size winding [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 N 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.","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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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\nNN\ntheories. Specifically we examine SU(N)SU(N)\nmatrix quantum mechanics and 3-rank tensor\nO(N)^3O(N)3\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\nNN\nanalysis and do not appeal to a particular form of Hamiltonian or\nholography.","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":1729266847992},"doi:10.1103/PhysRev.58.1098":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T21:39:24Z","timestamp":1700602764269},"reference-count":18,"publisher":"American Physical Society (APS)","issue":"12","license":[{"start":{"date-parts":[[1940,12,15]],"date-time":"1940-12-15T00:00:00Z","timestamp":-916617600000},"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/physrev.58.1098","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T00:30:13Z","timestamp":1027729813000},"page":"1098-1113","source":"Crossref","is-referenced-by-count":3119,"title":"Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet","prefix":"10.1103","volume":"58","author":[{"given":"T.","family":"Holstein","sequence":"first","affiliation":[]},{"given":"H.","family":"Primakoff","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[1940,12,15]]},"reference":[],"container-title":"Physical Review","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRev.58.1098","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRev.58.1098/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,12,10]],"date-time":"2017-12-10T03:56:03Z","timestamp":1512878163000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRev.58.1098"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1940,12,15]]},"references-count":18,"journal-issue":{"issue":"12","published-print":{"date-parts":[[1940,12]]}},"URL":"http://dx.doi.org/10.1103/PhysRev.58.1098","relation":{},"ISSN":["0031-899X"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. 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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. 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Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state of H1 at inverse temperature &#x03B2;&#x2265;0 starting from a purification of the thermal state of H0. The complexity of the quantum algorithm, given by the number of uses of certain unitaries, is O&#x007E;(e&#x03B2;(&#x0394;A&#x2212;wl)/2), where &#x0394;A is the free-energy difference between H1 and H0, and wl is a work cutoff that depends on the properties of the work distribution and the approximation error &#x03F5;&#x003E;0. If the non-equilibrium process is trivial, this complexity is exponential in &#x03B2;&#x2016;V&#x2016;, where &#x2016;V&#x2016; is the spectral norm of V. This represents a significant improvement of prior quantum algorithms that have complexity exponential in &#x03B2;&#x2016;H1&#x2016; in the regime where &#x2016;V&#x2016;&#x226A;&#x2016;H1&#x2016;. The dependence of the complexity in &#x03F5; varies according to the structure of the quantum systems. It can be exponential in 1/&#x03F5; in general, but we show it to be sublinear in 1/&#x03F5; if H0 and H1 commute, or polynomial in 1/&#x03F5; if H0 and H1 are local spin systems. The possibility of applying a unitary that drives the system out of equilibrium allows one to increase the value of wl 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.","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":1731787379100},"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":"We present the problem of approximating the time-evolution operatore−iH^tto errorϵ, where the HamiltonianH^=(⟨G|⊗I^)U^(|G⟩⊗I^)is the projection of a unitary oracleU^onto the state|G⟩created by another unitary oracle. Our algorithm solves this with a query complexityO(t+log⁡(1/ϵ))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 ared-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 whereH^is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed anyH^in an invariantSU(2)subspace. A large class of operator functions ofH^can then be computed with optimal query complexity, of whiche−iH^tis a special case.","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":1731787380055},"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":1731787381054},"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":"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 Z4(1) 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.","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":1731787382151},"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":1731787383081},"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":"\n Subsystem codes are a generalization of noiseless subsystems, decoherence-free subspaces and stabilizer codes. We generalize the quantum Singleton bound to\n \n \n \n \n q\n \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 \n \n \n \n q\n \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 ","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&M University, College StationTexas, TX 77843, USA"}]},{"given":"Pradeep Kiran","family":"Sarvepalli","sequence":"additional","affiliation":[{"name":"Texas A&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":1731787384133},"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":1731787385055},"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":"AbstractTopological 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.","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":1731787386068},"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":"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 &#x2248;0.2&#x0025; to &#x2248;0.4&#x0025;, and also that they improve the teraquop footprint at a 0.1&#x0025; physical gate error rate from &#x2248;6000 qubits to &#x2248;3000 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 &#x2248;0.03&#x0025; (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 &#x2248;0.8&#x0025;, teraquop footprint at 0.1&#x0025; of &#x2248;1000).","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":1731787387060},"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":1731787388060},"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":"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.","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":1731787389062},"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":"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 O(d3) and mean runtime subquadratic in the surface code distance d. 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.","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":1731787390154},"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 inD=3and 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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The graphical formulation is based on the diagrammatic tools of the ZX-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 \n \n \n [\n n\n ,\n k\n ,\n 3\n ]\n \n \n codes into a \n \n \n [\n [\n 2\n n\n −\n k\n +\n 2\n ,\n k\n ,\n 3\n ]\n ]\n \n \n 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. 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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":1731787394146},"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":"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. 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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). 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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":"2019-04-12","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2019-07-18","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2019-07-30","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/ab3372","_hash":"f17650087a535f5057117518a4e946afabf6d178c38e8fd0bc36a0123fda6f67"},"expire":1732264933982},"doi:10.1103/PhysRevA.81.032301":{"value":{"indexed":{"date-parts":[[2023,10,12]],"date-time":"2023-10-12T16:21:13Z","timestamp":1697127673857},"reference-count":31,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2010,3,3]],"date-time":"2010-03-03T00:00:00Z","timestamp":1267574400000},"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.81.032301","type":"journal-article","created":{"date-parts":[[2010,3,3]],"date-time":"2010-03-03T21:03:43Z","timestamp":1267650223000},"source":"Crossref","is-referenced-by-count":72,"title":"Topological subsystem codes","prefix":"10.1103","volume":"81","author":[{"given":"H.","family":"Bombin","sequence":"first","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2010,3,3]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.81.032301","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.81.032301/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,19]],"date-time":"2017-06-19T05:46:49Z","timestamp":1497851209000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.81.032301"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,3,3]]},"references-count":31,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2010,3]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.81.032301","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. 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A: Math. Theor. 48, 215302 (2015)], with a specific implementation of the boundary. It utilizes weight-six (XYZXYZ) plaquette stabilizers and weight-two (XX) link stabilizers on a planar hexagonal grid composed of 2d2 qubits for code distance d, 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 X, Y, or Z 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 d2 only for pure Y noise, here the code distance is 2d2 for both pure Z and pure Y noise. Thresholds for noise with finite Z 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.","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":3,"title":"The XYZ2 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":"76381d0fcf9ef68ce93daabb1d3d6c55d43a4f1d0c8ba037dde694f90d067de1"},"expire":1732264953989},"doi:10.1088/1751-8121/ac7a75":{"value":{"indexed":{"date-parts":[[2023,5,1]],"date-time":"2023-05-01T14:08:55Z","timestamp":1682950135412},"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":"Abstract\n 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%.","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":2,"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":"9ed870c4e6560b49078c9adb8a3133aad84eeafa9577ba7ef85c4ccf4008149e"},"expire":1732264955065},"doi:10.1109/TIT.2016.2555700":{"value":{"indexed":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T05:01:06Z","timestamp":1698987666321},"reference-count":32,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","license":[{"start":{"date-parts":[[2016,6,1]],"date-time":"2016-06-01T00:00:00Z","timestamp":1464739200000},"content-version":"vor","delay-in-days":0,"URL":"https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html"}],"funder":[{"name":"EU Program QALGO"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2016,6]]},"DOI":"10.1109/tit.2016.2555700","type":"journal-article","created":{"date-parts":[[2016,4,21]],"date-time":"2016-04-21T18:18:34Z","timestamp":1461262714000},"page":"3731-3744","source":"Crossref","is-referenced-by-count":55,"title":"Constructions and Noise Threshold of Hyperbolic Surface Codes","prefix":"10.1109","volume":"62","author":[{"given":"Nikolas P.","family":"Breuckmann","sequence":"first","affiliation":[]},{"given":"Barbara M.","family":"Terhal","sequence":"additional","affiliation":[]}],"member":"263","reference":[],"container-title":"IEEE Transactions on Information Theory","original-title":[],"link":[{"URL":"http://xplorestaging.ieee.org/ielx7/18/7473802/07456305.pdf?arnumber=7456305","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,12]],"date-time":"2022-01-12T16:44:00Z","timestamp":1642005840000},"score":1,"resource":{"primary":{"URL":"http://ieeexplore.ieee.org/document/7456305/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,6]]},"references-count":32,"journal-issue":{"issue":"6"},"URL":"http://dx.doi.org/10.1109/TIT.2016.2555700","relation":{},"ISSN":["0018-9448","1557-9654"],"subject":["Library and Information Sciences","Computer Science Applications","Information Systems"],"container-title-short":"IEEE Trans. Inform. Theory","published":{"date-parts":[[2016,6]]},"id":"doi:10.1109/TIT.2016.2555700","_hash":"9c10d1c6308b77ac7d9f093112e2ed36b67c7819e015a97c65f49f19ecb7dd95"},"expire":1732264955985},"doi:10.1038/s41534-020-00330-w":{"value":{"indexed":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T20:09:24Z","timestamp":1696622964302},"reference-count":55,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,12,14]],"date-time":"2020-12-14T00:00:00Z","timestamp":1607904000000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2020,12,14]],"date-time":"2020-12-14T00:00:00Z","timestamp":1607904000000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"name":"European Research Council, under the Engineering Quantum Error Correction Project. 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},"abstract":"AbstractLeakage 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.","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":24,"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":"https://www.nature.com/articles/s41534-020-00330-w.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-020-00330-w","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-020-00330-w.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,6]],"date-time":"2022-12-06T02:55:14Z","timestamp":1670295314000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-020-00330-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,14]]},"references-count":55,"journal-issue":{"issue":"1","published-online":{"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":[[2020,12,14]]},"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":"be086ef15f13555ee11f1e8e62e15ca0bd6c37529d0faa5237224b347400f37d"},"expire":1732264957216},"doi:10.1038/s41534-017-0039-x":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T18:21:08Z","timestamp":1700590868910},"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":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2017,9,25]],"date-time":"2017-09-25T00:00:00Z","timestamp":1506297600000},"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":"AbstractWe present a density-matrix simulation of the quantum memory and computing performance of the distance-3 logical qubit Surface-17, following a recently proposed quantum circuit and using experimental error parameters for transmon qubits in a planar circuit QED architecture. We use this simulation to optimize components of the QEC scheme (e.g., trading off stabilizer measurement infidelity for reduced cycle time) and to investigate the benefits of feedback harnessing the fundamental asymmetry of relaxation-dominated error in the constituent transmons. A lower-order approximate calculation extends these predictions to the distance-5 Surface-49. These results clearly indicate error rates below the fault-tolerance threshold of the surface code, and the potential for Surface-17 to perform beyond the break-even point of quantum memory. However, Surface-49 is required to surpass the break-even point of computation at state-of-the-art qubit relaxation times and readout speeds.","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":53,"title":"Density-matrix simulation of small surface codes under current and projected experimental noise","prefix":"10.1038","volume":"3","author":[{"given":"T. E.","family":"O’Brien","sequence":"first","affiliation":[]},{"given":"B.","family":"Tarasinski","sequence":"additional","affiliation":[]},{"given":"L.","family":"DiCarlo","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,9,25]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41534-017-0039-x.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-017-0039-x","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-017-0039-x.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,23]],"date-time":"2022-12-23T01:46:21Z","timestamp":1671759981000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-017-0039-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,9,25]]},"references-count":38,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,12]]}},"alternative-id":["39"],"URL":"http://dx.doi.org/10.1038/s41534-017-0039-x","relation":{},"ISSN":["2056-6387"],"subject":["Computational 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Rev. Lett.","published":{"date-parts":[[2007,5,11]]},"article-number":"190504","id":"doi:10.1103/PhysRevLett.98.190504","_hash":"81fdfd4c80a9b971aeea9c521875ab8b4f0d0263b234373d72fc2d7503e56f45"},"expire":1732264965002},"doi:10.22331/q-2021-07-06-497":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T18:20:19Z","timestamp":1701109219079},"reference-count":30,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2021,7,6]],"date-time":"2021-07-06T00:00:00Z","timestamp":1625529600000},"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":"This paper presents “Stim\", a fast simulator for quantum stabilizer circuits. The paper explains how Stim works and compares it to existing tools. With no foreknowledge, Stim can analyze a distance 100 surface code circuit (20 thousand qubits, 8 million gates, 1 million measurements) in 15 seconds and then begin sampling full circuit shots at a rate of 1 kHz. Stim uses a stabilizer tableau representation, similar to Aaronson and Gottesman's CHP simulator, but with three main improvements. First, Stim improves the asymptotic complexity of deterministic measurement from quadratic to linear by tracking the inverse of the circuit's stabilizer tableau. Second, Stim improves the constant factors of the algorithm by using a cache-friendly data layout and 256 bit wide SIMD instructions. Third, Stim only uses expensive stabilizer tableau simulation to create an initial reference sample. Further samples are collected in bulk by using that sample as a reference for batches of Pauli frames propagating through the circuit.","DOI":"10.22331/q-2021-07-06-497","type":"journal-article","created":{"date-parts":[[2021,7,6]],"date-time":"2021-07-06T09:26:56Z","timestamp":1625563616000},"page":"497","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":44,"title":"Stim: a fast stabilizer circuit simulator","prefix":"10.22331","volume":"5","author":[{"given":"Craig","family":"Gidney","sequence":"first","affiliation":[{"name":"Google Inc., Santa Barbara, California 93117, USA"}]}],"member":"9598","published-online":{"date-parts":[[2021,7,6]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2021-07-06-497/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2021,7,6]],"date-time":"2021-07-06T09:27:37Z","timestamp":1625563657000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2021-07-06-497/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,6]]},"references-count":30,"URL":"http://dx.doi.org/10.22331/q-2021-07-06-497","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2021,7,6]]},"article-number":"497","id":"doi:10.22331/q-2021-07-06-497","_hash":"c86ef9f8cbd68ff86d56a19c2724da369c3ef91e320cbb0d16aaa5172f8bd2f2"},"expire":1732264966059},"doi:10.22331/q-2020-10-28-352":{"value":{"indexed":{"date-parts":[[2023,11,18]],"date-time":"2023-11-18T10:37:25Z","timestamp":1700303845266},"reference-count":20,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2020,10,28]],"date-time":"2020-10-28T00:00:00Z","timestamp":1603843200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"The surface code is a prominent topological error-correcting code exhibiting high fault-tolerance accuracy thresholds. 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 only 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.","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":18,"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, 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The error suppression achieved by the surface code is usually estimated by simulating toy noise models describing random Pauli errors. However, Pauli noise models fail to capture coherent processes such as systematic unitary errors caused by imperfect control pulses. Here we report the first large-scale simulation of quantum error correction protocols based on the surface code in the presence of coherent noise. We observe that the standard Pauli approximation provides an accurate estimate of the error threshold but underestimates the logical error rate in the sub-threshold regime. We find that for large code size the logical-level noise is well approximated by random Pauli errors even though the physical-level noise is coherent. Our work demonstrates that coherent effects do not significantly change the error correcting threshold of surface codes. This gives more confidence in the viability of the fault-tolerance architecture pursued by several experimental groups.","DOI":"10.1038/s41534-018-0106-y","type":"journal-article","created":{"date-parts":[[2018,10,25]],"date-time":"2018-10-25T10:09:29Z","timestamp":1540462169000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":54,"title":"Correcting coherent errors with surface codes","prefix":"10.1038","volume":"4","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Matthias","family":"Englbrecht","sequence":"additional","affiliation":[]},{"given":"Robert","family":"König","sequence":"additional","affiliation":[]},{"given":"Nolan","family":"Peard","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,10,31]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41534-018-0106-y.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-018-0106-y","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-018-0106-y.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T23:02:50Z","timestamp":1671577370000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-018-0106-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,31]]},"references-count":46,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,12]]}},"alternative-id":["106"],"URL":"http://dx.doi.org/10.1038/s41534-018-0106-y","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":[[2018,10,31]]},"assertion":[{"value":"9 July 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 September 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 October 2018","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 October 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":"55","id":"doi:10.1038/s41534-018-0106-y","_hash":"648cb68eaa264f22db74a2003fcfb83af257233d98d1d4ba8875148f4239e228"},"expire":1732264972150},"doi:10.1109/ISIT.2012.6284206":{"value":{"indexed":{"date-parts":[[2023,10,24]],"date-time":"2023-10-24T17:11:52Z","timestamp":1698167512934},"reference-count":24,"publisher":"IEEE","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2012,7]]},"DOI":"10.1109/isit.2012.6284206","type":"proceedings-article","created":{"date-parts":[[2012,8,30]],"date-time":"2012-08-30T20:57:57Z","timestamp":1346360277000},"source":"Crossref","is-referenced-by-count":20,"title":"Improved quantum hypergraph-product LDPC codes","prefix":"10.1109","author":[{"given":"Alexey A.","family":"Kovalev","sequence":"first","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"member":"263","reference":[],"event":"2012 IEEE International Symposium on Information Theory - 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However, measuring this entanglement in real materials is extremely tricky. Now, two groups take a different approach and turn to synthetic systems to engineer the topological order of the so-called toric code type (see the Perspective by Bartlett). Satzinger\n et al\n . used a quantum processor to study the ground state and excitations of the toric code. 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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 generation3–5. We use this architecture to realize programmable generation of entangled graph states, such as cluster states and a seven-qubit Steane code state6,7. Furthermore, we shuttle entangled ancilla arrays to realize a surface code state with thirteen data and six ancillary qubits8and a toric code state on a torus with sixteen data and eight ancillary qubits9. Finally, we use this architecture to realize a hybrid analogue–digital evolution2and use it for measuring entanglement entropy in quantum simulations10–12, experimentally observing non-monotonic entanglement dynamics associated with quantum many-body scars13,14. Realizing a long-standing goal, these results provide a route towards scalable quantum processing and enable applications ranging from simulation to metrology.","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":153,"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":[[2023,2,2]],"date-time":"2023-02-02T19:35:27Z","timestamp":1675366527000},"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":"6ed5f0b3bafd70d97cff6e1faa51b80cf0df04877b516698240b09ab75f5a4d9"},"expire":1732264976088},"doi:10.1103/PhysRevLett.102.200501":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T22:37:40Z","timestamp":1700606260025},"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":91,"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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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":"2017-01-12","name":"date_received","label":"Date received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2017-05-02","name":"date_accepted","label":"Date accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2017-06-06","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/aa7099","_hash":"303f18ef4e958049c28364a93adc9839f427e5a02eeb9f13bdaa09f5dfa772ab"},"expire":1732265000003},"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":1732265000725},"doi:10.1103/PhysRevA.92.032309":{"value":{"indexed":{"date-parts":[[2023,11,16]],"date-time":"2023-11-16T17:21:07Z","timestamp":1700155267064},"reference-count":33,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2015,9,8]],"date-time":"2015-09-08T00:00:00Z","timestamp":1441670400000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"funder":[{"DOI":"10.13039/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP/G037043/1","EP/K022512/1"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.92.032309","type":"journal-article","created":{"date-parts":[[2015,9,8]],"date-time":"2015-09-08T17:08:24Z","timestamp":1441732104000},"source":"Crossref","is-referenced-by-count":29,"title":"Fast fault-tolerant decoder for qubit and qudit surface codes","prefix":"10.1103","volume":"92","author":[{"given":"Fern H. 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Rev. A","published":{"date-parts":[[2015,9,8]]},"article-number":"032309","id":"doi:10.1103/PhysRevA.92.032309","_hash":"b6f5922e636b9767fe768f5e9b64e1e6b33ae894bc92f0b4fcf971902581e283"},"expire":1732265002004},"doi:10.22331/q-2021-12-02-595":{"value":{"indexed":{"date-parts":[[2023,11,14]],"date-time":"2023-11-14T14:04:20Z","timestamp":1699970660419},"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":"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 O(nα(n)), where n is the number of physical qubits and α is the inverse of Ackermann's function, which is very slowly growing. For all practical purposes, α(n)≤3. We prove that our algorithm performs optimally for errors of weight up to (d−1)/2 and for loss of up to d−1 qubits, where d is the minimum distance of the code. Numerically, we obtain a threshold of 9.9% for the 2d-toric code with perfect syndrome measurements and 2.6% with faulty measurements.","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":42,"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":"ee85d8a611176be409984b86e4104d0571ece7695f73c537742bd5042e64e76e"},"expire":1732265003030},"doi:10.4153/CJM-1965-045-4":{"value":{"indexed":{"date-parts":[[2023,11,26]],"date-time":"2023-11-26T02:13:25Z","timestamp":1700964805942},"reference-count":9,"publisher":"Canadian Mathematical Society","license":[{"start":{"date-parts":[[2018,11,20]],"date-time":"2018-11-20T00:00:00Z","timestamp":1542672000000},"content-version":"unspecified","delay-in-days":19681,"URL":"https://www.cambridge.org/core/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[1965]]},"abstract":"A graph G for purposes here is a finite set of elements called vertices and a finite set of elements called edges such that each edge meets exactly two vertices, called the end-points of the edge. An edge is said to join its end-points.A matching in G 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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The goal of this work is to demonstrate that this is not true. By writing distillation circuits in a form that separates qubits that are capable of error detection from those that are not, most logical qubits used for distillation can be encoded at a very low code distance. This significantly reduces the space-time cost of distillation, as well as the number of qubits. In extreme cases, it can cost less to distill a magic state than to perform a logical Clifford gate on full-distance logical qubits.","DOI":"10.22331/q-2019-12-02-205","type":"journal-article","created":{"date-parts":[[2019,12,2]],"date-time":"2019-12-02T13:45:29Z","timestamp":1575294329000},"page":"205","source":"Crossref","is-referenced-by-count":62,"title":"Magic State Distillation: Not as Costly as You Think","prefix":"10.22331","volume":"3","author":[{"given":"Daniel","family":"Litinski","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2019,12,2]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2019-12-02-205/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2021,2,9]],"date-time":"2021-02-09T21:59:50Z","timestamp":1612907990000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2019-12-02-205/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,2]]},"references-count":42,"URL":"http://dx.doi.org/10.22331/q-2019-12-02-205","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2019,12,2]]},"article-number":"205","id":"doi:10.22331/q-2019-12-02-205","_hash":"9f9f37051e1870f2a108ee7941255614370122bd0b55819a978ef1bcc60f559d"},"expire":1732265006895},"doi:10.1103/PhysRevResearch.4.023090":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T18:21:48Z","timestamp":1701109308806},"reference-count":42,"publisher":"American Physical Society (APS)","issue":"2","license":[{"start":{"date-parts":[[2022,5,2]],"date-time":"2022-05-02T00:00:00Z","timestamp":1651449600000},"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/physrevresearch.4.023090","type":"journal-article","created":{"date-parts":[[2022,5,2]],"date-time":"2022-05-02T14:03:25Z","timestamp":1651500205000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":7,"title":"Circuit-level protocol and analysis for twist-based lattice surgery","prefix":"10.1103","volume":"4","author":[{"ORCID":"http://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"given":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2022,5,2]]},"reference":[],"container-title":"Physical Review Research","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevResearch.4.023090","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevResearch.4.023090/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,5,2]],"date-time":"2022-05-02T14:03:27Z","timestamp":1651500207000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevResearch.4.023090"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,2]]},"references-count":42,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,5]]}},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.4.023090","relation":{},"ISSN":["2643-1564"],"subject":["General Engineering"],"container-title-short":"Phys. 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Research","published":{"date-parts":[[2022,5,2]]},"article-number":"023090","id":"doi:10.1103/PhysRevResearch.4.023090","_hash":"4c88f1c0e36e1fcbcc64f5bca7e61283a62359dbddb2b0630e6fd89b556585b2"},"expire":1732265007901},"doi:10.1103/PRXQuantum.3.010331":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T18:20:03Z","timestamp":1701109203758},"reference-count":66,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[2022,2,25]],"date-time":"2022-02-25T00:00:00Z","timestamp":1645747200000},"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.3.010331","type":"journal-article","created":{"date-parts":[[2022,3,3]],"date-time":"2022-03-03T14:27:34Z","timestamp":1646317654000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":19,"title":"Universal Quantum Computing with Twist-Free and Temporally Encoded Lattice Surgery","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":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2022,2,25]]},"reference":[],"container-title":"PRX Quantum","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PRXQuantum.3.010331","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PRXQuantum.3.010331/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,3,3]],"date-time":"2022-03-03T14:28:24Z","timestamp":1646317704000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PRXQuantum.3.010331"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,25]]},"references-count":66,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,2]]}},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.010331","relation":{},"ISSN":["2691-3399"],"subject":["General Medicine"],"container-title-short":"PRX Quantum","published":{"date-parts":[[2022,2,25]]},"article-number":"010331","id":"doi:10.1103/PRXQuantum.3.010331","_hash":"b5faf37a8acad61802e622f34e367002d570104020ce38caa15a8755eb9f386e"},"expire":1732265009026},"doi:10.22331/q-2019-03-05-128":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T18:21:50Z","timestamp":1701109310795},"reference-count":65,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2019,3,5]],"date-time":"2019-03-05T00:00:00Z","timestamp":1551744000000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"Given a quantum gate circuit, how does one execute it in a fault-tolerant architecture with as little overhead as possible? 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. Therefore, no knowledge of quantum error correction is necessary to understand the schemes in this paper, but only the concepts of qubits and measurements.","DOI":"10.22331/q-2019-03-05-128","type":"journal-article","created":{"date-parts":[[2019,3,5]],"date-time":"2019-03-05T19:50:11Z","timestamp":1551815411000},"page":"128","source":"Crossref","is-referenced-by-count":118,"title":"A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery","prefix":"10.22331","volume":"3","author":[{"given":"Daniel","family":"Litinski","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2019,3,5]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2019-03-05-128/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,9,13]],"date-time":"2022-09-13T01:04:56Z","timestamp":1663031096000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2019-03-05-128/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,5]]},"references-count":65,"URL":"http://dx.doi.org/10.22331/q-2019-03-05-128","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2019,3,5]]},"article-number":"128","id":"doi:10.22331/q-2019-03-05-128","_hash":"faa3de0304abe494902c663c22b7b85b57e117ed151839d3916d52d2ce371317"},"expire":1732265010006},"doi:10.22331/q-2018-05-04-62":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T18:19:26Z","timestamp":1701109166183},"reference-count":27,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2018,5,4]],"date-time":"2018-05-04T00:00:00Z","timestamp":1525392000000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"We present a planar surface-code-based scheme for fault-tolerant quantum computation which eliminates the time overhead of single-qubit Clifford gates, and implements long-range multi-target CNOT gates with a time overhead that scales only logarithmically with the control-target separation. This is done by replacing hardware operations for single-qubit Clifford gates with a classical tracking protocol. Inter-qubit communication is added via a modified lattice surgery protocol that employs twist defects of the surface code. The long-range multi-target CNOT gates facilitate magic state distillation, which renders our scheme fault-tolerant and universal.","DOI":"10.22331/q-2018-05-04-62","type":"journal-article","created":{"date-parts":[[2018,5,4]],"date-time":"2018-05-04T10:19:18Z","timestamp":1525429158000},"page":"62","source":"Crossref","is-referenced-by-count":38,"title":"Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes","prefix":"10.22331","volume":"2","author":[{"given":"Daniel","family":"Litinski","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"}]},{"given":"Felix von","family":"Oppen","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2018,5,4]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","deposited":{"date-parts":[[2019,10,17]],"date-time":"2019-10-17T13:49:48Z","timestamp":1571320188000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2018-05-04-62/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,4]]},"references-count":27,"URL":"http://dx.doi.org/10.22331/q-2018-05-04-62","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2018,5,4]]},"article-number":"62","id":"doi:10.22331/q-2018-05-04-62","_hash":"494612d3c2db955ad0cfe1a7256d49ab865f1e288f437d9c0c9a1cf598e097b9"},"expire":1732265010900},"doi:10.1088/1367-2630/14/12/123011":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T18:19:20Z","timestamp":1701109160586},"reference-count":32,"publisher":"IOP Publishing","issue":"12","license":[{"start":{"date-parts":[[2012,12,7]],"date-time":"2012-12-07T00:00:00Z","timestamp":1354838400000},"content-version":"vor","delay-in-days":6,"URL":"http://creativecommons.org/licenses/by-nc-sa/3.0/"},{"start":{"date-parts":[[2012,12,7]],"date-time":"2012-12-07T00:00:00Z","timestamp":1354838400000},"content-version":"tdm","delay-in-days":6,"URL":"https://iopscience.iop.org/info/page/text-and-data-mining"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2012,12,1]]},"abstract":"Abstract\n In recent years, surface codes have become a leading method for quantum error correction in theoretical large-scale computational and communications architecture designs. Their comparatively high fault-tolerant thresholds and their natural two-dimensional nearest-neighbour (2DNN) structure make them an obvious choice for large scale designs in experimentally realistic systems. While fundamentally based on the toric code of Kitaev, there are many variants, two of which are the planar- and defect-based codes. Planar codes require fewer qubits to implement (for the same strength of error correction), but are restricted to encoding a single qubit of information. Interactions between encoded qubits are achieved via transversal operations, thus destroying the inherent 2DNN nature of the code. In this paper we introduce a new technique enabling the coupling of two planar codes without transversal operations, maintaining the 2DNN of the encoded computer. Our lattice surgery technique comprises splitting and merging planar code surfaces, and enables us to perform universal quantum computation (including magic state injection) while removing the need for braided logic in a strictly 2DNN design, and hence reduces the overall qubit resources for logic operations. Those resources are further reduced by the use of a rotated lattice for the planar encoding. We show how lattice surgery allows us to distribute encoded GHZ states in a more direct (and overhead friendly) manner, and how a demonstration of an encoded CNOT between two distance-3 logical states is possible with 53 physical qubits, half of that required in any other known construction in 2D.","DOI":"10.1088/1367-2630/14/12/123011","type":"journal-article","created":{"date-parts":[[2012,12,7]],"date-time":"2012-12-07T16:14:11Z","timestamp":1354896851000},"page":"123011","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":243,"title":"Surface code quantum computing by lattice surgery","prefix":"10.1088","volume":"14","author":[{"given":"Dominic","family":"Horsman","sequence":"first","affiliation":[]},{"given":"Austin G","family":"Fowler","sequence":"additional","affiliation":[]},{"given":"Simon","family":"Devitt","sequence":"additional","affiliation":[]},{"given":"Rodney Van","family":"Meter","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2012,12,7]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/14/12/123011","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/14/12/123011/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/14/12/123011/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/14/12/123011/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,30]],"date-time":"2023-05-30T12:53:56Z","timestamp":1685451236000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/14/12/123011"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,12,1]]},"references-count":32,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2012,12,7]]},"published-print":{"date-parts":[[2012,12,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/14/12/123011","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. Phys.","published":{"date-parts":[[2012,12,1]]},"assertion":[{"value":"Surface code quantum computing by lattice surgery","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":"© IOP Publishing and Deutsche Physikalische Gesellschaft","name":"copyright_information","label":"Copyright Information"},{"value":"2012-05-01","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2012-12-07","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/14/12/123011","_hash":"1b16f8c146133a4dcbb19b7226528cbc2051c65e977dfbd887e1b5639fa03d1b"},"expire":1732265011922},"doi:10.1038/srep08975":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T18:22:15Z","timestamp":1700590935831},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2015,3,10]],"date-time":"2015-03-10T00:00:00Z","timestamp":1425945600000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2015,3,10]],"date-time":"2015-03-10T00:00:00Z","timestamp":1425945600000},"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":"AbstractWe present a scheme for encoding and decoding an unknown state for CSS codes, based on syndrome measurements. We illustrate our method by means of Kitaev toric code, defected-lattice code, topological subsystem code and 3D Haah code. The protocol is local whenever in a given code the crossings between the logical operators consist of next neighbour pairs, which holds for the above codes. For subsystem code we also present scheme in a noisy case, where we allow for bit and phase-flip errors on qubits as well as state preparation and syndrome measurement errors. Similar scheme can be built for two other codes. We show that the fidelity of the protected qubit in the noisy scenario in a large code size limit is of \"Equation missing\", where p is a probability of error on a single qubit per time step. 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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 L, however the most efficient known method for encoding an unknown state, introduced by Dennis et al. (2002), has O(L2) time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly 2L time steps to encode an unknown state in a distance L planar code. We further show how an O(L) complexity local unitary encoder for the toric code can be found by enforcing locality in the O(log⁡L)-depth non-local renormalisation encoder. We relate these techniques by providing an O(L) 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.","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":1732265019912},"doi:10.1088/1367-2630/9/6/199":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T15:59:35Z","timestamp":1700582375607},"reference-count":38,"publisher":"IOP Publishing","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1088/1367-2630/9/6/199","type":"journal-article","created":{"date-parts":[[2007,6,30]],"date-time":"2007-06-30T03:14:43Z","timestamp":1183173283000},"page":"199-199","source":"Crossref","is-referenced-by-count":399,"title":"Topological fault-tolerance in cluster state quantum computation","prefix":"10.1088","volume":"9","author":[{"given":"R","family":"Raussendorf","sequence":"first","affiliation":[]},{"given":"J","family":"Harrington","sequence":"additional","affiliation":[]},{"given":"K","family":"Goyal","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2007,6,29]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"deposited":{"date-parts":[[2021,8,16]],"date-time":"2021-08-16T08:53:05Z","timestamp":1629103985000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/9/6/199"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,6,29]]},"references-count":38,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2007,6,1]]}},"alternative-id":["S1367-2630(07)44251-3"],"URL":"http://dx.doi.org/10.1088/1367-2630/9/6/199","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. Phys.","published":{"date-parts":[[2007,6,29]]},"id":"doi:10.1088/1367-2630/9/6/199","_hash":"d1079095e48f02c84343ce92e0b57778aabf76a10e060c8af6cb7550e52dcdf6"},"expire":1732265020918},"doi:10.1088/1367-2630/ab8e5c":{"value":{"indexed":{"date-parts":[[2023,11,16]],"date-time":"2023-11-16T17:20:21Z","timestamp":1700155221166},"reference-count":37,"publisher":"IOP Publishing","issue":"7","license":[{"start":{"date-parts":[[2020,7,31]],"date-time":"2020-07-31T00:00:00Z","timestamp":1596153600000},"content-version":"vor","delay-in-days":30,"URL":"https://creativecommons.org/licenses/by/4.0/"},{"start":{"date-parts":[[2020,7,31]],"date-time":"2020-07-31T00:00:00Z","timestamp":1596153600000},"content-version":"tdm","delay-in-days":30,"URL":"https://iopscience.iop.org/info/page/text-and-data-mining"}],"funder":[{"name":"Army Research Office and Laboratory for Physical Sciences","award":["W911NF-18- 1-0103"]},{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["PHY-1733907"]}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2020,7,1]]},"abstract":"Abstract\n We study the effectiveness of quantum error correction against coherent noise. 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.","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":20,"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":"b60cad6da53062c6454c117c2333343cd38704261dc49448abeb8cbb2a239a56"},"expire":1732265021862},"doi:10.1007/978-1-4615-5923-8_19":{"value":{"indexed":{"date-parts":[[2023,8,26]],"date-time":"2023-08-26T19:39:20Z","timestamp":1693078760074},"publisher-location":"Boston, MA","reference-count":11,"publisher":"Springer US","isbn-type":[{"value":"9781461377160","type":"print"},{"value":"9781461559238","type":"electronic"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[1997]]},"DOI":"10.1007/978-1-4615-5923-8_19","type":"book-chapter","created":{"date-parts":[[2011,9,12]],"date-time":"2011-09-12T17:37:40Z","timestamp":1315849060000},"page":"181-188","source":"Crossref","is-referenced-by-count":47,"title":"Quantum Error Correction with Imperfect Gates","prefix":"10.1007","author":[{"given":"A. Yu.","family":"Kitaev","sequence":"first","affiliation":[]}],"member":"297","reference":[],"container-title":"Quantum Communication, Computing, and Measurement","original-title":[],"link":[{"URL":"http://link.springer.com/content/pdf/10.1007/978-1-4615-5923-8_19.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,5,1]],"date-time":"2021-05-01T11:23:21Z","timestamp":1619868201000},"score":1,"resource":{"primary":{"URL":"http://link.springer.com/10.1007/978-1-4615-5923-8_19"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997]]},"ISBN":["9781461377160","9781461559238"],"references-count":11,"URL":"http://dx.doi.org/10.1007/978-1-4615-5923-8_19","relation":{},"published":{"date-parts":[[1997]]},"id":"doi:10.1007/978-1-4615-5923-8_19","_hash":"bce5798b2291907a334b222ef51d0cd2bd76d6378ad048ebf97ef5fb82e467bd"},"expire":1732265022938},"doi:10.1070/RM1997v052n06ABEH002155":{"value":{"indexed":{"date-parts":[[2023,11,22]],"date-time":"2023-11-22T16:58:11Z","timestamp":1700672291749},"reference-count":0,"publisher":"Steklov Mathematical Institute","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[1997,12,31]]},"DOI":"10.1070/rm1997v052n06abeh002155","type":"journal-article","created":{"date-parts":[[2005,11,8]],"date-time":"2005-11-08T14:31:52Z","timestamp":1131460312000},"page":"1191-1249","source":"Crossref","is-referenced-by-count":678,"title":"Quantum computations: algorithms and error correction","prefix":"10.4213","volume":"52","author":[{"given":"A Yu","family":"Kitaev","sequence":"first","affiliation":[]}],"member":"2731","published-online":{"date-parts":[[2007,10,17]]},"container-title":"Russian Mathematical Surveys","original-title":[],"link":[{"URL":"http://stacks.iop.org/0036-0279/52/i=6/a=R02/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,30]],"date-time":"2023-05-30T04:45:07Z","timestamp":1685421907000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1070/RM1997v052n06ABEH002155"},"secondary":[{"URL":"https://www.mathnet.ru/eng/rm892"}]},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,12,31]]},"references-count":0,"journal-issue":{"issue":"6","published-print":{"date-parts":[[1997,12,31]]}},"URL":"http://dx.doi.org/10.1070/RM1997v052n06ABEH002155","relation":{},"ISSN":["0036-0279","1468-4829"],"subject":["General Mathematics"],"container-title-short":"Russ. 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Phys.","published":{"date-parts":[[2021,12,16]]},"assertion":[{"value":"31 March 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 October 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 December 2021","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/s41567-021-01423-9","_hash":"09f120e8e46779d0f23d6b0997baa06f955a3ccf899b46d706d93e5084282d4d"},"expire":1732265024848},"doi:10.1038/s41586-021-03588-y":{"value":{"indexed":{"date-parts":[[2023,11,22]],"date-time":"2023-11-22T13:19:12Z","timestamp":1700659152712},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"7867","license":[{"start":{"date-parts":[[2021,7,14]],"date-time":"2021-07-14T00:00:00Z","timestamp":1626220800000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2021,7,14]],"date-time":"2021-07-14T00:00:00Z","timestamp":1626220800000},"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":[[2021,7,15]]},"abstract":"AbstractRealizing the potential of quantum computing requires sufficiently low logical error rates1. Many applications call for error rates as low as 10−15 (refs. 2–9), but state-of-the-art quantum platforms typically have physical error rates near 10−3 (refs. 10–14). Quantum error correction15–17 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 device18,19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits.","DOI":"10.1038/s41586-021-03588-y","type":"journal-article","created":{"date-parts":[[2021,7,14]],"date-time":"2021-07-14T16:13:39Z","timestamp":1626279219000},"page":"383-387","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":142,"title":"Exponential suppression of bit or phase errors with cyclic error correction","prefix":"10.1038","volume":"595","author":[{"name":"Google Quantum AI","sequence":"first","affiliation":[]},{"given":"Zijun","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Kevin J.","family":"Satzinger","sequence":"additional","affiliation":[]},{"given":"Juan","family":"Atalaya","sequence":"additional","affiliation":[]},{"given":"Alexander 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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.","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":32,"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":"81dfa871ee2a05092165f2d3edf3f334323f9d77f51060bcc4b54293af03063e"},"expire":1732265029859},"doi:10.1103/PhysRevLett.124.130501":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T17:21:52Z","timestamp":1701105712621},"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":58,"title":"Fault-Tolerant Thresholds for the Surface Code in Excess of \n5%\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. Lett.","published":{"date-parts":[[2020,3,30]]},"article-number":"130501","id":"doi:10.1103/PhysRevLett.124.130501","_hash":"47f2dd9c00b169b032029098a9189e12a0f6cc0fcbf3feeaa11cde83eae2314a"},"expire":1732265030939},"doi:10.1038/s41467-021-22274-1":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T18:19:32Z","timestamp":1701109172579},"reference-count":82,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,4,12]],"date-time":"2021-04-12T00:00:00Z","timestamp":1618185600000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2021,4,12]],"date-time":"2021-04-12T00:00:00Z","timestamp":1618185600000},"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":"AbstractPerforming large calculations with a quantum computer will likely require a fault-tolerant architecture based on quantum error-correcting codes. 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.","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":85,"title":"The XZZX surface code","prefix":"10.1038","volume":"12","author":[{"ORCID":"http://orcid.org/0000-0001-5518-7907","authenticated-orcid":false,"given":"J. Pablo","family":"Bonilla Ataides","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-3776-2864","authenticated-orcid":false,"given":"David K.","family":"Tuckett","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4387-670X","authenticated-orcid":false,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-8060-8109","authenticated-orcid":false,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,4,12]]},"reference":[],"container-title":"Nature 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X","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevX.9.041031","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevX.9.041031/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,4]],"date-time":"2022-10-04T13:02:05Z","timestamp":1664888525000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevX.9.041031"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,12]]},"references-count":28,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2019,11]]}},"URL":"http://dx.doi.org/10.1103/PhysRevX.9.041031","relation":{},"ISSN":["2160-3308"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. 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X","published":{"date-parts":[[2019,11,12]]},"article-number":"041031","id":"doi:10.1103/PhysRevX.9.041031","_hash":"84be184ff27fcd616460785114e209fa2e431f5a9e4f0ee7bbad0fcd327ce02c"},"expire":1732265032846},"doi:10.1103/PhysRevLett.120.050505":{"value":{"indexed":{"date-parts":[[2023,11,18]],"date-time":"2023-11-18T10:36:27Z","timestamp":1700303787266},"reference-count":34,"publisher":"American Physical Society (APS)","issue":"5","license":[{"start":{"date-parts":[[2018,1,31]],"date-time":"2018-01-31T00:00:00Z","timestamp":1517356800000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"},{"start":{"date-parts":[[2019,1,31]],"date-time":"2019-01-31T00:00:00Z","timestamp":1548892800000},"content-version":"am","delay-in-days":365,"URL":"https://link.aps.org/licenses/aps-default-accepted-manuscript-license"}],"funder":[{"DOI":"10.13039/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["CE110001013"]},{"DOI":"10.13039/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["W911NF-14-1-0098","W911NF-14-1-0103"]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevlett.120.050505","type":"journal-article","created":{"date-parts":[[2018,1,31]],"date-time":"2018-01-31T18:44:25Z","timestamp":1517424265000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":109,"title":"Ultrahigh Error Threshold for Surface Codes with Biased Noise","prefix":"10.1103","volume":"120","author":[{"given":"David K.","family":"Tuckett","sequence":"first","affiliation":[]},{"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2018,1,31]]},"reference":[],"container-title":"Physical Review Letters","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/accepted/10.1103/PhysRevLett.120.050505","content-type":"application/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"https://link.aps.org/article/10.1103/PhysRevLett.120.050505","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.120.050505/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,13]],"date-time":"2022-08-13T17:52:33Z","timestamp":1660413153000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevLett.120.050505"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,31]]},"references-count":34,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2018,1]]}},"URL":"http://dx.doi.org/10.1103/PhysRevLett.120.050505","relation":{},"ISSN":["0031-9007","1079-7114"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. 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Homological codes from surfaces, i.e., surface codes, have also been suggested as a possible way to construct stable quantum memory and fault-tolerant computation. It has been conjectured that all homological codes have a square root bound on there distance and therefore cannot produce good codes. This claim has been disputed in dimension four using the geometric property of systolic freedom. 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Here, we consider an algorithm which maps the search for CWS codes to a problem of identifying maximum cliques in a graph. While solving this problem is in general very hard, we provide three structure theorems which reduce the search space, specifying certain admissible and optimal ((n,K,d)) additive codes. In particular, we find that the re does not exist any ((7,3,3)) CWS code though the linear programming bound does not rule it out. The complexity of the CWS-search algorithm is compared with the contrasting method introduced by Aggarwal and Calderbank [IEEE Trans. Inf. 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This 10th anniversary edition includes an introduction from the authors setting the work in context. This comprehensive textbook describes such remarkable effects as fast quantum algorithms, quantum teleportation, quantum cryptography and quantum error-correction. Quantum mechanics and computer science are introduced before moving on to describe what a quantum computer is, how it can be used to solve problems faster than 'classical' computers and its real-world implementation. It concludes with an in-depth treatment of quantum information. Containing a wealth of figures and exercises, this well-known textbook is ideal for courses on the subject, and will interest beginning graduate students and researchers in physics, computer science, mathematics, and electrical engineering.","DOI":"10.1017/cbo9780511976667","type":"monograph","created":{"date-parts":[[2012,6,18]],"date-time":"2012-06-18T17:58:14Z","timestamp":1340042294000},"source":"Crossref","is-referenced-by-count":3318,"title":"Quantum Computation and Quantum Information","prefix":"10.1017","author":[{"given":"Michael A.","family":"Nielsen","sequence":"first","affiliation":[]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2012,6,5]]},"container-title":[],"original-title":[],"deposited":{"date-parts":[[2022,6,11]],"date-time":"2022-06-11T21:26:40Z","timestamp":1654982800000},"score":1,"resource":{"primary":{"URL":"https://www.cambridge.org/core/product/identifier/9780511976667/type/book"}},"subtitle":["10th Anniversary Edition"],"short-title":[],"issued":{"date-parts":[[2012,6,5]]},"ISBN":["9781107002173","9780511976667"],"references-count":0,"URL":"http://dx.doi.org/10.1017/CBO9780511976667","relation":{},"published":{"date-parts":[[2012,6,5]]},"reference":[],"id":"doi:10.1017/CBO9780511976667","_hash":"483210dc82a22a3321ade6e915e5e56a1d558ded6618c15db02247e2442d02dc"},"expire":1732265082231},"doi:10.22331/q-2022-09-22-815":{"value":{"indexed":{"date-parts":[[2023,11,14]],"date-time":"2023-11-14T16:12:10Z","timestamp":1699978330537},"reference-count":26,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,9,22]],"date-time":"2022-09-22T00:00:00Z","timestamp":1663804800000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/501100000923","name":"Australian Research Council","doi-asserted-by":"crossref","award":["CE170100009"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"We propose an extension to the Pauli stabiliser formalism that includes fractional 2&#x03C0;/N rotations around the Z axis for some integer N. 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. 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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 &#x0398;(N2/3). 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. 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November 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 April 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 May 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"T.M., R.B. and T.F. are connected to Alpine Quantum Technologies, a commercially oriented quantum computing company.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"id":"doi:10.1038/s41586-022-04721-1","_hash":"4a07cbbf5f11f82adcc456294e8e6325ad6c506109eb807bfde686e97066c854"},"expire":1732265124252},"doi:10.1103/Physics.14.184":{"value":{"indexed":{"date-parts":[[2023,5,6]],"date-time":"2023-05-06T05:35:06Z","timestamp":1683351306979},"reference-count":2,"publisher":"American Physical Society 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Attempts to process and manipulate quantum states can destroy the encoded information. Nigg\n et al.\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 \n \n Science\n , this issue p.\n 302\n ","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":276,"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":"455d88658f40700ebcfb0f575d7b3a6b349e00d7fb88cb8e8963c92427037a1d"},"expire":1732265126381},"doi:10.1088/2058-9565/abc6f4":{"value":{"indexed":{"date-parts":[[2023,9,22]],"date-time":"2023-09-22T08:18:47Z","timestamp":1695370727007},"reference-count":20,"publisher":"IOP Publishing","issue":"1","license":[{"start":{"date-parts":[[2020,11,24]],"date-time":"2020-11-24T00:00:00Z","timestamp":1606176000000},"content-version":"vor","delay-in-days":0,"URL":"https://iopscience.iop.org/page/copyright"},{"start":{"date-parts":[[2020,11,24]],"date-time":"2020-11-24T00:00:00Z","timestamp":1606176000000},"content-version":"tdm","delay-in-days":0,"URL":"https://iopscience.iop.org/info/page/text-and-data-mining"}],"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":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2021,1,1]]},"abstract":"Abstract\n Steane’s seven-qubit quantum code is a natural choice for fault-tolerance experiments because it is small and just two extra qubits are enough to correct errors. 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.","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":21,"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. Technol.","published":{"date-parts":[[2020,11,24]]},"assertion":[{"value":"Fault-tolerant quantum error correction for Steane’s seven-qubit color code with few or no extra qubits","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":"© 2020 IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2020-08-10","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-11-02","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-11-24","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/2058-9565/abc6f4","_hash":"630df5e9326895fcf572f90683c593e429420bb9d614e780ebf4ed1af618d54b"},"expire":1732265127328},"doi:10.1103/PhysRevLett.121.050502":{"value":{"indexed":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T16:41:05Z","timestamp":1699893665332},"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":104,"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":"f6c2f6c4b157ce69649bb5b666a77be2f78489384f0e0a3024daaf99f92dc60c"},"expire":1732265128251},"doi:10.22331/q-2017-04-25-2":{"value":{"indexed":{"date-parts":[[2023,10,24]],"date-time":"2023-10-24T17:10:06Z","timestamp":1698167406500},"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":"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.","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":67,"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":[[2023,8,23]],"date-time":"2023-08-23T07:52:12Z","timestamp":1692777132000},"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 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Realizations"],"short-title":[],"issued":{"date-parts":[[2008,3,11]]},"ISBN":["9780429146510"],"references-count":0,"URL":"http://dx.doi.org/10.1201/9781420012293","relation":{},"published":{"date-parts":[[2008,3,11]]},"reference":[],"id":"doi:10.1201/9781420012293","_hash":"9f43ccb87bf8fc7e85c7e8bc853a7765ec5f03c847754805f9195ae4d6b6efb1"},"expire":1732265130377},"doi:10.1103/PRXQuantum.3.030319":{"value":{"indexed":{"date-parts":[[2023,10,12]],"date-time":"2023-10-12T10:30:24Z","timestamp":1697106624500},"reference-count":65,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2022,8,8]],"date-time":"2022-08-08T00:00:00Z","timestamp":1659916800000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/501100000023","name":"Government of Canada","doi-asserted-by":"publisher"},{"DOI":"10.13039/100011332","name":"Innovation, Science and Economic Development 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Quantum","published":{"date-parts":[[2022,8,8]]},"article-number":"030319","id":"doi:10.1103/PRXQuantum.3.030319","_hash":"fa3ea84b86257ee5526987ef7522be2429b19637b2a66b4168f754cc20f72897"},"expire":1732265131318},"doi:10.1038/s41586-022-04819-6":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T22:42:17Z","timestamp":1700606537161},"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":"AbstractSolid-state spin qubits is a promising platform for quantum computation and quantum networks1,2. Recent experiments have demonstrated high-quality control over multi-qubit systems3–8, elementary quantum algorithms8–11 and non-fault-tolerant error correction12–14. Large-scale systems will require using error-corrected logical qubits that are operated fault tolerantly, so that reliable computation becomes possible despite noisy operations15–18. Overcoming imperfections in this way remains an important outstanding challenge for quantum science15,19–27. 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 qubits28–30. 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.","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":52,"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":"20a5bb1e2780caed34392cd2b48063a80244d3f4831a231805ef58582afa8a90"},"expire":1732265132375},"doi:10.1093/nsr/nwab011":{"value":{"indexed":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T15:57:25Z","timestamp":1696607845718},"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":"Abstract\n 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.","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":19,"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 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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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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":"0f6548aeec52018d0ddc6061a5d7058bfd699981a0ea4b77a17ac97bb5fc7a2a"},"expire":1732265146859},"doi:10.22331/q-2017-10-03-31":{"value":{"indexed":{"date-parts":[[2023,8,10]],"date-time":"2023-08-10T02:27:14Z","timestamp":1691634434941},"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":"We present an infinite family of protocols to distill magic states for T-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 T-gate depth of the circuit scale linearly in the logarithm of the target error δ (up to log⁡log⁡1/δ). 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. 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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 \n \n\t\n\t \n\t 1\n\t \n\t\n\t×\n\t\n\t \n\t 2\n\t \n\t\n \n based on the Shor code state, the effectiveness of our loss-tolerant scheme and the loss tolerance of the encoded RGS are respectively verified. 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Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2014,2,2]],"date-time":"2014-02-02T00:00:00Z","timestamp":1391299200000},"content-version":"tdm","delay-in-days":0,"URL":"http://www.springer.com/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2014,3]]},"DOI":"10.1038/nnano.2014.2","type":"journal-article","created":{"date-parts":[[2014,2,1]],"date-time":"2014-02-01T04:54:19Z","timestamp":1391230459000},"page":"171-176","source":"Crossref","is-referenced-by-count":288,"title":"Universal control and error correction in multi-qubit spin registers in diamond","prefix":"10.1038","volume":"9","author":[{"given":"T. H.","family":"Taminiau","sequence":"first","affiliation":[]},{"given":"J.","family":"Cramer","sequence":"additional","affiliation":[]},{"given":"T.","family":"van der Sar","sequence":"additional","affiliation":[]},{"given":"V. V.","family":"Dobrovitski","sequence":"additional","affiliation":[]},{"given":"R.","family":"Hanson","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2014,2,2]]},"reference":[],"container-title":"Nature Nanotechnology","original-title":[],"language":"en","link":[{"URL":"http://www.nature.com/articles/nnano.2014.2.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/nnano.2014.2","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/nnano.2014.2.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T23:40:56Z","timestamp":1684453256000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/nnano.2014.2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,2,2]]},"references-count":32,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2014,3]]}},"alternative-id":["BFnnano20142"],"URL":"http://dx.doi.org/10.1038/nnano.2014.2","relation":{},"ISSN":["1748-3387","1748-3395"],"subject":["Electrical and Electronic Engineering","Condensed Matter Physics","General Materials Science","Biomedical Engineering","Atomic and Molecular Physics, and Optics","Bioengineering"],"container-title-short":"Nature Nanotech","published":{"date-parts":[[2014,2,2]]},"id":"doi:10.1038/nnano.2014.2","_hash":"77a09ca5fcf4eb79cf2b643b8d3977377b7c2c9684763cd7a7a74c157520a3d4"},"expire":1732265154849},"doi:10.1038/s42005-022-00875-6":{"value":{"indexed":{"date-parts":[[2023,11,20]],"date-time":"2023-11-20T16:00:10Z","timestamp":1700496010663},"reference-count":53,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T00:00:00Z","timestamp":1651017600000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T00:00:00Z","timestamp":1651017600000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/501100001691","name":"MEXT | Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["20H05661","20K2044120"]},{"DOI":"10.13039/501100002241","name":"MEXT | Japan Science and Technology Agency","doi-asserted-by":"publisher","award":["JPMJCR1773","JPMJMS2062"]},{"DOI":"10.13039/501100009105","name":"Ministry of Internal Affairs and Communications","doi-asserted-by":"publisher","award":["JPMI00316"]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"AbstractFault-tolerant quantum memory plays a key role in interfacing quantum computers with quantum networks to construct quantum computer networks. 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.","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":6,"title":"Quantum error correction of spin quantum memories in diamond under a zero magnetic field","prefix":"10.1038","volume":"5","author":[{"given":"Takaya","family":"Nakazato","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-4880-1654","authenticated-orcid":false,"given":"Raustin","family":"Reyes","sequence":"additional","affiliation":[]},{"given":"Nobuaki","family":"Imaike","sequence":"additional","affiliation":[]},{"given":"Kazuyasu","family":"Matsuda","sequence":"additional","affiliation":[]},{"given":"Kazuya","family":"Tsurumoto","sequence":"additional","affiliation":[]},{"given":"Yuhei","family":"Sekiguchi","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-3778-7236","authenticated-orcid":false,"given":"Hideo","family":"Kosaka","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,27]]},"reference":[],"container-title":"Communications 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Physics and Astronomy"],"container-title-short":"Commun Phys","published":{"date-parts":[[2022,4,27]]},"assertion":[{"value":"3 December 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 March 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 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":"102","id":"doi:10.1038/s42005-022-00875-6","_hash":"d3803a5b72b50f77296e99af358f73c5577c7f6be0fe22ef5997060f7f4724e7"},"expire":1732265155891},"doi:10.1038/nature12919":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T22:39:15Z","timestamp":1700606355262},"reference-count":39,"publisher":"Springer Science and Business Media 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register","prefix":"10.1038","volume":"506","author":[{"given":"G.","family":"Waldherr","sequence":"first","affiliation":[]},{"given":"Y.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"S.","family":"Zaiser","sequence":"additional","affiliation":[]},{"given":"M.","family":"Jamali","sequence":"additional","affiliation":[]},{"given":"T.","family":"Schulte-Herbrüggen","sequence":"additional","affiliation":[]},{"given":"H.","family":"Abe","sequence":"additional","affiliation":[]},{"given":"T.","family":"Ohshima","sequence":"additional","affiliation":[]},{"given":"J.","family":"Isoya","sequence":"additional","affiliation":[]},{"given":"J. F.","family":"Du","sequence":"additional","affiliation":[]},{"given":"P.","family":"Neumann","sequence":"additional","affiliation":[]},{"given":"J.","family":"Wrachtrup","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2014,2,12]]},"reference":[],"container-title":"Nature","original-title":[],"language":"en","link":[{"URL":"http://www.nature.com/articles/nature12919.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/nature12919","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/nature12919.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T18:18:17Z","timestamp":1684433897000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/nature12919"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,2,12]]},"references-count":39,"journal-issue":{"issue":"7487","published-print":{"date-parts":[[2014,2,13]]}},"alternative-id":["BFnature12919"],"URL":"http://dx.doi.org/10.1038/nature12919","relation":{},"ISSN":["0028-0836","1476-4687"],"subject":["Multidisciplinary"],"container-title-short":"Nature","published":{"date-parts":[[2014,2,12]]},"assertion":[{"value":"2 September 2013","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 November 2013","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 February 2014","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 February 2014","order":4,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":5,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A new reference (31) has been added to the main-text reference list and all subsequent references have been renumbered.","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/nature12919","_hash":"77393642a755cc5d1d88604416f98c4e599f09677d4b5309bdad5ccc19bb866f"},"expire":1732265156884},"doi:10.1038/s41586-022-04986-6":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T22:40:08Z","timestamp":1700606408537},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"7924","license":[{"start":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T00:00:00Z","timestamp":1661299200000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T00:00:00Z","timestamp":1661299200000},"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":[[2022,8,25]]},"abstract":"AbstractFuture large-scale quantum computers will rely on quantum error correction (QEC) to protect the fragile quantum information during computation1,2. 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 computers3–5. Recent advances in silicon-based qubits have enabled the implementations of high-quality one-qubit and two-qubit systems6–8. However, the demonstration of QEC, which requires three or more coupled qubits1, and involves a three-qubit gate9–11 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.","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":30,"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":"0596baf0d34fa7b9497612ae8313e8c255e80c5b29fecbe0f0ab1fb5298acf2a"},"expire":1732265157875},"doi:10.1038/s41467-022-29906-0":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T19:02:11Z","timestamp":1700593331212},"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":"AbstractThe 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.","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":12,"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":"7a28eedb58e169835c97647dab4034f09121c927e8e739a1261b4e9ae7babf38"},"expire":1732265159054},"doi:10.1038/nature14270":{"value":{"indexed":{"date-parts":[[2023,11,20]],"date-time":"2023-11-20T15:29:08Z","timestamp":1700494148977},"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":647,"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 topological_codes module of Qiskit-Ignis, which is designed to provide the tools necessary to perform such tests. Specifically, we use the RepetitionCode and GraphDecoder 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 Rochester device is presented.","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":16,"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":"ae8e417086060b9fdf6a053d814c6336bd66d6003c2d460c56384b7c87a5d565"},"expire":1732265160809},"doi:10.1103/PhysRevA.97.052313":{"value":{"indexed":{"date-parts":[[2023,10,5]],"date-time":"2023-10-05T12:36:27Z","timestamp":1696509387569},"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":37,"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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Quantum error correction protects quantum states by encoding a logical quantum bit (qubit) in multiple physical qubits. To be compatible with universal fault-tolerant computations, it is essential that states remain encoded at all times and that errors are actively corrected. Here we demonstrate such active error correction on a continuously protected logical qubit using a diamond quantum processor. We encode the logical qubit in three long-lived nuclear spins, repeatedly detect phase errors by non-destructive measurements, and apply corrections by real-time feedback. The actively error-corrected qubit is robust against errors and encoded quantum superposition states are preserved beyond the natural dephasing time of the best physical qubit in the encoding. These results establish a powerful platform to investigate error correction under different types of noise and mark an important step towards fault-tolerant quantum information processing.","DOI":"10.1038/ncomms11526","type":"journal-article","created":{"date-parts":[[2016,5,5]],"date-time":"2016-05-05T11:03:40Z","timestamp":1462446220000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":158,"title":"Repeated quantum error correction on a continuously encoded qubit by real-time feedback","prefix":"10.1038","volume":"7","author":[{"given":"J.","family":"Cramer","sequence":"first","affiliation":[]},{"given":"N.","family":"Kalb","sequence":"additional","affiliation":[]},{"given":"M. A.","family":"Rol","sequence":"additional","affiliation":[]},{"given":"B.","family":"Hensen","sequence":"additional","affiliation":[]},{"given":"M. S.","family":"Blok","sequence":"additional","affiliation":[]},{"given":"M.","family":"Markham","sequence":"additional","affiliation":[]},{"given":"D. J.","family":"Twitchen","sequence":"additional","affiliation":[]},{"given":"R.","family":"Hanson","sequence":"additional","affiliation":[]},{"given":"T. H.","family":"Taminiau","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2016,5,5]]},"reference":[],"container-title":"Nature Communications","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/ncomms11526.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/ncomms11526","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/ncomms11526.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,4]],"date-time":"2023-01-04T11:28:11Z","timestamp":1672831691000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/ncomms11526"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,5,5]]},"references-count":38,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,9,1]]}},"alternative-id":["BFncomms11526"],"URL":"http://dx.doi.org/10.1038/ncomms11526","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":[[2016,5,5]]},"assertion":[{"value":"22 December 2015","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 April 2016","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 May 2016","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":"11526","id":"doi:10.1038/ncomms11526","_hash":"bceaf54667213cc99ecde7605c3b932ee327c4f093e4899068dd27d8ecbca408"},"expire":1732265162974},"doi:10.1038/ncomms7983":{"value":{"indexed":{"date-parts":[[2023,11,20]],"date-time":"2023-11-20T15:34:34Z","timestamp":1700494474429},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2015,4,29]],"date-time":"2015-04-29T00:00:00Z","timestamp":1430265600000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2015,4,29]],"date-time":"2015-04-29T00:00:00Z","timestamp":1430265600000},"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":"AbstractQuantum data are susceptible to decoherence induced by the environment and to errors in the hardware processing it. A future fault-tolerant quantum computer will use quantum error correction to actively protect against both. In the smallest error correction codes, the information in one logical qubit is encoded in a two-dimensional subspace of a larger Hilbert space of multiple physical qubits. For each code, a set of non-demolition multi-qubit measurements, termed stabilizers, can discretize and signal physical qubit errors without collapsing the encoded information. Here using a five-qubit superconducting processor, we realize the two parity measurements comprising the stabilizers of the three-qubit repetition code protecting one logical qubit from physical bit-flip errors. While increased physical qubit coherence times and shorter quantum error correction blocks are required to actively safeguard the quantum information, this demonstration is a critical step towards larger codes based on multiple parity measurements.","DOI":"10.1038/ncomms7983","type":"journal-article","created":{"date-parts":[[2015,4,29]],"date-time":"2015-04-29T13:13:35Z","timestamp":1430313215000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":207,"title":"Detecting bit-flip errors in a logical qubit using stabilizer measurements","prefix":"10.1038","volume":"6","author":[{"given":"D.","family":"Ristè","sequence":"first","affiliation":[]},{"given":"S.","family":"Poletto","sequence":"additional","affiliation":[]},{"given":"M.-Z.","family":"Huang","sequence":"additional","affiliation":[]},{"given":"A.","family":"Bruno","sequence":"additional","affiliation":[]},{"given":"V.","family":"Vesterinen","sequence":"additional","affiliation":[]},{"given":"O.-P.","family":"Saira","sequence":"additional","affiliation":[]},{"given":"L.","family":"DiCarlo","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2015,4,29]]},"reference":[],"container-title":"Nature Communications","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/ncomms7983.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/ncomms7983","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/ncomms7983.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,5]],"date-time":"2023-01-05T11:42:59Z","timestamp":1672918979000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/ncomms7983"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,4,29]]},"references-count":31,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,11,3]]}},"alternative-id":["BFncomms7983"],"URL":"http://dx.doi.org/10.1038/ncomms7983","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":[[2015,4,29]]},"assertion":[{"value":"19 February 2015","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 March 2015","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 April 2015","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":"6983","id":"doi:10.1038/ncomms7983","_hash":"9d81de71803b4caf82fdc52d334ddacd25244f7993c6a5ed972edbb542792acd"},"expire":1732265163888},"doi:10.1038/nature10786":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T18:22:54Z","timestamp":1700590974667},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"7385","license":[{"start":{"date-parts":[[2012,2,1]],"date-time":"2012-02-01T00:00:00Z","timestamp":1328054400000},"content-version":"tdm","delay-in-days":0,"URL":"http://www.springer.com/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2012,2]]},"DOI":"10.1038/nature10786","type":"journal-article","created":{"date-parts":[[2012,1,31]],"date-time":"2012-01-31T14:02:15Z","timestamp":1328018535000},"page":"382-385","source":"Crossref","is-referenced-by-count":449,"title":"Realization of three-qubit quantum error correction with superconducting circuits","prefix":"10.1038","volume":"482","author":[{"given":"M. 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However, quantum sensors lose their sensitivity in the presence of noise. To protect them, the use of quantum error-correcting codes has been proposed. Trapped ions are an excellent technological platform for both quantum sensing and quantum error correction. Here we present a quantum error correction scheme that harnesses dissipation to stabilize a trapped-ion qubit. In our approach, always-on couplings to an engineered environment protect the qubit against spin-flips or phase-flips. Our dissipative error correction scheme operates in a continuous manner without the need to perform measurements or feedback operations. We show that the resulting enhanced coherence time translates into a significantly enhanced precision for quantum measurements. Our work constitutes a stepping stone towards the paradigm of self-correcting quantum information processing.","DOI":"10.1038/s41467-017-01895-5","type":"journal-article","created":{"date-parts":[[2017,11,22]],"date-time":"2017-11-22T14:29:39Z","timestamp":1511360979000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":76,"title":"Dissipative quantum error correction and application to quantum sensing with trapped ions","prefix":"10.1038","volume":"8","author":[{"ORCID":"http://orcid.org/0000-0002-5217-3064","authenticated-orcid":false,"given":"F.","family":"Reiter","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-1337-9163","authenticated-orcid":false,"given":"A. S.","family":"Sørensen","sequence":"additional","affiliation":[]},{"given":"P.","family":"Zoller","sequence":"additional","affiliation":[]},{"given":"C. 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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.","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":6,"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":"d8bb9e5048f741bde476007246da8b0addab0dc612da4753fea06647ec54e715"},"expire":1732265194802},"doi:10.22331/q-2021-12-20-605":{"value":{"indexed":{"date-parts":[[2023,11,16]],"date-time":"2023-11-16T15:35:20Z","timestamp":1700148920383},"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":"Recently, Hastings & 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 0.2&#x0025;&#x2212;0.3&#x0025; for the honeycomb code compared to a threshold of 0.5&#x0025;&#x2212;0.7&#x0025; 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 1.5&#x0025;&#x003C;p&#x003C;2.0&#x0025;, where p 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 10&#x2212;3, we project that the honeycomb code can reach the teraquop regime with only 600 physical qubits. 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We discuss small instances of the code.","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":14,"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":"393c7d35a888030994255b012376817219765000872ad52d0ff888c4434ed4d9"},"expire":1732265197809},"doi:10.26081/K6F65V":{"value":{"type":"article","id":"doi:10.26081/K6F65V","author":[{"family":"Brown","given":"Ben"}],"issued":{"date-parts":[[2022]]},"DOI":"10.26081/K6F65V","publisher":"Kavli Institute for Theoretical Physics","title":"Anyon condensation and the color code","URL":"https://online.kitp.ucsb.edu/online/dynisq-c22/brown/","reference":[],"_hash":"93e96e1f8c954b6b2bcfd00c23db64624f313f80b16e8ac1a894b53904f05bee"},"expire":1732265198677},"doi:10.1103/PhysRevResearch.2.033042":{"value":{"indexed":{"date-parts":[[2023,11,19]],"date-time":"2023-11-19T03:32:50Z","timestamp":1700364770400},"reference-count":34,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2020,7,9]],"date-time":"2020-07-09T00:00:00Z","timestamp":1594252800000},"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-1125565"]},{"DOI":"10.13039/100000936","name":"Gordon and Betty Moore Foundation","doi-asserted-by":"publisher","award":["GBMF-2644"]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevresearch.2.033042","type":"journal-article","created":{"date-parts":[[2020,7,9]],"date-time":"2020-07-09T18:14:26Z","timestamp":1594318466000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":26,"title":"Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel","prefix":"10.1103","volume":"2","author":[{"given":"Nicolas","family":"Delfosse","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-6041-9554","authenticated-orcid":true,"given":"Gilles","family":"Zémor","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2020,7,9]]},"reference":[],"container-title":"Physical Review Research","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevResearch.2.033042","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevResearch.2.033042/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,24]],"date-time":"2021-03-24T03:48:08Z","timestamp":1616557688000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevResearch.2.033042"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,9]]},"references-count":34,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,7]]}},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.033042","relation":{},"ISSN":["2643-1564"],"subject":["General Engineering"],"container-title-short":"Phys. 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Many schemes for fault-tolerant quantum information processing have been developed so far, one of which, called topological quantum computation, makes use of degrees of freedom that are inherently insensitive to local errors. However, this scheme is not so reliable against thermal errors. Other fault-tolerant schemes achieve better reliability through active error correction, but incur a substantial overhead cost. Thus, it is of practical importance and theoretical interest to design and assess fault-tolerant schemes that work well at finite temperature without active error correction.\n\nIn this thesis, a three-dimensional gapped lattice spin model is found which demonstrates for the first time that a reliable quantum memory at finite temperature is possible, at least to some extent. When quantum information is encoded into a highly entangled ground state of this model and subjected to thermal errors, the errors remain easily correctable for a long time without any active intervention, because a macroscopic energy barrier keeps the errors well localized. As a result, stored quantum information can be retrieved faithfully for a memory time which grows exponentially with the square of the inverse temperature. In contrast, for previously known types of topological quantum storage in three or fewer spatial dimensions the memory time scales exponentially with the inverse temperature, rather than its square.\n\nThis spin model exhibits a previously unexpected topological quantum order, in which ground states are locally indistinguishable, pointlike excitations are immobile, and the immobility is not affected by small perturbations of the Hamiltonian. The degeneracy of the ground state, though also insensitive to perturbations, is a complicated number-theoretic function of the system size, and the system bifurcates into multiple noninteracting copies of itself under real-space renormalization group transformations. The degeneracy, the excitations, and the renormalization group flow can be analyzed using a framework that exploits the spin model's symmetry and some associated free resolutions of modules over polynomial algebras.","DOI":"10.7907/GCYW-ZE58","publisher":"California Institute of Technology","title":"Lattice Quantum Codes and Exotic Topological Phases of Matter","URL":"https://resolver.caltech.edu/CaltechTHESIS:05292013-140541902","copyright":"No commercial reproduction, distribution, display or performance rights in this work are provided.","version":"Final","reference":[],"_hash":"7e9320ff258d493d1014350a30e5178046fac94201b8fdfaaa45d4989b91c470"},"expire":1732265201691},"doi:10.1007/s00220-013-1810-2":{"value":{"indexed":{"date-parts":[[2023,10,10]],"date-time":"2023-10-10T12:49:38Z","timestamp":1696942178409},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2013,10,10]],"date-time":"2013-10-10T00:00:00Z","timestamp":1381363200000},"content-version":"tdm","delay-in-days":0,"URL":"http://www.springer.com/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2013,12]]},"DOI":"10.1007/s00220-013-1810-2","type":"journal-article","created":{"date-parts":[[2013,10,9]],"date-time":"2013-10-09T06:33:03Z","timestamp":1381300383000},"page":"351-399","source":"Crossref","is-referenced-by-count":71,"title":"Commuting Pauli Hamiltonians as Maps between Free Modules","prefix":"10.1007","volume":"324","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2013,10,10]]},"reference":[],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","link":[{"URL":"http://link.springer.com/content/pdf/10.1007/s00220-013-1810-2.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://link.springer.com/article/10.1007/s00220-013-1810-2/fulltext.html","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http://link.springer.com/content/pdf/10.1007/s00220-013-1810-2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T10:57:05Z","timestamp":1558695425000},"score":1,"resource":{"primary":{"URL":"http://link.springer.com/10.1007/s00220-013-1810-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,10,10]]},"references-count":39,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2013,12]]}},"alternative-id":["1810"],"URL":"http://dx.doi.org/10.1007/s00220-013-1810-2","relation":{},"ISSN":["0010-3616","1432-0916"],"subject":["Mathematical Physics","Statistical and Nonlinear Physics"],"container-title-short":"Commun. 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In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in D-dimensional hyperbolic space.","DOI":"10.22331/q-2022-05-13-711","type":"journal-article","created":{"date-parts":[[2022,5,13]],"date-time":"2022-05-13T12:50:42Z","timestamp":1652446242000},"page":"711","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":"Connectivity constrains quantum codes","prefix":"10.22331","volume":"6","author":[{"given":"Nouédyn","family":"Baspin","sequence":"first","affiliation":[{"name":"Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1"}]},{"given":"Anirudh","family":"Krishna","sequence":"additional","affiliation":[{"name":"Stanford University, Stanford, CA, USA, 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Phys.","published":{"date-parts":[[2016,1,25]]},"assertion":[{"value":"New Journal of Physics","name":"journal_title","label":"Journal title"},{"value":"paper","name":"article_type","label":"Article type"},{"value":"A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)","name":"article_title","label":"Article title"},{"value":"© 2016 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft","name":"copyright_information","label":"Copyright information"},{"value":"cc-by Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. 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":"2015-08-12","name":"date_received","label":"Date received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2015-11-24","name":"date_accepted","label":"Date accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2016-01-25","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/18/1/013050","_hash":"0859a650f912ddd41a6feed730d18de81b39177fcf9457e9d5a7b2b279948e5c"},"expire":1732265221873},"doi:10.1038/s41534-017-0019-1":{"value":{"indexed":{"date-parts":[[2023,11,22]],"date-time":"2023-11-22T13:58:59Z","timestamp":1700661539069},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2017,4,19]],"date-time":"2017-04-19T00:00:00Z","timestamp":1492560000000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2017,4,19]],"date-time":"2017-04-19T00:00:00Z","timestamp":1492560000000},"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":"AbstractPhotonic cat states stored in high-Q resonators show great promise for hardware efficient universal quantum computing. We propose an approach to efficiently prepare such cat states in a Kerr-nonlinear resonator by the use of a two-photon drive. Significantly, we show that this preparation is robust against single-photon loss. An outcome of this observation is that a two-photon drive can eliminate undesirable phase evolution induced by a Kerr nonlinearity. By exploiting the concept of transitionless quantum driving, we moreover demonstrate how non-adiabatic initialization of cat states is possible. Finally, we present a universal set of quantum logical gates that can be performed on the engineered eigenspace of such a two-photon driven resonator and discuss a possible realization using superconducting circuits. The robustness of the engineered subspace to higher-order circuit nonlinearities makes this implementation favorable for scalable quantum computation.","DOI":"10.1038/s41534-017-0019-1","type":"journal-article","created":{"date-parts":[[2017,4,13]],"date-time":"2017-04-13T13:23:14Z","timestamp":1492089794000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":172,"title":"Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving","prefix":"10.1038","volume":"3","author":[{"given":"Shruti","family":"Puri","sequence":"first","affiliation":[]},{"given":"Samuel","family":"Boutin","sequence":"additional","affiliation":[]},{"given":"Alexandre","family":"Blais","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,4,19]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41534-017-0019-1.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-017-0019-1","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-017-0019-1.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,23]],"date-time":"2022-12-23T01:44:11Z","timestamp":1671759851000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-017-0019-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,19]]},"references-count":43,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,12]]}},"alternative-id":["19"],"URL":"http://dx.doi.org/10.1038/s41534-017-0019-1","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":[[2017,4,19]]},"assertion":[{"value":"2 November 2016","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 March 2017","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 March 2017","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 April 2017","order":4,"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"}}],"article-number":"18","id":"doi:10.1038/s41534-017-0019-1","_hash":"e472969f32578a9aebdc749761858f49b2171f53490a63d457e5be6dcb47596f"},"expire":1732265222852},"doi:10.1098/rsta.2011.0485":{"value":{"indexed":{"date-parts":[[2023,8,18]],"date-time":"2023-08-18T23:10:25Z","timestamp":1692400225945},"reference-count":30,"publisher":"The Royal Society","issue":"1979","license":[{"start":{"date-parts":[[2012,11,28]],"date-time":"2012-11-28T00:00:00Z","timestamp":1354060800000},"content-version":"tdm","delay-in-days":0,"URL":"https://royalsociety.org/journals/ethics-policies/data-sharing-mining/"}],"content-domain":{"domain":["royalsocietypublishing.org"],"crossmark-restriction":true},"published-print":{"date-parts":[[2012,11,28]]},"abstract":"\n We provide a solution to the problem of determining whether a target pure state can be asymptotically prepared using dissipative Markovian dynamics under\n fixed\n locality constraints. Besides recovering existing results for a large class of physically relevant entangled states, our approach has the advantage of providing an explicit stabilization test solely based on the input state and constraints of the problem. Connections with the formalism of frustration-free parent Hamiltonians are discussed, as well as control implementations in terms of a switching output-feedback law.\n ","DOI":"10.1098/rsta.2011.0485","type":"journal-article","created":{"date-parts":[[2012,10,22]],"date-time":"2012-10-22T07:32:58Z","timestamp":1350891178000},"page":"5259-5269","update-policy":"http://dx.doi.org/10.1098/crossmark-policy","source":"Crossref","is-referenced-by-count":49,"title":"Stabilizing entangled states with quasi-local quantum dynamical semigroups","prefix":"10.1098","volume":"370","author":[{"given":"Francesco","family":"Ticozzi","sequence":"first","affiliation":[{"name":"Dipartimento di Ingegneria dell'Informazione, Università di Padova, via Gradenigo 6/B, 35131 Padova, Italy"},{"name":"Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA"}]},{"given":"Lorenza","family":"Viola","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA"}]}],"member":"175","published-online":{"date-parts":[[2012,11,28]]},"reference":[],"container-title":"Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","link":[{"URL":"https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2011.0485","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://royalsocietypublishing.org/doi/full-xml/10.1098/rsta.2011.0485","content-type":"application/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2011.0485","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,20]],"date-time":"2021-02-20T23:54:14Z","timestamp":1613865254000},"score":1,"resource":{"primary":{"URL":"https://royalsocietypublishing.org/doi/10.1098/rsta.2011.0485"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,11,28]]},"references-count":30,"journal-issue":{"issue":"1979","published-print":{"date-parts":[[2012,11,28]]}},"alternative-id":["10.1098/rsta.2011.0485"],"URL":"http://dx.doi.org/10.1098/rsta.2011.0485","relation":{},"ISSN":["1364-503X","1471-2962"],"subject":["General Physics and Astronomy","General Engineering","General Mathematics"],"container-title-short":"Phil. Trans. R. Soc. 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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. 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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.","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":23,"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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(miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2020,7,2]]},"article-number":"288","id":"doi:10.22331/q-2020-07-02-288","_hash":"40a5979ff3a10f463658847bb7b976f70bb8958b3d187020da4c050b0bef9d20"},"expire":1732265267877},"doi:10.1103/PhysRevLett.126.150503":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T20:37:03Z","timestamp":1700599023596},"reference-count":63,"publisher":"American Physical Society (APS)","issue":"15","license":[{"start":{"date-parts":[[2021,4,15]],"date-time":"2021-04-15T00:00:00Z","timestamp":1618444800000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"funder":[{"DOI":"10.13039/100000893","name":"Simons Foundation","doi-asserted-by":"publisher"},{"DOI":"10.13039/501100000023","name":"Government of Canada","doi-asserted-by":"publisher"},{"DOI":"10.13039/100011332","name":"Innovation, Science and Economic 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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. We also present a type of covariant codes which nearly saturates these lower bounds.","DOI":"10.22331/q-2021-08-09-521","type":"journal-article","created":{"date-parts":[[2021,8,9]],"date-time":"2021-08-09T11:21:59Z","timestamp":1628508119000},"page":"521","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":19,"title":"New perspectives on covariant quantum error correction","prefix":"10.22331","volume":"5","author":[{"given":"Sisi","family":"Zhou","sequence":"first","affiliation":[{"name":"Department of Physics, Yale University, New Haven, Connecticut 06511, USA"},{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Illinois 60637, USA"}]},{"given":"Zi-Wen","family":"Liu","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada"}]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Illinois 60637, USA"}]}],"member":"9598","published-online":{"date-parts":[[2021,8,9]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2021-08-09-521/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,1,6]],"date-time":"2023-01-06T23:31:04Z","timestamp":1673047864000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2021-08-09-521/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,9]]},"references-count":75,"URL":"http://dx.doi.org/10.22331/q-2021-08-09-521","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2021,8,9]]},"article-number":"521","id":"doi:10.22331/q-2021-08-09-521","_hash":"66042970da09f32ffae04932212f5a41e3402769aacf7eba0bd91e573c6c2291"},"expire":1732265269874},"doi:10.1109/TIT.2018.2873764":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T17:20:31Z","timestamp":1701105631307},"reference-count":91,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","license":[{"start":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T00:00:00Z","timestamp":1554076800000},"content-version":"vor","delay-in-days":0,"URL":"https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html"},{"start":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T00:00:00Z","timestamp":1554076800000},"content-version":"am","delay-in-days":0,"URL":"https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html"},{"start":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T00:00:00Z","timestamp":1554076800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https://doi.org/10.15223/policy-029"},{"start":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T00:00:00Z","timestamp":1554076800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https://doi.org/10.15223/policy-037"}],"funder":[{"name":"ARL-CDQI","award":["W911NF-15-2-0067"]},{"name":"ARO","award":["W911NF-14-1-0011","W911NF-14-1-0563","W911NF-18-1-0020","W911NF-18-1-0212"]},{"name":"ARO MURI","award":["W911NF-16-1-0349"]},{"name":"AFOSR MURI","award":["FA9550-14-1-0052","FA9550-15-1-0015"]},{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["EFMA-1640959"]},{"DOI":"10.13039/100000879","name":"Alfred P. 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Hence determining the quantum capacity of these channels is an outstanding open problem for quantum computation and communication. Here we derive several upper bounds on the quantum capacity of qubit and bosonic thermal attenuators. We introduce an extended version of such channels which is degradable and hence has a single-letter quantum capacity, bounding that of the original thermal attenuators. Another bound for bosonic attenuators is given by the bottleneck inequality applied to a particular channel decomposition. With respect to previously known bounds we report better results in a broad range of attenuation and noise: we can now approximate the quantum capacity up to a negligible uncertainty for most practical applications, e.g., for low thermal noise.","DOI":"10.1038/s41467-018-06848-0","type":"journal-article","created":{"date-parts":[[2018,10,12]],"date-time":"2018-10-12T14:05:36Z","timestamp":1539353136000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":"Narrow bounds for the quantum capacity of thermal attenuators","prefix":"10.1038","volume":"9","author":[{"ORCID":"http://orcid.org/0000-0002-8972-2936","authenticated-orcid":false,"given":"Matteo","family":"Rosati","sequence":"first","affiliation":[]},{"given":"Andrea","family":"Mari","sequence":"additional","affiliation":[]},{"given":"Vittorio","family":"Giovannetti","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,10,18]]},"reference":[],"container-title":"Nature Communications","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41467-018-06848-0","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-018-06848-0.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-018-06848-0.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T19:18:00Z","timestamp":1671563880000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41467-018-06848-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,18]]},"references-count":53,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,12]]}},"alternative-id":["6848"],"URL":"http://dx.doi.org/10.1038/s41467-018-06848-0","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":[[2018,10,18]]},"assertion":[{"value":"5 June 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 September 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 October 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 January 2019","order":4,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":5,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The original version of this Article contained an error in Equation (40). The numerator of the fraction inside the logarithm was missing an overall minus sign. 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We present general results that we illustrate through examples taken from different classes of codes, including scaled self-dual GKP codes and the concatenated surface-GKP code.","DOI":"10.22331/q-2022-02-10-648","type":"journal-article","created":{"date-parts":[[2022,2,10]],"date-time":"2022-02-10T11:00:14Z","timestamp":1644490814000},"page":"648","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":7,"title":"Gottesman-Kitaev-Preskill codes: A lattice perspective","prefix":"10.22331","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0001-6120-9930","authenticated-orcid":false,"given":"Jonathan","family":"Conrad","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"},{"name":"Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, 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EEC-1941583, OMA-2137642"]},{"name":"NTT Research","award":["NTT Research"]},{"DOI":"10.13039/100000008","name":"Packard Foundation","doi-asserted-by":"crossref","award":["2020-71479"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"Quantum error correction has recently been shown to benefit greatly from specific physical encodings of the code qubits. 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.","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":8,"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, 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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. 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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 distancedthrough 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 ofO(d/log⁡d). 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For one of these problems, namely[Formula: see text]where dpq is the Euclidean distance between points P and Q and m is the number of points, we discuss the results for m ≤ 16 and 1 ≤ n ≤ ∞. For the cases m = 5, 11, 13–16 we find hitherto undiscovered solutions. Our solutions for m = 5 and 11 correct earlier results in the literature. We also sharpen the existing literature results for m = 7 and 10. 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Comput.","published":{"date-parts":[[1998,6]]},"id":"doi:10.1137/S0097539796302531","_hash":"8a6be0a2edcf4f436caf0be798e0555367bbed2a0bb2e9ea9c0b56939a7f739c"},"expire":1732265728872},"doi:10.1145/502090.502098":{"value":{"indexed":{"date-parts":[[2023,11,28]],"date-time":"2023-11-28T05:27:04Z","timestamp":1701149224835},"reference-count":32,"publisher":"Association for Computing Machinery (ACM)","issue":"4","content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"published-print":{"date-parts":[[2001,7]]},"abstract":"\n We prove optimal, up to an arbitrary ε > 0, inapproximability results for Max-E\n k\n -Sat for\n k\n ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime\n p\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 ","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","update-policy":"http://dx.doi.org/10.1145/crossmark-policy","source":"Crossref","is-referenced-by-count":773,"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","published-online":{"date-parts":[[2001,7]]},"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":[[2023,1,2]],"date-time":"2023-01-02T21:40:24Z","timestamp":1672695624000},"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":"d6089c0143c32c32141b4f17359d02b92c6bf47aba112f2538c6af052bef9870"},"expire":1732265730902},"doi:10.1145/3051093":{"value":{"indexed":{"date-parts":[[2023,8,29]],"date-time":"2023-08-29T22:53:55Z","timestamp":1693349635987},"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":390,"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":["dl.acm.org"],"crossmark-restriction":true},"published-print":{"date-parts":[[2017,4,30]]},"abstract":"\n Locally correctable codes (LCCs) and locally testable codes (LTCs) are error-correcting codes that admit\n local\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 \n \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 n\n , constant rate (which can even be taken arbitrarily close to 1), and constant relative distance, whose query complexity is exp(Õ(√log\n n\n )) (for LCCs) and (log\n n\n )\n \n O\n (log log\n n\n )\n \n (for LTCs).\n \n \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 o\n (\n n\n ) query complexity. Our results on LCCs also immediately give locally decodable codes with the same parameters.\n ","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","update-policy":"http://dx.doi.org/10.1145/crossmark-policy","source":"Crossref","is-referenced-by-count":20,"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","published-online":{"date-parts":[[2017,5,25]]},"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":[[2022,12,31]],"date-time":"2022-12-31T09:47:48Z","timestamp":1672480068000},"score":1,"resource":{"primary":{"URL":"https://dl.acm.org/doi/10.1145/3051093"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,30]]},"references-count":69,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2017,4,30]]}},"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":[[2017,4,30]]},"assertion":[{"value":"2016-05-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2017-02-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2017-05-25","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"id":"doi:10.1145/3051093","_hash":"deaca14d109a2689ae98627787190a7748fc6ba32efe87c52fef6408d5986575"},"expire":1732265731878},"doi:10.1145/1162349.1162351":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T18:22:46Z","timestamp":1700590966966},"reference-count":32,"publisher":"Association for Computing Machinery (ACM)","issue":"4","content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"published-print":{"date-parts":[[2006,7]]},"abstract":"\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 k\n information bits are encoded by a codeword of length\n k\n ⋅ exp(Õ(√(log\n k\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 n\n ⋅ exp(Õ(√(log\n n\n ))) and verification in performed by a constant number (i.e., 19) of queries, as opposed to previous results that used proof length\n n\n \n (1 +\n O\n (1/\n q\n ))\n \n for verification by\n q\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 ","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","update-policy":"http://dx.doi.org/10.1145/crossmark-policy","source":"Crossref","is-referenced-by-count":84,"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","published-online":{"date-parts":[[2006,7]]},"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":[[2022,12,28]],"date-time":"2022-12-28T19:56:08Z","timestamp":1672257368000},"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 largest travel time differences for any solar phenomena are observed.Conclusions.With sufficient modeling effort, these should lead to better understanding of sunspot structure.","DOI":"10.1051/0004-6361/201732424","type":"journal-article","created":{"date-parts":[[2018,1,31]],"date-time":"2018-01-31T08:48:16Z","timestamp":1517388496000},"page":"A73","source":"Crossref","is-referenced-by-count":4,"title":"Probing sunspots with two-skip time–distance helioseismology","prefix":"10.1051","volume":"613","author":[{"suffix":"Jr.","given":"Thomas L.","family":"Duvall","sequence":"first","affiliation":[]},{"given":"Paul S.","family":"Cally","sequence":"additional","affiliation":[]},{"given":"Damien","family":"Przybylski","sequence":"additional","affiliation":[]},{"given":"Kaori","family":"Nagashima","sequence":"additional","affiliation":[]},{"given":"Laurent","family":"Gizon","sequence":"additional","affiliation":[]}],"member":"250","published-online":{"date-parts":[[2018,6,4]]},"reference":[],"container-title":"Astronomy & Astrophysics","original-title":[],"link":[{"URL":"https://www.aanda.org/10.1051/0004-6361/201732424/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,13]],"date-time":"2022-08-13T15:39:52Z","timestamp":1660405192000},"score":1,"resource":{"primary":{"URL":"https://www.aanda.org/10.1051/0004-6361/201732424"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5]]},"references-count":36,"alternative-id":["aa32424-17"],"URL":"http://dx.doi.org/10.1051/0004-6361/201732424","relation":{},"ISSN":["0004-6361","1432-0746"],"subject":["Space and Planetary Science","Astronomy and Astrophysics"],"container-title-short":"A&A","published":{"date-parts":[[2018,5]]},"id":"doi:10.1051/0004-6361/201732424","_hash":"684444e104b819e95f029ee0306a38de104ba1b472b459709f5f4ccfe41824c5"},"expire":1732286485686},"doi:10.26421/QIC14.9-10-1":{"value":{"indexed":{"date-parts":[[2023,8,23]],"date-time":"2023-08-23T23:32:32Z","timestamp":1692833552065},"reference-count":0,"publisher":"Rinton Press","issue":"9&10","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2014,7]]},"abstract":"We present a three-dimensional generalization of a renormalization group decoding algorithm for topological codes with Abelian anyonic excitations that we introduced for two dimensions in \\cite{DP09a,DP10a}. 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Provided with input magic states, our protocol operates on a two-dimensional square grid by measurements of ZZ on horizontal pairs of qubits, XX on vertical pairs, and Z,X on single qubits.","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":1,"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":"793809ddb352ada0f94f1227933c3fb6d00c3b1552b28a55faff64e78430a3d9"},"expire":1734264356128},"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":"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.","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":1734264356922},"doi:10.1103/PhysRevA.57.127":{"value":{"indexed":{"date-parts":[[2023,12,20]],"date-time":"2023-12-20T17:00:50Z","timestamp":1703091650237},"reference-count":13,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[1998,1,1]],"date-time":"1998-01-01T00:00:00Z","timestamp":883612800000},"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.57.127","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T01:39:22Z","timestamp":1027733962000},"page":"127-137","source":"Crossref","is-referenced-by-count":490,"title":"Theory of fault-tolerant quantum computation","prefix":"10.1103","volume":"57","author":[{"given":"Daniel","family":"Gottesman","sequence":"first","affiliation":[]}],"member":"16","published-online":{"date-parts":[[1998,1,1]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.57.127","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.57.127/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,15]],"date-time":"2017-06-15T08:21:12Z","timestamp":1497514872000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.57.127"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,1,1]]},"references-count":13,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1998,1]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.57.127","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. 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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.","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":70,"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 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circuits","prefix":"10.1103","volume":"65","author":[{"given":"Barbara M.","family":"Terhal","sequence":"first","affiliation":[]},{"given":"David P.","family":"DiVincenzo","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2002,3,1]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.65.032325","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.65.032325/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,15]],"date-time":"2017-06-15T07:35:44Z","timestamp":1497512144000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.65.032325"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,3,1]]},"references-count":9,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2002,3]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.65.032325","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. Rev. A","published":{"date-parts":[[2002,3,1]]},"article-number":"032325","id":"doi:10.1103/PhysRevA.65.032325","_hash":"f04ea05d72e6da785d55e8b60fafc003f485f437b3db33087ca04c3b62fe421c"},"expire":1734264360836},"doi:10.22331/q-2017-04-25-4":{"value":{"indexed":{"date-parts":[[2023,8,10]],"date-time":"2023-08-10T11:50:21Z","timestamp":1691668221215},"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":"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 n, the number of encoded qubits k, the code distance d, the accuracy parameter &#x03B4; that quantifies how well the erasure channel can be reversed, and the locality parameter &#x2113; that specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracy &#x03B4; that is exponentially small in &#x2113;, which is the case for perturbations of local commuting projector codes, our bound reads kd2D&#x2212;1&#x2264;O(n(log&#x2061;n)2DD&#x2212;1) for codes on D-dimensional lattices of Euclidean metric. We also find that the code distance of any local approximate code cannot exceed O(&#x2113;n(D&#x2212;1)/D) if &#x03B4;&#x2264;O(&#x2113;n&#x2212;1/D). As a corollary of our formulation of correctability in terms of logical operator avoidance, we show that the code distance d and the size d&#x007E; of a minimal region that can support all approximate logical operators satisfies d&#x007E;d1D&#x2212;1&#x2264;O(n&#x2113;DD&#x2212;1), where the logical operators are accurate up to O((n&#x03B4;/d)1/2) 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.","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":19,"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. J. Watson Research Center, Yorktown Heights, New York, USA"},{"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":[[2017,4,25]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2017-04-25-4/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,2,7]],"date-time":"2022-02-07T13:13:17Z","timestamp":1644239597000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2017-04-25-4/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,25]]},"references-count":56,"URL":"http://dx.doi.org/10.22331/q-2017-04-25-4","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":"4","id":"doi:10.22331/q-2017-04-25-4","_hash":"315ea08cc0ae082500e19cb236880c49a8c9be9d5a76a24c63593d47ea4c277f"},"expire":1734264361947},"doi:10.1098/rsta.1995.0106":{"value":{"indexed":{"date-parts":[[2023,11,4]],"date-time":"2023-11-04T01:56:56Z","timestamp":1699063016140},"reference-count":45,"publisher":"The Royal Society","issue":"1703","license":[{"start":{"date-parts":[[1995,12,15]],"date-time":"1995-12-15T00:00:00Z","timestamp":818985600000},"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":[[1995,12,15]]},"abstract":"\n Technologies differ in their explicit utilization of quantum mechanical behaviour. 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 localization\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 ","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":102,"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. 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Specifically, we investigate the quantum phase-flip repetition code as a quantum memory and theoretically demonstrate that it can preserve arbitrary quantum information longer than the lifetime of a single idle qubit in a dephasing-time-limited system, e.g. in semiconductor qubits. Our circuit-based analytical calculations show the efficiency of the phase-flip code as a quantum memory in the presence of relaxation, dephasing, and faulty quantum gates. Moreover, we identify the optimal repetition number of quantum error correction cycles required to reach the break-even point by considering the gate error probabilities of current platforms for quantum computing. Our results provide guidelines for developing quantum memories in semiconductor quantum devices.","DOI":"10.1088/1367-2630/acfba5","type":"journal-article","created":{"date-parts":[[2023,9,20]],"date-time":"2023-09-20T22:28:00Z","timestamp":1695248880000},"page":"103004","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":"Break-even point of the phase-flip error correcting code","prefix":"10.1088","volume":"25","author":[{"given":"Áron","family":"Rozgonyi","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4178-5759","authenticated-orcid":true,"given":"Gábor","family":"Széchenyi","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2023,10,5]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,5]],"date-time":"2023-10-05T11:57:00Z","timestamp":1696507020000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,1]]},"references-count":40,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10,5]]},"published-print":{"date-parts":[[2023,10,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/acfba5","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":"2023-05-15","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2023-09-20","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2023-10-05","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/acfba5","_hash":"cd36bd2f1a52af022ead4eef16300d136cc60dfad25c6d98466cd54418b0bcc9"},"expire":1734778369500},"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":"Abstract\n \n 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(n\n 1−ν ) for any ν ∈ (0, 1) and Ω(log n) 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 d-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.","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. 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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":1734778370842},"doi:10.1017/fms.2023.98":{"value":{"indexed":{"date-parts":[[2023,11,29]],"date-time":"2023-11-29T01:01:49Z","timestamp":1701219709765},"reference-count":40,"publisher":"Cambridge University Press (CUP)","license":[{"start":{"date-parts":[[2023,11,28]],"date-time":"2023-11-28T00:00:00Z","timestamp":1701129600000},"content-version":"unspecified","delay-in-days":331,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"published-print":{"date-parts":[[2023]]},"abstract":"Abstract\n\t We show that every ergodic Davies generator associated to any 2D Kitaev’s quantum double model has a nonvanishing spectral gap in the thermodynamic limit. This validates rigorously the extended belief that those models are useless as self-correcting quantum memories, even in the non-abelian case. The proof uses recent ideas and results regarding the characterization of the spectral gap for parent Hamiltonians associated to Projected Entangled Pair States in terms of a bulk-boundary correspondence.","DOI":"10.1017/fms.2023.98","type":"journal-article","created":{"date-parts":[[2023,11,28]],"date-time":"2023-11-28T10:01:08Z","timestamp":1701165668000},"update-policy":"http://dx.doi.org/10.1017/policypage","source":"Crossref","is-referenced-by-count":0,"title":"Thermalization in Kitaev’s quantum double models via tensor network techniques","prefix":"10.1017","volume":"11","author":[{"ORCID":"http://orcid.org/0000-0003-1709-1220","authenticated-orcid":false,"given":"Angelo","family":"Lucia","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-2990-791X","authenticated-orcid":false,"given":"David","family":"Pérez-García","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-8600-7083","authenticated-orcid":false,"given":"Antonio","family":"Pérez-Hernández","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,11,28]]},"reference":[],"container-title":"Forum of Mathematics, Sigma","original-title":[],"language":"en","link":[{"URL":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S2050509423000981","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,28]],"date-time":"2023-11-28T10:01:13Z","timestamp":1701165673000},"score":1,"resource":{"primary":{"URL":"https://www.cambridge.org/core/product/identifier/S2050509423000981/type/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023]]},"references-count":40,"alternative-id":["S2050509423000981"],"URL":"http://dx.doi.org/10.1017/fms.2023.98","relation":{},"ISSN":["2050-5094"],"subject":["Computational Mathematics","Discrete Mathematics and Combinatorics","Geometry and Topology","Mathematical Physics","Statistics and Probability","Algebra and Number Theory","Theoretical Computer Science","Analysis"],"container-title-short":"Forum of Mathematics, Sigma","published":{"date-parts":[[2023]]},"assertion":[{"value":"© The Author(s), 2023. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}],"article-number":"e107","id":"doi:10.1017/fms.2023.98","_hash":"78b79372db2887a939619d6094efb4ec696bd1300af70ed95bc09e50548dc6ff"},"expire":1734778371826},"doi:10.1007/JHEP01(2022)170":{"value":{"indexed":{"date-parts":[[2023,10,28]],"date-time":"2023-10-28T23:25:55Z","timestamp":1698535555058},"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":"Abstract\n 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 N and a large gap in the spectrum of operators in the emergence of complementary recovery.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.","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":8,"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. Phys.","published":{"date-parts":[[2022,1]]},"assertion":[{"value":"4 July 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 December 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 January 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"170","id":"doi:10.1007/JHEP01(2022)170","_hash":"04e87b4a3419aa06427adab3e265d4bae557a83fdf0df9e461e31c190807b756"},"expire":1734778372738},"doi:10.1038/s41534-023-00788-4":{"value":{"indexed":{"date-parts":[[2023,11,22]],"date-time":"2023-11-22T00:25:46Z","timestamp":1700612746143},"reference-count":97,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T00:00:00Z","timestamp":1700524800000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T00:00:00Z","timestamp":1700524800000},"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":"AbstractQuantum error correction (QEC) is a key concept in quantum computation as well as many areas of physics. There are fundamental tensions between continuous symmetries and QEC. One vital situation is unfolded by the Eastin–Knill theorem, which forbids the existence of QEC codes that admit transversal continuous symmetry actions (transformations). Here, we systematically study the competition between continuous symmetries and QEC in a quantitative manner. We first define a series of meaningful measures of approximate symmetries motivated from different perspectives, and then establish a series of trade-off bounds between them and QEC accuracy utilizing multiple different methods. Remarkably, the results allow us to derive general quantitative limitations of transversally implementable logical gates, an important topic in fault-tolerant quantum computation. As concrete examples, we showcase two explicit types of quantum codes, obtained from quantum Reed–Muller codes and thermodynamic codes, respectively, that nearly saturate our bounds. Finally, we discuss several potential applications of our results in physics.","DOI":"10.1038/s41534-023-00788-4","type":"journal-article","created":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T20:05:28Z","timestamp":1700597128000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":"Approximate symmetries and quantum error correction","prefix":"10.1038","volume":"9","author":[{"ORCID":"http://orcid.org/0000-0002-3402-9763","authenticated-orcid":false,"given":"Zi-Wen","family":"Liu","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4618-8590","authenticated-orcid":false,"given":"Sisi","family":"Zhou","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,11,21]]},"reference":[],"container-title":"npj Quantum 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Theory and Mathematics","Computer Networks and Communications","Statistical and Nonlinear Physics","Computer Science (miscellaneous)"],"container-title-short":"npj Quantum Inf","published":{"date-parts":[[2023,11,21]]},"assertion":[{"value":"25 April 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 November 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 November 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 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Multimode Gottesman-Kitaev-Preskill Codes","prefix":"10.1103","volume":"4","author":[{"ORCID":"http://orcid.org/0009-0002-6102-5730","authenticated-orcid":true,"given":"Mao","family":"Lin","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-6318-8472","authenticated-orcid":true,"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2023,12,1]]},"reference":[],"container-title":"PRX Quantum","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PRXQuantum.4.040334","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PRXQuantum.4.040334/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,1]],"date-time":"2023-12-01T16:53:44Z","timestamp":1701449624000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PRXQuantum.4.040334"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,1]]},"references-count":73,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,12]]}},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.040334","relation":{},"ISSN":["2691-3399"],"subject":["General Physics and Astronomy","Mathematical Physics","Applied Mathematics","Electronic, Optical and Magnetic Materials","Electrical and Electronic Engineering","General Computer Science"],"container-title-short":"PRX Quantum","published":{"date-parts":[[2023,12,1]]},"article-number":"040334","id":"doi:10.1103/PRXQuantum.4.040334","_hash":"7a503229967ae75fe64c4c71ef626043a5c7add45b575c20e1936a9591901c93"},"expire":1734778375613},"doi:10.1103/PhysRevLett.98.130501":{"value":{"indexed":{"date-parts":[[2023,12,23]],"date-time":"2023-12-23T06:19:21Z","timestamp":1703312361800},"reference-count":21,"publisher":"American Physical Society (APS)","issue":"13","license":[{"start":{"date-parts":[[2007,3,26]],"date-time":"2007-03-26T00:00:00Z","timestamp":1174867200000},"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.98.130501","type":"journal-article","created":{"date-parts":[[2007,3,27]],"date-time":"2007-03-27T15:29:54Z","timestamp":1175009394000},"source":"Crossref","is-referenced-by-count":125,"title":"Quantum Capacities of Bosonic Channels","prefix":"10.1103","volume":"98","author":[{"given":"Michael M.","family":"Wolf","sequence":"first","affiliation":[]},{"given":"David","family":"Pérez-García","sequence":"additional","affiliation":[]},{"given":"Geza","family":"Giedke","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2007,3,26]]},"reference":[],"container-title":"Physical Review Letters","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevLett.98.130501","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.98.130501/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,4,5]],"date-time":"2017-04-05T21:14:56Z","timestamp":1491426896000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevLett.98.130501"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,3,26]]},"references-count":21,"journal-issue":{"issue":"13","published-print":{"date-parts":[[2007,3]]}},"URL":"http://dx.doi.org/10.1103/PhysRevLett.98.130501","relation":{},"ISSN":["0031-9007","1079-7114"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. Rev. Lett.","published":{"date-parts":[[2007,3,26]]},"article-number":"130501","id":"doi:10.1103/PhysRevLett.98.130501","_hash":"a8fc501f5d2954c7a55f1d0903c1cc33c89c274ebd21f4a20b832f7d90252ba9"},"expire":1734778376895},"doi:10.22331/q-2022-09-29-821":{"value":{"indexed":{"date-parts":[[2023,11,18]],"date-time":"2023-11-18T10:31:00Z","timestamp":1700303460293},"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":"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. 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Such a pair of Ising models can be interpreted as a two-chain complex with k being the rank of the first homology group. Our focus is on the case where k is extensive, that is, scales linearly with the number of bonds n. Flipping any of these additional spins introduces a homologically nontrivial defect (generalized domain wall). In the presence of bond disorder, we prove the existence of a low-temperature weak-disorder region where additional summation over the defects has no effect on the free energy density f(T) in the thermodynamical limit and of a high-temperature region where an extensive homological defect does not affect f(T). We also discuss the convergence of the high- and low-temperature series for the free energy density, prove the analyticity of limiting f(T) at high and low temperatures, and construct inequalities for the critical point(s) where analyticity is lost. 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Bosonic phase noise channels, a class of non-Gaussian channels, have emerged as a relevant noise model in quantum information and optical communication. However, while the fundamental limits for communication over Gaussian channels have been extensively studied, the properties of communication over Bosonic phase noise channels are not well understood. Here we propose and demonstrate experimentally the concept of optimized communication strategies for communication over phase noise channels to enhance information transfer beyond what is possible with conventional methods of modulation and detection. Two key ingredients are generalized constellations of coherent states that interpolate between standard on-off keying and binary phase-shift keying formats, and non-Gaussian measurements based on photon number resolving detection of the coherently displaced signal. 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We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are \n\n \n m\n m\n \n\n distances between distinct points in it and it is a spherical \n\n \n \n (\n 2\n m\n −\n 1\n )\n \n (2m-1)\n \n\n-design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the \n\n \n \n E\n 8\n \n E_8\n \n\n and Leech lattices. We also prove the same result for the vertices of the \n\n \n 600\n 600\n \n\n-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.

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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. 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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\nc_-c−\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.","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 K-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.","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 *","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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It starts with a demonstration of how,in these electrodynamic systems, both the superconducting gap and the long-range Coulomb interactions contribute to the existence of collective modesthat have extremely low dissipationand that can be quantized. The chapter also gives in-depth coverage of quantum harmonic oscillators in all their glory from various perspectives. Superconducting qubits are then introduced, starting with the fundamental theory of the Josephson effect and moving on to a presentation of the variety of existing superconducting qubits. Particular attention is given to the dispersive readout of qubits through their interaction with a cavity into which flying modes can be scattered. 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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.","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. 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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 non-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.","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":1712590663927},"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":"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.","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":1712590665010},"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":" 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. ","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 codes2–8, 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 scenarios9–17. 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 architecture18, where the logical qubit is binomially encoded in photon-number states of a microwave cavity8, 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. Our work illustrates the potential of hardware-efficient discrete-variable encodings for fault-tolerant quantum computation19.","DOI":"10.1038/s41586-023-05784-4","type":"journal-article","created":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T17:03:38Z","timestamp":1679504618000},"page":"56-60","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":"Beating the break-even point with a discrete-variable-encoded logical qubit","prefix":"10.1038","volume":"616","author":[{"given":"Zhongchu","family":"Ni","sequence":"first","affiliation":[]},{"given":"Sai","family":"Li","sequence":"additional","affiliation":[]},{"given":"Xiaowei","family":"Deng","sequence":"additional","affiliation":[]},{"given":"Yanyan","family":"Cai","sequence":"additional","affiliation":[]},{"given":"Libo","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-1510-772X","authenticated-orcid":false,"given":"Weiting","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-3964-4714","authenticated-orcid":false,"given":"Zhen-Biao","family":"Yang","sequence":"additional","affiliation":[]},{"given":"Haifeng","family":"Yu","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-4674-2806","authenticated-orcid":false,"given":"Fei","family":"Yan","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-2340-4556","authenticated-orcid":false,"given":"Song","family":"Liu","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-2484-7292","authenticated-orcid":false,"given":"Chang-Ling","family":"Zou","sequence":"additional","affiliation":[]},{"given":"Luyan","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Shi-Biao","family":"Zheng","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-2603-0721","authenticated-orcid":false,"given":"Yuan","family":"Xu","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-0672-8233","authenticated-orcid":false,"given":"Dapeng","family":"Yu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,3,22]]},"reference":[],"container-title":"Nature","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41586-023-05784-4.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41586-023-05784-4","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41586-023-05784-4.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,5]],"date-time":"2023-04-05T13:07:06Z","timestamp":1680700026000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41586-023-05784-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,22]]},"references-count":44,"journal-issue":{"issue":"7955","published-print":{"date-parts":[[2023,4,6]]}},"alternative-id":["5784"],"URL":"http://dx.doi.org/10.1038/s41586-023-05784-4","relation":{},"ISSN":["0028-0836","1476-4687"],"subject":["Multidisciplinary"],"container-title-short":"Nature","published":{"date-parts":[[2023,3,22]]},"assertion":[{"value":"16 November 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 February 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 March 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"}}],"id":"doi:10.1038/s41586-023-05784-4","_hash":"076123c1958be266ec1f311b99bbee5b16b66d59d28b8b4d4a59afabe6254ded"},"expire":1712590668017},"doi:10.1038/s41586-023-05782-6":{"value":{"indexed":{"date-parts":[[2023,4,6]],"date-time":"2023-04-06T04:51:51Z","timestamp":1680756711266},"reference-count":51,"publisher":"Springer Science and Business Media 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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 bepSTC ≈ 1.045%. Such a high threshold together with local parity-check measurements make the 3D STC particularly appealing for realizing fault-tolerant quantum computing.","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":1715612576851},"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":"AbstractElectrons 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.","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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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.","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":1718831603960},"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":"AbstractQuantum 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.","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":1718831605131},"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":"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 17% under independent bit-and phase-flip data noise (to be compared to the ideal threshold of 20.6%) and a threshold value of 14% under depolarizing data noise (compared to the ideal threshold of 18.9%), which are on par with thresholds achieved by non-BP post-processing methods.","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":1718831606055},"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":1718831607022},"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":"We present magic state factory constructions for producing|CCZ⟩states and|T⟩states. For the|CCZ⟩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 of12d×6d(wheredis the code distance) and produces one|CCZ⟩every5.5dsurface code cycles. Our|T⟩state factory uses the|CCZ⟩factory's output and a catalyst|T⟩state to exactly transform one|CCZ⟩state into two|T⟩states. It has a footprint25%smaller than the factory in \\cite{fowler2018} but outputs|T⟩states twice as quickly. We show how to generalize the catalyzed transformation to arbitrary phase angles, and note that the caseθ=22.5∘produces a particularly efficient circuit for producing|T⟩states. Compared to using the12d×8d×6.5d|T⟩factory of \\cite{fowler2018}, our|CCZ⟩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 of10−3, our CCZ factory can produce∼1010states 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∼1012by increasing the factory footprint∼20%.","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|CCZ⟩to2|T⟩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":1718831608057},"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":"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. We give a first taste of the power of the calculus as a language for lattice surgery by considering two operations (T gates and producing a CNOT) and show how ZX diagram re-write rules give lattice surgery procedures for these operations that are novel, efficient, and highly configurable.","DOI":"10.22331/q-2020-01-09-218","type":"journal-article","created":{"date-parts":[[2020,1,9]],"date-time":"2020-01-09T19:54:16Z","timestamp":1578599656000},"page":"218","source":"Crossref","is-referenced-by-count":13,"title":"The ZX calculus is a language for surface code lattice surgery","prefix":"10.22331","volume":"4","author":[{"ORCID":"http://orcid.org/0000-0001-9549-5146","authenticated-orcid":false,"given":"Niel","family":"de Beaudrap","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Oxford, Parks Road, Oxford, OX1 3QD"}]},{"ORCID":"http://orcid.org/0000-0003-4965-0584","authenticated-orcid":false,"given":"Dominic","family":"Horsman","sequence":"additional","affiliation":[{"name":"Department of Physics, Durham University, South Road, Durham, DH1 1LE Department of Computer Science, University of Oxford, Parks Road, Oxford, OX1 3QD"}]}],"member":"9598","published-online":{"date-parts":[[2020,1,9]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2020-01-09-218/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2020,6,5]],"date-time":"2020-06-05T18:19:50Z","timestamp":1591381190000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2020-01-09-218/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,9]]},"references-count":47,"URL":"http://dx.doi.org/10.22331/q-2020-01-09-218","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2020,1,9]]},"article-number":"218","id":"doi:10.22331/q-2020-01-09-218","_hash":"5dd2287e91568ba6d4ffc89ce0afc2402586ba6ca11c443b1f830231880b4dfc"},"expire":1718831608957},"doi:10.1088/1751-8113/42/9/095302":{"value":{"indexed":{"date-parts":[[2023,6,13]],"date-time":"2023-06-13T17:51:13Z","timestamp":1686678673142},"reference-count":18,"publisher":"IOP Publishing","issue":"9","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2009,3,6]]},"DOI":"10.1088/1751-8113/42/9/095302","type":"journal-article","created":{"date-parts":[[2009,2,5]],"date-time":"2009-02-05T04:15:17Z","timestamp":1233807317000},"page":"095302","source":"Crossref","is-referenced-by-count":79,"title":"Quantum measurements and gates by code deformation","prefix":"10.1088","volume":"42","author":[{"given":"H","family":"Bombin","sequence":"first","affiliation":[]},{"given":"M A","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2009,2,4]]},"reference":[],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"link":[{"URL":"http://stacks.iop.org/1751-8121/42/i=9/a=095302/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,1]],"date-time":"2021-10-01T03:26:38Z","timestamp":1633058798000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1751-8113/42/9/095302"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,2,4]]},"references-count":18,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2009,3,6]]}},"alternative-id":["S1751-8113(09)86826-1"],"URL":"http://dx.doi.org/10.1088/1751-8113/42/9/095302","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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Theor.","published":{"date-parts":[[2009,2,4]]},"id":"doi:10.1088/1751-8113/42/9/095302","_hash":"125ec259e206ce78c5591e149b311124aece02b018574439ddb2f8a66d9daa2f"},"expire":1718831609951},"doi:10.21468/SciPostPhysLectNotes.70":{"value":{"indexed":{"date-parts":[[2023,6,13]],"date-time":"2023-06-13T04:28:45Z","timestamp":1686630525727},"reference-count":132,"publisher":"Stichting SciPost","license":[{"start":{"date-parts":[[2023,6,12]],"date-time":"2023-06-12T00:00:00Z","timestamp":1686528000000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["W911NF-18-1-0212"]},{"DOI":"10.13039/100005326","name":"Yale University","doi-asserted-by":"publisher"}],"content-domain":{"domain":["scipost.org"],"crossmark-restriction":false},"abstract":"These lecture notes from the 2019 Les Houches Summer School on “Quantum Information Machines” are intended to provide an introduction to classical and quantum error correction with bits and qubits, and with continuous variable systems (harmonic oscillators). The focus on the latter will be on practical examples that can be realized today or in the near future with a modular architecture based on superconducting electrical circuits and microwave photons. The goal and vision is “hardware-efficient” quantum error correction that does not require exponentially large hardware overhead in order to achieve practical and useful levels of fault tolerance and circuit depth.","DOI":"10.21468/scipostphyslectnotes.70","type":"journal-article","created":{"date-parts":[[2023,6,12]],"date-time":"2023-06-12T14:52:35Z","timestamp":1686581555000},"update-policy":"http://dx.doi.org/10.21468/scipost.crossmarkpolicy","source":"Crossref","is-referenced-by-count":0,"title":"Introduction to quantum error correction and fault tolerance","prefix":"10.21468","author":[{"given":"Steven M.","family":"Girvin","sequence":"first","affiliation":[{"name":"Yale University"}]}],"member":"8907","published-online":{"date-parts":[[2023,6,12]]},"reference":[],"container-title":"SciPost Physics Lecture Notes","original-title":[],"link":[{"URL":"https://scipost.org/10.21468/SciPostPhysLectNotes.70/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://scipost.org/10.21468/SciPostPhysLectNotes.70/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,12]],"date-time":"2023-06-12T14:52:48Z","timestamp":1686581568000},"score":1,"resource":{"primary":{"URL":"https://scipost.org/10.21468/SciPostPhysLectNotes.70"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,12]]},"references-count":132,"URL":"http://dx.doi.org/10.21468/SciPostPhysLectNotes.70","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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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 2T of the quaternions. The resulting 2T-qutrit naturally inherits the algebraic properties of the group 2T 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.","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 2T-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":1718831618966},"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":"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.","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":1718831619965},"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 α of the black hole entropy. In the Schrödinger picture, the boundary subregion encodes the α-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 α-bit codes. Unintuitively, Hawking radiation always reveals the α-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/N.","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.","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":1721576483986},"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":"In this work, we compute the number of [[n,k]]d stabilizer codes made up of d-dimensional qudits, for arbitrary positive integers d. In a seminal work by Gross \\cite{Gross2006} the number of [[n,k]]d stabilizer codes was computed for the case when d is a prime (or the power of a prime, i.e., d=pm, 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 [[n,k]]d codes, and use this in conjunction with the Chinese remainder theorem to count the number of [[n,k]]d codes. Our work overlaps with \\cite{Gross2006} when d 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 d 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.","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 d = 17 surface code volume, we show a \n \n \n \n 10\n 6\n \n ×\n \n \n speedup of the minimum-weight perfect matching decoder. Lastly, we estimate the cost of implementing our local decoders on field programmable gate arrays.","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":1728583705599},"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":"Recently, a class of fractal surface codes (FSCs), has been constructed on fractal lattices with Hausdorff dimension 2+&#x03F5;, 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 2+&#x03F5;. 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 (&#x223C;1.7&#x0025;) under phenomenological noise for the sweep decoder and the code capacity threshold (lower bounded by 2.95&#x0025;) for the MWPM decoder for a particular FSC with Hausdorff dimension DH&#x2248;2.966. 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.","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'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":1728583706613},"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":"flip 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 flip, 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 flip 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 flip we observe a threshold of &#x223C;5.5&#x0025; under phenomenological noise. This is comparable to the best known threshold for this code (&#x223C;7.1&#x0025;) which was achieved using belief propagation and ordered statistics decoding [Higgott and Breuckmann, 2022], a strategy with a runtime of O(n3) as opposed to the O(n) (O(1) 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.","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. R.","family":"Scruby","sequence":"first","affiliation":[{"name":"Okinawa Institute of Science and Technology, Okinawa, 904-0495, Japan"}]},{"given":"K.","family":"Nemoto","sequence":"additional","affiliation":[{"name":"Okinawa Institute of Science and Technology, Okinawa, 904-0495, Japan"}]}],"member":"9598","published-online":{"date-parts":[[2023,8,29]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2023-08-29-1093/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,8,29]],"date-time":"2023-08-29T11:18:57Z","timestamp":1693307937000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2023-08-29-1093/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,29]]},"references-count":34,"URL":"http://dx.doi.org/10.22331/q-2023-08-29-1093","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2023,8,29]]},"article-number":"1093","id":"doi:10.22331/q-2023-08-29-1093","_hash":"6ea5e213eafec4ed7fe8e9dcbe5e817b287878b74d155e0f7ef28c9e9d3028b9"},"expire":1728583707706},"doi:10.22331/q-2023-08-08-1075":{"value":{"indexed":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T17:11:44Z","timestamp":1697044304821},"reference-count":40,"publisher":"Verein 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":"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 d that can correct up to t=&#x230A;(d&#x2212;1)/2&#x230B; errors, the traditional Shor scheme handles ancilla preparation and measurement faults by performing syndrome measurements until the syndromes are repeated t+1 times in a row; in the worst-case scenario, (t+1)2 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 (t+3)2/4&#x2212;1 rounds and (t+3)2/4&#x2212;2 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. 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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 price 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 detailed size winding [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 N 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.","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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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\nNN\ntheories. Specifically we examine SU(N)SU(N)\nmatrix quantum mechanics and 3-rank tensor\nO(N)^3O(N)3\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\nNN\nanalysis and do not appeal to a particular form of Hamiltonian or\nholography.","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":1729266847994},"doi:10.1103/PhysRev.58.1098":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T21:39:24Z","timestamp":1700602764269},"reference-count":18,"publisher":"American Physical Society (APS)","issue":"12","license":[{"start":{"date-parts":[[1940,12,15]],"date-time":"1940-12-15T00:00:00Z","timestamp":-916617600000},"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/physrev.58.1098","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T00:30:13Z","timestamp":1027729813000},"page":"1098-1113","source":"Crossref","is-referenced-by-count":3119,"title":"Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet","prefix":"10.1103","volume":"58","author":[{"given":"T.","family":"Holstein","sequence":"first","affiliation":[]},{"given":"H.","family":"Primakoff","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[1940,12,15]]},"reference":[],"container-title":"Physical Review","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRev.58.1098","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRev.58.1098/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,12,10]],"date-time":"2017-12-10T03:56:03Z","timestamp":1512878163000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRev.58.1098"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1940,12,15]]},"references-count":18,"journal-issue":{"issue":"12","published-print":{"date-parts":[[1940,12]]}},"URL":"http://dx.doi.org/10.1103/PhysRev.58.1098","relation":{},"ISSN":["0031-899X"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. 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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. 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Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state of H1 at inverse temperature &#x03B2;&#x2265;0 starting from a purification of the thermal state of H0. The complexity of the quantum algorithm, given by the number of uses of certain unitaries, is O&#x007E;(e&#x03B2;(&#x0394;A&#x2212;wl)/2), where &#x0394;A is the free-energy difference between H1 and H0, and wl is a work cutoff that depends on the properties of the work distribution and the approximation error &#x03F5;&#x003E;0. If the non-equilibrium process is trivial, this complexity is exponential in &#x03B2;&#x2016;V&#x2016;, where &#x2016;V&#x2016; is the spectral norm of V. This represents a significant improvement of prior quantum algorithms that have complexity exponential in &#x03B2;&#x2016;H1&#x2016; in the regime where &#x2016;V&#x2016;&#x226A;&#x2016;H1&#x2016;. The dependence of the complexity in &#x03F5; varies according to the structure of the quantum systems. It can be exponential in 1/&#x03F5; in general, but we show it to be sublinear in 1/&#x03F5; if H0 and H1 commute, or polynomial in 1/&#x03F5; if H0 and H1 are local spin systems. The possibility of applying a unitary that drives the system out of equilibrium allows one to increase the value of wl 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.","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":1731787379103},"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":"We present the problem of approximating the time-evolution operatore−iH^tto errorϵ, where the HamiltonianH^=(⟨G|⊗I^)U^(|G⟩⊗I^)is the projection of a unitary oracleU^onto the state|G⟩created by another unitary oracle. Our algorithm solves this with a query complexityO(t+log⁡(1/ϵ))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 ared-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 whereH^is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed anyH^in an invariantSU(2)subspace. A large class of operator functions ofH^can then be computed with optimal query complexity, of whiche−iH^tis a special case.","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":1731787380058},"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":1731787381057},"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":"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 Z4(1) 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.","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":1731787382154},"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":1731787383084},"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":"\n Subsystem codes are a generalization of noiseless subsystems, decoherence-free subspaces and stabilizer codes. We generalize the quantum Singleton bound to\n \n \n \n \n q\n \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 \n \n \n \n q\n \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 ","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&M University, College StationTexas, TX 77843, USA"}]},{"given":"Pradeep Kiran","family":"Sarvepalli","sequence":"additional","affiliation":[{"name":"Texas A&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":1731787384136},"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":1731787385058},"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":"AbstractTopological 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.","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":1731787386071},"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":"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 &#x2248;0.2&#x0025; to &#x2248;0.4&#x0025;, and also that they improve the teraquop footprint at a 0.1&#x0025; physical gate error rate from &#x2248;6000 qubits to &#x2248;3000 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 &#x2248;0.03&#x0025; (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 &#x2248;0.8&#x0025;, teraquop footprint at 0.1&#x0025; of &#x2248;1000).","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":1731787387063},"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":1731787388063},"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":"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.","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":1731787389065},"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":"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 O(d3) and mean runtime subquadratic in the surface code distance d. 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.","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":1731787390157},"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 inD=3and 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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The graphical formulation is based on the diagrammatic tools of the ZX-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 \n \n \n [\n n\n ,\n k\n ,\n 3\n ]\n \n \n codes into a \n \n \n [\n [\n 2\n n\n −\n k\n +\n 2\n ,\n k\n ,\n 3\n ]\n ]\n \n \n 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. 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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":1731787394150},"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":"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. 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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). 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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":"2019-04-12","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2019-07-18","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2019-07-30","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/ab3372","_hash":"f17650087a535f5057117518a4e946afabf6d178c38e8fd0bc36a0123fda6f67"},"expire":1732264933986},"doi:10.1103/PhysRevA.81.032301":{"value":{"indexed":{"date-parts":[[2023,10,12]],"date-time":"2023-10-12T16:21:13Z","timestamp":1697127673857},"reference-count":31,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2010,3,3]],"date-time":"2010-03-03T00:00:00Z","timestamp":1267574400000},"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.81.032301","type":"journal-article","created":{"date-parts":[[2010,3,3]],"date-time":"2010-03-03T21:03:43Z","timestamp":1267650223000},"source":"Crossref","is-referenced-by-count":72,"title":"Topological subsystem codes","prefix":"10.1103","volume":"81","author":[{"given":"H.","family":"Bombin","sequence":"first","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2010,3,3]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.81.032301","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.81.032301/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,19]],"date-time":"2017-06-19T05:46:49Z","timestamp":1497851209000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.81.032301"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,3,3]]},"references-count":31,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2010,3]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.81.032301","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. 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A: Math. Theor. 48, 215302 (2015)], with a specific implementation of the boundary. It utilizes weight-six (XYZXYZ) plaquette stabilizers and weight-two (XX) link stabilizers on a planar hexagonal grid composed of 2d2 qubits for code distance d, 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 X, Y, or Z 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 d2 only for pure Y noise, here the code distance is 2d2 for both pure Z and pure Y noise. Thresholds for noise with finite Z 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.","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":3,"title":"The XYZ2 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":"76381d0fcf9ef68ce93daabb1d3d6c55d43a4f1d0c8ba037dde694f90d067de1"},"expire":1732264953993},"doi:10.1088/1751-8121/ac7a75":{"value":{"indexed":{"date-parts":[[2023,5,1]],"date-time":"2023-05-01T14:08:55Z","timestamp":1682950135412},"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":"Abstract\n 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%.","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":2,"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":"9ed870c4e6560b49078c9adb8a3133aad84eeafa9577ba7ef85c4ccf4008149e"},"expire":1732264955069},"doi:10.1109/TIT.2016.2555700":{"value":{"indexed":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T05:01:06Z","timestamp":1698987666321},"reference-count":32,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","license":[{"start":{"date-parts":[[2016,6,1]],"date-time":"2016-06-01T00:00:00Z","timestamp":1464739200000},"content-version":"vor","delay-in-days":0,"URL":"https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html"}],"funder":[{"name":"EU Program QALGO"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2016,6]]},"DOI":"10.1109/tit.2016.2555700","type":"journal-article","created":{"date-parts":[[2016,4,21]],"date-time":"2016-04-21T18:18:34Z","timestamp":1461262714000},"page":"3731-3744","source":"Crossref","is-referenced-by-count":55,"title":"Constructions and Noise Threshold of Hyperbolic Surface Codes","prefix":"10.1109","volume":"62","author":[{"given":"Nikolas P.","family":"Breuckmann","sequence":"first","affiliation":[]},{"given":"Barbara M.","family":"Terhal","sequence":"additional","affiliation":[]}],"member":"263","reference":[],"container-title":"IEEE Transactions on Information Theory","original-title":[],"link":[{"URL":"http://xplorestaging.ieee.org/ielx7/18/7473802/07456305.pdf?arnumber=7456305","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,12]],"date-time":"2022-01-12T16:44:00Z","timestamp":1642005840000},"score":1,"resource":{"primary":{"URL":"http://ieeexplore.ieee.org/document/7456305/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,6]]},"references-count":32,"journal-issue":{"issue":"6"},"URL":"http://dx.doi.org/10.1109/TIT.2016.2555700","relation":{},"ISSN":["0018-9448","1557-9654"],"subject":["Library and Information Sciences","Computer Science Applications","Information Systems"],"container-title-short":"IEEE Trans. Inform. Theory","published":{"date-parts":[[2016,6]]},"id":"doi:10.1109/TIT.2016.2555700","_hash":"9c10d1c6308b77ac7d9f093112e2ed36b67c7819e015a97c65f49f19ecb7dd95"},"expire":1732264955989},"doi:10.1038/s41534-020-00330-w":{"value":{"indexed":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T20:09:24Z","timestamp":1696622964302},"reference-count":55,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,12,14]],"date-time":"2020-12-14T00:00:00Z","timestamp":1607904000000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2020,12,14]],"date-time":"2020-12-14T00:00:00Z","timestamp":1607904000000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"name":"European Research Council, under the Engineering Quantum Error Correction Project. 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},"abstract":"AbstractLeakage 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.","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":24,"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":"https://www.nature.com/articles/s41534-020-00330-w.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-020-00330-w","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-020-00330-w.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,6]],"date-time":"2022-12-06T02:55:14Z","timestamp":1670295314000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-020-00330-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,14]]},"references-count":55,"journal-issue":{"issue":"1","published-online":{"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":[[2020,12,14]]},"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":"be086ef15f13555ee11f1e8e62e15ca0bd6c37529d0faa5237224b347400f37d"},"expire":1732264957220},"doi:10.1038/s41534-017-0039-x":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T18:21:08Z","timestamp":1700590868910},"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":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2017,9,25]],"date-time":"2017-09-25T00:00:00Z","timestamp":1506297600000},"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":"AbstractWe present a density-matrix simulation of the quantum memory and computing performance of the distance-3 logical qubit Surface-17, following a recently proposed quantum circuit and using experimental error parameters for transmon qubits in a planar circuit QED architecture. We use this simulation to optimize components of the QEC scheme (e.g., trading off stabilizer measurement infidelity for reduced cycle time) and to investigate the benefits of feedback harnessing the fundamental asymmetry of relaxation-dominated error in the constituent transmons. A lower-order approximate calculation extends these predictions to the distance-5 Surface-49. These results clearly indicate error rates below the fault-tolerance threshold of the surface code, and the potential for Surface-17 to perform beyond the break-even point of quantum memory. However, Surface-49 is required to surpass the break-even point of computation at state-of-the-art qubit relaxation times and readout speeds.","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":53,"title":"Density-matrix simulation of small surface codes under current and projected experimental noise","prefix":"10.1038","volume":"3","author":[{"given":"T. E.","family":"O’Brien","sequence":"first","affiliation":[]},{"given":"B.","family":"Tarasinski","sequence":"additional","affiliation":[]},{"given":"L.","family":"DiCarlo","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,9,25]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41534-017-0039-x.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-017-0039-x","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-017-0039-x.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,23]],"date-time":"2022-12-23T01:46:21Z","timestamp":1671759981000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-017-0039-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,9,25]]},"references-count":38,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,12]]}},"alternative-id":["39"],"URL":"http://dx.doi.org/10.1038/s41534-017-0039-x","relation":{},"ISSN":["2056-6387"],"subject":["Computational 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Rev. Lett.","published":{"date-parts":[[2007,5,11]]},"article-number":"190504","id":"doi:10.1103/PhysRevLett.98.190504","_hash":"81fdfd4c80a9b971aeea9c521875ab8b4f0d0263b234373d72fc2d7503e56f45"},"expire":1732264965006},"doi:10.22331/q-2021-07-06-497":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T18:20:19Z","timestamp":1701109219079},"reference-count":30,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2021,7,6]],"date-time":"2021-07-06T00:00:00Z","timestamp":1625529600000},"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":"This paper presents “Stim\", a fast simulator for quantum stabilizer circuits. The paper explains how Stim works and compares it to existing tools. With no foreknowledge, Stim can analyze a distance 100 surface code circuit (20 thousand qubits, 8 million gates, 1 million measurements) in 15 seconds and then begin sampling full circuit shots at a rate of 1 kHz. Stim uses a stabilizer tableau representation, similar to Aaronson and Gottesman's CHP simulator, but with three main improvements. First, Stim improves the asymptotic complexity of deterministic measurement from quadratic to linear by tracking the inverse of the circuit's stabilizer tableau. Second, Stim improves the constant factors of the algorithm by using a cache-friendly data layout and 256 bit wide SIMD instructions. Third, Stim only uses expensive stabilizer tableau simulation to create an initial reference sample. Further samples are collected in bulk by using that sample as a reference for batches of Pauli frames propagating through the circuit.","DOI":"10.22331/q-2021-07-06-497","type":"journal-article","created":{"date-parts":[[2021,7,6]],"date-time":"2021-07-06T09:26:56Z","timestamp":1625563616000},"page":"497","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":44,"title":"Stim: a fast stabilizer circuit simulator","prefix":"10.22331","volume":"5","author":[{"given":"Craig","family":"Gidney","sequence":"first","affiliation":[{"name":"Google Inc., Santa Barbara, California 93117, USA"}]}],"member":"9598","published-online":{"date-parts":[[2021,7,6]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2021-07-06-497/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2021,7,6]],"date-time":"2021-07-06T09:27:37Z","timestamp":1625563657000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2021-07-06-497/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,6]]},"references-count":30,"URL":"http://dx.doi.org/10.22331/q-2021-07-06-497","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2021,7,6]]},"article-number":"497","id":"doi:10.22331/q-2021-07-06-497","_hash":"c86ef9f8cbd68ff86d56a19c2724da369c3ef91e320cbb0d16aaa5172f8bd2f2"},"expire":1732264966063},"doi:10.22331/q-2020-10-28-352":{"value":{"indexed":{"date-parts":[[2023,11,18]],"date-time":"2023-11-18T10:37:25Z","timestamp":1700303845266},"reference-count":20,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2020,10,28]],"date-time":"2020-10-28T00:00:00Z","timestamp":1603843200000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"The surface code is a prominent topological error-correcting code exhibiting high fault-tolerance accuracy thresholds. 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 only 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.","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":18,"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, 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The error suppression achieved by the surface code is usually estimated by simulating toy noise models describing random Pauli errors. However, Pauli noise models fail to capture coherent processes such as systematic unitary errors caused by imperfect control pulses. Here we report the first large-scale simulation of quantum error correction protocols based on the surface code in the presence of coherent noise. We observe that the standard Pauli approximation provides an accurate estimate of the error threshold but underestimates the logical error rate in the sub-threshold regime. We find that for large code size the logical-level noise is well approximated by random Pauli errors even though the physical-level noise is coherent. Our work demonstrates that coherent effects do not significantly change the error correcting threshold of surface codes. This gives more confidence in the viability of the fault-tolerance architecture pursued by several experimental groups.","DOI":"10.1038/s41534-018-0106-y","type":"journal-article","created":{"date-parts":[[2018,10,25]],"date-time":"2018-10-25T10:09:29Z","timestamp":1540462169000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":54,"title":"Correcting coherent errors with surface codes","prefix":"10.1038","volume":"4","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Matthias","family":"Englbrecht","sequence":"additional","affiliation":[]},{"given":"Robert","family":"König","sequence":"additional","affiliation":[]},{"given":"Nolan","family":"Peard","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,10,31]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41534-018-0106-y.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-018-0106-y","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-018-0106-y.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T23:02:50Z","timestamp":1671577370000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-018-0106-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,31]]},"references-count":46,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,12]]}},"alternative-id":["106"],"URL":"http://dx.doi.org/10.1038/s41534-018-0106-y","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":[[2018,10,31]]},"assertion":[{"value":"9 July 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 September 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 October 2018","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 October 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":"55","id":"doi:10.1038/s41534-018-0106-y","_hash":"648cb68eaa264f22db74a2003fcfb83af257233d98d1d4ba8875148f4239e228"},"expire":1732264972154},"doi:10.1109/ISIT.2012.6284206":{"value":{"indexed":{"date-parts":[[2023,10,24]],"date-time":"2023-10-24T17:11:52Z","timestamp":1698167512934},"reference-count":24,"publisher":"IEEE","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2012,7]]},"DOI":"10.1109/isit.2012.6284206","type":"proceedings-article","created":{"date-parts":[[2012,8,30]],"date-time":"2012-08-30T20:57:57Z","timestamp":1346360277000},"source":"Crossref","is-referenced-by-count":20,"title":"Improved quantum hypergraph-product LDPC codes","prefix":"10.1109","author":[{"given":"Alexey A.","family":"Kovalev","sequence":"first","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"member":"263","reference":[],"event":"2012 IEEE International Symposium on Information Theory - 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However, measuring this entanglement in real materials is extremely tricky. Now, two groups take a different approach and turn to synthetic systems to engineer the topological order of the so-called toric code type (see the Perspective by Bartlett). Satzinger\n et al\n . used a quantum processor to study the ground state and excitations of the toric code. 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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 generation3–5. We use this architecture to realize programmable generation of entangled graph states, such as cluster states and a seven-qubit Steane code state6,7. Furthermore, we shuttle entangled ancilla arrays to realize a surface code state with thirteen data and six ancillary qubits8and a toric code state on a torus with sixteen data and eight ancillary qubits9. Finally, we use this architecture to realize a hybrid analogue–digital evolution2and use it for measuring entanglement entropy in quantum simulations10–12, experimentally observing non-monotonic entanglement dynamics associated with quantum many-body scars13,14. Realizing a long-standing goal, these results provide a route towards scalable quantum processing and enable applications ranging from simulation to metrology.","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":153,"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":[[2023,2,2]],"date-time":"2023-02-02T19:35:27Z","timestamp":1675366527000},"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":"6ed5f0b3bafd70d97cff6e1faa51b80cf0df04877b516698240b09ab75f5a4d9"},"expire":1732264976092},"doi:10.1103/PhysRevLett.102.200501":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T22:37:40Z","timestamp":1700606260025},"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":91,"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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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) CCZ&#x00AF; 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 CCZ should be applied. We use these slices to simulate the performance of a simple JIT decoder against stochastic X and measurement errors and find evidence for a threshold pc&#x223C;0.1&#x0025; in all three codes. 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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":"2017-01-12","name":"date_received","label":"Date received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2017-05-02","name":"date_accepted","label":"Date accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2017-06-06","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/aa7099","_hash":"303f18ef4e958049c28364a93adc9839f427e5a02eeb9f13bdaa09f5dfa772ab"},"expire":1732265000007},"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":1732265000729},"doi:10.1103/PhysRevA.92.032309":{"value":{"indexed":{"date-parts":[[2023,11,16]],"date-time":"2023-11-16T17:21:07Z","timestamp":1700155267064},"reference-count":33,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2015,9,8]],"date-time":"2015-09-08T00:00:00Z","timestamp":1441670400000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"}],"funder":[{"DOI":"10.13039/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP/G037043/1","EP/K022512/1"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.92.032309","type":"journal-article","created":{"date-parts":[[2015,9,8]],"date-time":"2015-09-08T17:08:24Z","timestamp":1441732104000},"source":"Crossref","is-referenced-by-count":29,"title":"Fast fault-tolerant decoder for qubit and qudit surface codes","prefix":"10.1103","volume":"92","author":[{"given":"Fern H. 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Rev. A","published":{"date-parts":[[2015,9,8]]},"article-number":"032309","id":"doi:10.1103/PhysRevA.92.032309","_hash":"b6f5922e636b9767fe768f5e9b64e1e6b33ae894bc92f0b4fcf971902581e283"},"expire":1732265002008},"doi:10.22331/q-2021-12-02-595":{"value":{"indexed":{"date-parts":[[2023,11,14]],"date-time":"2023-11-14T14:04:20Z","timestamp":1699970660419},"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":"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 O(nα(n)), where n is the number of physical qubits and α is the inverse of Ackermann's function, which is very slowly growing. For all practical purposes, α(n)≤3. We prove that our algorithm performs optimally for errors of weight up to (d−1)/2 and for loss of up to d−1 qubits, where d is the minimum distance of the code. Numerically, we obtain a threshold of 9.9% for the 2d-toric code with perfect syndrome measurements and 2.6% with faulty measurements.","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":42,"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":"ee85d8a611176be409984b86e4104d0571ece7695f73c537742bd5042e64e76e"},"expire":1732265003034},"doi:10.4153/CJM-1965-045-4":{"value":{"indexed":{"date-parts":[[2023,11,26]],"date-time":"2023-11-26T02:13:25Z","timestamp":1700964805942},"reference-count":9,"publisher":"Canadian Mathematical Society","license":[{"start":{"date-parts":[[2018,11,20]],"date-time":"2018-11-20T00:00:00Z","timestamp":1542672000000},"content-version":"unspecified","delay-in-days":19681,"URL":"https://www.cambridge.org/core/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[1965]]},"abstract":"A graph G for purposes here is a finite set of elements called vertices and a finite set of elements called edges such that each edge meets exactly two vertices, called the end-points of the edge. An edge is said to join its end-points.A matching in G 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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The goal of this work is to demonstrate that this is not true. By writing distillation circuits in a form that separates qubits that are capable of error detection from those that are not, most logical qubits used for distillation can be encoded at a very low code distance. This significantly reduces the space-time cost of distillation, as well as the number of qubits. In extreme cases, it can cost less to distill a magic state than to perform a logical Clifford gate on full-distance logical qubits.","DOI":"10.22331/q-2019-12-02-205","type":"journal-article","created":{"date-parts":[[2019,12,2]],"date-time":"2019-12-02T13:45:29Z","timestamp":1575294329000},"page":"205","source":"Crossref","is-referenced-by-count":62,"title":"Magic State Distillation: Not as Costly as You Think","prefix":"10.22331","volume":"3","author":[{"given":"Daniel","family":"Litinski","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2019,12,2]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2019-12-02-205/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2021,2,9]],"date-time":"2021-02-09T21:59:50Z","timestamp":1612907990000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2019-12-02-205/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,2]]},"references-count":42,"URL":"http://dx.doi.org/10.22331/q-2019-12-02-205","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2019,12,2]]},"article-number":"205","id":"doi:10.22331/q-2019-12-02-205","_hash":"9f9f37051e1870f2a108ee7941255614370122bd0b55819a978ef1bcc60f559d"},"expire":1732265006899},"doi:10.1103/PhysRevResearch.4.023090":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T18:21:48Z","timestamp":1701109308806},"reference-count":42,"publisher":"American Physical Society (APS)","issue":"2","license":[{"start":{"date-parts":[[2022,5,2]],"date-time":"2022-05-02T00:00:00Z","timestamp":1651449600000},"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/physrevresearch.4.023090","type":"journal-article","created":{"date-parts":[[2022,5,2]],"date-time":"2022-05-02T14:03:25Z","timestamp":1651500205000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":7,"title":"Circuit-level protocol and analysis for twist-based lattice surgery","prefix":"10.1103","volume":"4","author":[{"ORCID":"http://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"given":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2022,5,2]]},"reference":[],"container-title":"Physical Review Research","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevResearch.4.023090","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevResearch.4.023090/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,5,2]],"date-time":"2022-05-02T14:03:27Z","timestamp":1651500207000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevResearch.4.023090"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,2]]},"references-count":42,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,5]]}},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.4.023090","relation":{},"ISSN":["2643-1564"],"subject":["General Engineering"],"container-title-short":"Phys. 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Research","published":{"date-parts":[[2022,5,2]]},"article-number":"023090","id":"doi:10.1103/PhysRevResearch.4.023090","_hash":"4c88f1c0e36e1fcbcc64f5bca7e61283a62359dbddb2b0630e6fd89b556585b2"},"expire":1732265007905},"doi:10.1103/PRXQuantum.3.010331":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T18:20:03Z","timestamp":1701109203758},"reference-count":66,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[2022,2,25]],"date-time":"2022-02-25T00:00:00Z","timestamp":1645747200000},"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.3.010331","type":"journal-article","created":{"date-parts":[[2022,3,3]],"date-time":"2022-03-03T14:27:34Z","timestamp":1646317654000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":19,"title":"Universal Quantum Computing with Twist-Free and Temporally Encoded Lattice Surgery","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":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2022,2,25]]},"reference":[],"container-title":"PRX Quantum","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PRXQuantum.3.010331","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PRXQuantum.3.010331/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,3,3]],"date-time":"2022-03-03T14:28:24Z","timestamp":1646317704000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PRXQuantum.3.010331"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,25]]},"references-count":66,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,2]]}},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.010331","relation":{},"ISSN":["2691-3399"],"subject":["General Medicine"],"container-title-short":"PRX Quantum","published":{"date-parts":[[2022,2,25]]},"article-number":"010331","id":"doi:10.1103/PRXQuantum.3.010331","_hash":"b5faf37a8acad61802e622f34e367002d570104020ce38caa15a8755eb9f386e"},"expire":1732265009030},"doi:10.22331/q-2019-03-05-128":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T18:21:50Z","timestamp":1701109310795},"reference-count":65,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2019,3,5]],"date-time":"2019-03-05T00:00:00Z","timestamp":1551744000000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"Given a quantum gate circuit, how does one execute it in a fault-tolerant architecture with as little overhead as possible? 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. Therefore, no knowledge of quantum error correction is necessary to understand the schemes in this paper, but only the concepts of qubits and measurements.","DOI":"10.22331/q-2019-03-05-128","type":"journal-article","created":{"date-parts":[[2019,3,5]],"date-time":"2019-03-05T19:50:11Z","timestamp":1551815411000},"page":"128","source":"Crossref","is-referenced-by-count":118,"title":"A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery","prefix":"10.22331","volume":"3","author":[{"given":"Daniel","family":"Litinski","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2019,3,5]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2019-03-05-128/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,9,13]],"date-time":"2022-09-13T01:04:56Z","timestamp":1663031096000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2019-03-05-128/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,5]]},"references-count":65,"URL":"http://dx.doi.org/10.22331/q-2019-03-05-128","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2019,3,5]]},"article-number":"128","id":"doi:10.22331/q-2019-03-05-128","_hash":"faa3de0304abe494902c663c22b7b85b57e117ed151839d3916d52d2ce371317"},"expire":1732265010010},"doi:10.22331/q-2018-05-04-62":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T18:19:26Z","timestamp":1701109166183},"reference-count":27,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2018,5,4]],"date-time":"2018-05-04T00:00:00Z","timestamp":1525392000000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"We present a planar surface-code-based scheme for fault-tolerant quantum computation which eliminates the time overhead of single-qubit Clifford gates, and implements long-range multi-target CNOT gates with a time overhead that scales only logarithmically with the control-target separation. This is done by replacing hardware operations for single-qubit Clifford gates with a classical tracking protocol. Inter-qubit communication is added via a modified lattice surgery protocol that employs twist defects of the surface code. The long-range multi-target CNOT gates facilitate magic state distillation, which renders our scheme fault-tolerant and universal.","DOI":"10.22331/q-2018-05-04-62","type":"journal-article","created":{"date-parts":[[2018,5,4]],"date-time":"2018-05-04T10:19:18Z","timestamp":1525429158000},"page":"62","source":"Crossref","is-referenced-by-count":38,"title":"Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes","prefix":"10.22331","volume":"2","author":[{"given":"Daniel","family":"Litinski","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"}]},{"given":"Felix von","family":"Oppen","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2018,5,4]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","deposited":{"date-parts":[[2019,10,17]],"date-time":"2019-10-17T13:49:48Z","timestamp":1571320188000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2018-05-04-62/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,4]]},"references-count":27,"URL":"http://dx.doi.org/10.22331/q-2018-05-04-62","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2018,5,4]]},"article-number":"62","id":"doi:10.22331/q-2018-05-04-62","_hash":"494612d3c2db955ad0cfe1a7256d49ab865f1e288f437d9c0c9a1cf598e097b9"},"expire":1732265010904},"doi:10.1088/1367-2630/14/12/123011":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T18:19:20Z","timestamp":1701109160586},"reference-count":32,"publisher":"IOP Publishing","issue":"12","license":[{"start":{"date-parts":[[2012,12,7]],"date-time":"2012-12-07T00:00:00Z","timestamp":1354838400000},"content-version":"vor","delay-in-days":6,"URL":"http://creativecommons.org/licenses/by-nc-sa/3.0/"},{"start":{"date-parts":[[2012,12,7]],"date-time":"2012-12-07T00:00:00Z","timestamp":1354838400000},"content-version":"tdm","delay-in-days":6,"URL":"https://iopscience.iop.org/info/page/text-and-data-mining"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2012,12,1]]},"abstract":"Abstract\n In recent years, surface codes have become a leading method for quantum error correction in theoretical large-scale computational and communications architecture designs. Their comparatively high fault-tolerant thresholds and their natural two-dimensional nearest-neighbour (2DNN) structure make them an obvious choice for large scale designs in experimentally realistic systems. While fundamentally based on the toric code of Kitaev, there are many variants, two of which are the planar- and defect-based codes. Planar codes require fewer qubits to implement (for the same strength of error correction), but are restricted to encoding a single qubit of information. Interactions between encoded qubits are achieved via transversal operations, thus destroying the inherent 2DNN nature of the code. In this paper we introduce a new technique enabling the coupling of two planar codes without transversal operations, maintaining the 2DNN of the encoded computer. Our lattice surgery technique comprises splitting and merging planar code surfaces, and enables us to perform universal quantum computation (including magic state injection) while removing the need for braided logic in a strictly 2DNN design, and hence reduces the overall qubit resources for logic operations. Those resources are further reduced by the use of a rotated lattice for the planar encoding. We show how lattice surgery allows us to distribute encoded GHZ states in a more direct (and overhead friendly) manner, and how a demonstration of an encoded CNOT between two distance-3 logical states is possible with 53 physical qubits, half of that required in any other known construction in 2D.","DOI":"10.1088/1367-2630/14/12/123011","type":"journal-article","created":{"date-parts":[[2012,12,7]],"date-time":"2012-12-07T16:14:11Z","timestamp":1354896851000},"page":"123011","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":243,"title":"Surface code quantum computing by lattice surgery","prefix":"10.1088","volume":"14","author":[{"given":"Dominic","family":"Horsman","sequence":"first","affiliation":[]},{"given":"Austin G","family":"Fowler","sequence":"additional","affiliation":[]},{"given":"Simon","family":"Devitt","sequence":"additional","affiliation":[]},{"given":"Rodney Van","family":"Meter","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2012,12,7]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/14/12/123011","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/14/12/123011/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/14/12/123011/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/14/12/123011/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,30]],"date-time":"2023-05-30T12:53:56Z","timestamp":1685451236000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/14/12/123011"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,12,1]]},"references-count":32,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2012,12,7]]},"published-print":{"date-parts":[[2012,12,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/14/12/123011","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. Phys.","published":{"date-parts":[[2012,12,1]]},"assertion":[{"value":"Surface code quantum computing by lattice surgery","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":"© IOP Publishing and Deutsche Physikalische Gesellschaft","name":"copyright_information","label":"Copyright Information"},{"value":"2012-05-01","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2012-12-07","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/14/12/123011","_hash":"1b16f8c146133a4dcbb19b7226528cbc2051c65e977dfbd887e1b5639fa03d1b"},"expire":1732265011926},"doi:10.1038/srep08975":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T18:22:15Z","timestamp":1700590935831},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2015,3,10]],"date-time":"2015-03-10T00:00:00Z","timestamp":1425945600000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2015,3,10]],"date-time":"2015-03-10T00:00:00Z","timestamp":1425945600000},"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":"AbstractWe present a scheme for encoding and decoding an unknown state for CSS codes, based on syndrome measurements. We illustrate our method by means of Kitaev toric code, defected-lattice code, topological subsystem code and 3D Haah code. The protocol is local whenever in a given code the crossings between the logical operators consist of next neighbour pairs, which holds for the above codes. For subsystem code we also present scheme in a noisy case, where we allow for bit and phase-flip errors on qubits as well as state preparation and syndrome measurement errors. Similar scheme can be built for two other codes. We show that the fidelity of the protected qubit in the noisy scenario in a large code size limit is of \"Equation missing\", where p is a probability of error on a single qubit per time step. 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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 L, however the most efficient known method for encoding an unknown state, introduced by Dennis et al. (2002), has O(L2) time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly 2L time steps to encode an unknown state in a distance L planar code. We further show how an O(L) complexity local unitary encoder for the toric code can be found by enforcing locality in the O(log⁡L)-depth non-local renormalisation encoder. We relate these techniques by providing an O(L) 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.","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":1732265019916},"doi:10.1088/1367-2630/9/6/199":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T15:59:35Z","timestamp":1700582375607},"reference-count":38,"publisher":"IOP Publishing","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1088/1367-2630/9/6/199","type":"journal-article","created":{"date-parts":[[2007,6,30]],"date-time":"2007-06-30T03:14:43Z","timestamp":1183173283000},"page":"199-199","source":"Crossref","is-referenced-by-count":399,"title":"Topological fault-tolerance in cluster state quantum computation","prefix":"10.1088","volume":"9","author":[{"given":"R","family":"Raussendorf","sequence":"first","affiliation":[]},{"given":"J","family":"Harrington","sequence":"additional","affiliation":[]},{"given":"K","family":"Goyal","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2007,6,29]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"deposited":{"date-parts":[[2021,8,16]],"date-time":"2021-08-16T08:53:05Z","timestamp":1629103985000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/9/6/199"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,6,29]]},"references-count":38,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2007,6,1]]}},"alternative-id":["S1367-2630(07)44251-3"],"URL":"http://dx.doi.org/10.1088/1367-2630/9/6/199","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. Phys.","published":{"date-parts":[[2007,6,29]]},"id":"doi:10.1088/1367-2630/9/6/199","_hash":"d1079095e48f02c84343ce92e0b57778aabf76a10e060c8af6cb7550e52dcdf6"},"expire":1732265020922},"doi:10.1088/1367-2630/ab8e5c":{"value":{"indexed":{"date-parts":[[2023,11,16]],"date-time":"2023-11-16T17:20:21Z","timestamp":1700155221166},"reference-count":37,"publisher":"IOP Publishing","issue":"7","license":[{"start":{"date-parts":[[2020,7,31]],"date-time":"2020-07-31T00:00:00Z","timestamp":1596153600000},"content-version":"vor","delay-in-days":30,"URL":"https://creativecommons.org/licenses/by/4.0/"},{"start":{"date-parts":[[2020,7,31]],"date-time":"2020-07-31T00:00:00Z","timestamp":1596153600000},"content-version":"tdm","delay-in-days":30,"URL":"https://iopscience.iop.org/info/page/text-and-data-mining"}],"funder":[{"name":"Army Research Office and Laboratory for Physical Sciences","award":["W911NF-18- 1-0103"]},{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["PHY-1733907"]}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2020,7,1]]},"abstract":"Abstract\n We study the effectiveness of quantum error correction against coherent noise. 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.","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":20,"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":"b60cad6da53062c6454c117c2333343cd38704261dc49448abeb8cbb2a239a56"},"expire":1732265021866},"doi:10.1007/978-1-4615-5923-8_19":{"value":{"indexed":{"date-parts":[[2023,8,26]],"date-time":"2023-08-26T19:39:20Z","timestamp":1693078760074},"publisher-location":"Boston, MA","reference-count":11,"publisher":"Springer US","isbn-type":[{"value":"9781461377160","type":"print"},{"value":"9781461559238","type":"electronic"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[1997]]},"DOI":"10.1007/978-1-4615-5923-8_19","type":"book-chapter","created":{"date-parts":[[2011,9,12]],"date-time":"2011-09-12T17:37:40Z","timestamp":1315849060000},"page":"181-188","source":"Crossref","is-referenced-by-count":47,"title":"Quantum Error Correction with Imperfect Gates","prefix":"10.1007","author":[{"given":"A. Yu.","family":"Kitaev","sequence":"first","affiliation":[]}],"member":"297","reference":[],"container-title":"Quantum Communication, Computing, and Measurement","original-title":[],"link":[{"URL":"http://link.springer.com/content/pdf/10.1007/978-1-4615-5923-8_19.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,5,1]],"date-time":"2021-05-01T11:23:21Z","timestamp":1619868201000},"score":1,"resource":{"primary":{"URL":"http://link.springer.com/10.1007/978-1-4615-5923-8_19"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997]]},"ISBN":["9781461377160","9781461559238"],"references-count":11,"URL":"http://dx.doi.org/10.1007/978-1-4615-5923-8_19","relation":{},"published":{"date-parts":[[1997]]},"id":"doi:10.1007/978-1-4615-5923-8_19","_hash":"bce5798b2291907a334b222ef51d0cd2bd76d6378ad048ebf97ef5fb82e467bd"},"expire":1732265022942},"doi:10.1070/RM1997v052n06ABEH002155":{"value":{"indexed":{"date-parts":[[2023,11,22]],"date-time":"2023-11-22T16:58:11Z","timestamp":1700672291749},"reference-count":0,"publisher":"Steklov Mathematical Institute","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[1997,12,31]]},"DOI":"10.1070/rm1997v052n06abeh002155","type":"journal-article","created":{"date-parts":[[2005,11,8]],"date-time":"2005-11-08T14:31:52Z","timestamp":1131460312000},"page":"1191-1249","source":"Crossref","is-referenced-by-count":678,"title":"Quantum computations: algorithms and error correction","prefix":"10.4213","volume":"52","author":[{"given":"A Yu","family":"Kitaev","sequence":"first","affiliation":[]}],"member":"2731","published-online":{"date-parts":[[2007,10,17]]},"container-title":"Russian Mathematical Surveys","original-title":[],"link":[{"URL":"http://stacks.iop.org/0036-0279/52/i=6/a=R02/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,30]],"date-time":"2023-05-30T04:45:07Z","timestamp":1685421907000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1070/RM1997v052n06ABEH002155"},"secondary":[{"URL":"https://www.mathnet.ru/eng/rm892"}]},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,12,31]]},"references-count":0,"journal-issue":{"issue":"6","published-print":{"date-parts":[[1997,12,31]]}},"URL":"http://dx.doi.org/10.1070/RM1997v052n06ABEH002155","relation":{},"ISSN":["0036-0279","1468-4829"],"subject":["General Mathematics"],"container-title-short":"Russ. 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Phys.","published":{"date-parts":[[2021,12,16]]},"assertion":[{"value":"31 March 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 October 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 December 2021","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/s41567-021-01423-9","_hash":"09f120e8e46779d0f23d6b0997baa06f955a3ccf899b46d706d93e5084282d4d"},"expire":1732265024852},"doi:10.1038/s41586-021-03588-y":{"value":{"indexed":{"date-parts":[[2023,11,22]],"date-time":"2023-11-22T13:19:12Z","timestamp":1700659152712},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"7867","license":[{"start":{"date-parts":[[2021,7,14]],"date-time":"2021-07-14T00:00:00Z","timestamp":1626220800000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2021,7,14]],"date-time":"2021-07-14T00:00:00Z","timestamp":1626220800000},"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":[[2021,7,15]]},"abstract":"AbstractRealizing the potential of quantum computing requires sufficiently low logical error rates1. Many applications call for error rates as low as 10−15 (refs. 2–9), but state-of-the-art quantum platforms typically have physical error rates near 10−3 (refs. 10–14). Quantum error correction15–17 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 device18,19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits.","DOI":"10.1038/s41586-021-03588-y","type":"journal-article","created":{"date-parts":[[2021,7,14]],"date-time":"2021-07-14T16:13:39Z","timestamp":1626279219000},"page":"383-387","update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":142,"title":"Exponential suppression of bit or phase errors with cyclic error correction","prefix":"10.1038","volume":"595","author":[{"name":"Google Quantum AI","sequence":"first","affiliation":[]},{"given":"Zijun","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Kevin J.","family":"Satzinger","sequence":"additional","affiliation":[]},{"given":"Juan","family":"Atalaya","sequence":"additional","affiliation":[]},{"given":"Alexander 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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.","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":32,"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":"81dfa871ee2a05092165f2d3edf3f334323f9d77f51060bcc4b54293af03063e"},"expire":1732265029863},"doi:10.1103/PhysRevLett.124.130501":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T17:21:52Z","timestamp":1701105712621},"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":58,"title":"Fault-Tolerant Thresholds for the Surface Code in Excess of \n5%\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. Lett.","published":{"date-parts":[[2020,3,30]]},"article-number":"130501","id":"doi:10.1103/PhysRevLett.124.130501","_hash":"47f2dd9c00b169b032029098a9189e12a0f6cc0fcbf3feeaa11cde83eae2314a"},"expire":1732265030943},"doi:10.1038/s41467-021-22274-1":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T18:19:32Z","timestamp":1701109172579},"reference-count":82,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,4,12]],"date-time":"2021-04-12T00:00:00Z","timestamp":1618185600000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2021,4,12]],"date-time":"2021-04-12T00:00:00Z","timestamp":1618185600000},"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":"AbstractPerforming large calculations with a quantum computer will likely require a fault-tolerant architecture based on quantum error-correcting codes. 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.","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":85,"title":"The XZZX surface code","prefix":"10.1038","volume":"12","author":[{"ORCID":"http://orcid.org/0000-0001-5518-7907","authenticated-orcid":false,"given":"J. Pablo","family":"Bonilla Ataides","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-3776-2864","authenticated-orcid":false,"given":"David K.","family":"Tuckett","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4387-670X","authenticated-orcid":false,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-8060-8109","authenticated-orcid":false,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,4,12]]},"reference":[],"container-title":"Nature 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X","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevX.9.041031","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevX.9.041031/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,4]],"date-time":"2022-10-04T13:02:05Z","timestamp":1664888525000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevX.9.041031"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,12]]},"references-count":28,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2019,11]]}},"URL":"http://dx.doi.org/10.1103/PhysRevX.9.041031","relation":{},"ISSN":["2160-3308"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. 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Homological codes from surfaces, i.e., surface codes, have also been suggested as a possible way to construct stable quantum memory and fault-tolerant computation. It has been conjectured that all homological codes have a square root bound on there distance and therefore cannot produce good codes. This claim has been disputed in dimension four using the geometric property of systolic freedom. 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Here, we consider an algorithm which maps the search for CWS codes to a problem of identifying maximum cliques in a graph. While solving this problem is in general very hard, we provide three structure theorems which reduce the search space, specifying certain admissible and optimal ((n,K,d)) additive codes. In particular, we find that the re does not exist any ((7,3,3)) CWS code though the linear programming bound does not rule it out. The complexity of the CWS-search algorithm is compared with the contrasting method introduced by Aggarwal and Calderbank [IEEE Trans. Inf. 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This 10th anniversary edition includes an introduction from the authors setting the work in context. This comprehensive textbook describes such remarkable effects as fast quantum algorithms, quantum teleportation, quantum cryptography and quantum error-correction. Quantum mechanics and computer science are introduced before moving on to describe what a quantum computer is, how it can be used to solve problems faster than 'classical' computers and its real-world implementation. It concludes with an in-depth treatment of quantum information. Containing a wealth of figures and exercises, this well-known textbook is ideal for courses on the subject, and will interest beginning graduate students and researchers in physics, computer science, mathematics, and electrical engineering.","DOI":"10.1017/cbo9780511976667","type":"monograph","created":{"date-parts":[[2012,6,18]],"date-time":"2012-06-18T17:58:14Z","timestamp":1340042294000},"source":"Crossref","is-referenced-by-count":3318,"title":"Quantum Computation and Quantum Information","prefix":"10.1017","author":[{"given":"Michael A.","family":"Nielsen","sequence":"first","affiliation":[]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2012,6,5]]},"container-title":[],"original-title":[],"deposited":{"date-parts":[[2022,6,11]],"date-time":"2022-06-11T21:26:40Z","timestamp":1654982800000},"score":1,"resource":{"primary":{"URL":"https://www.cambridge.org/core/product/identifier/9780511976667/type/book"}},"subtitle":["10th Anniversary Edition"],"short-title":[],"issued":{"date-parts":[[2012,6,5]]},"ISBN":["9781107002173","9780511976667"],"references-count":0,"URL":"http://dx.doi.org/10.1017/CBO9780511976667","relation":{},"published":{"date-parts":[[2012,6,5]]},"reference":[],"id":"doi:10.1017/CBO9780511976667","_hash":"483210dc82a22a3321ade6e915e5e56a1d558ded6618c15db02247e2442d02dc"},"expire":1732265082237},"doi:10.22331/q-2022-09-22-815":{"value":{"indexed":{"date-parts":[[2023,11,14]],"date-time":"2023-11-14T16:12:10Z","timestamp":1699978330537},"reference-count":26,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,9,22]],"date-time":"2022-09-22T00:00:00Z","timestamp":1663804800000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/501100000923","name":"Australian Research Council","doi-asserted-by":"crossref","award":["CE170100009"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"We propose an extension to the Pauli stabiliser formalism that includes fractional 2&#x03C0;/N rotations around the Z axis for some integer N. 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. 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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 &#x0398;(N2/3). 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. 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November 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 April 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 May 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"T.M., R.B. and T.F. are connected to Alpine Quantum Technologies, a commercially oriented quantum computing company.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"id":"doi:10.1038/s41586-022-04721-1","_hash":"4a07cbbf5f11f82adcc456294e8e6325ad6c506109eb807bfde686e97066c854"},"expire":1732265124258},"doi:10.1103/Physics.14.184":{"value":{"indexed":{"date-parts":[[2023,5,6]],"date-time":"2023-05-06T05:35:06Z","timestamp":1683351306979},"reference-count":2,"publisher":"American Physical Society 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Attempts to process and manipulate quantum states can destroy the encoded information. Nigg\n et al.\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 \n \n Science\n , this issue p.\n 302\n ","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":276,"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":"455d88658f40700ebcfb0f575d7b3a6b349e00d7fb88cb8e8963c92427037a1d"},"expire":1732265126387},"doi:10.1088/2058-9565/abc6f4":{"value":{"indexed":{"date-parts":[[2023,9,22]],"date-time":"2023-09-22T08:18:47Z","timestamp":1695370727007},"reference-count":20,"publisher":"IOP Publishing","issue":"1","license":[{"start":{"date-parts":[[2020,11,24]],"date-time":"2020-11-24T00:00:00Z","timestamp":1606176000000},"content-version":"vor","delay-in-days":0,"URL":"https://iopscience.iop.org/page/copyright"},{"start":{"date-parts":[[2020,11,24]],"date-time":"2020-11-24T00:00:00Z","timestamp":1606176000000},"content-version":"tdm","delay-in-days":0,"URL":"https://iopscience.iop.org/info/page/text-and-data-mining"}],"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":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2021,1,1]]},"abstract":"Abstract\n Steane’s seven-qubit quantum code is a natural choice for fault-tolerance experiments because it is small and just two extra qubits are enough to correct errors. 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.","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":21,"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. Technol.","published":{"date-parts":[[2020,11,24]]},"assertion":[{"value":"Fault-tolerant quantum error correction for Steane’s seven-qubit color code with few or no extra qubits","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":"© 2020 IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2020-08-10","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-11-02","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-11-24","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/2058-9565/abc6f4","_hash":"630df5e9326895fcf572f90683c593e429420bb9d614e780ebf4ed1af618d54b"},"expire":1732265127334},"doi:10.1103/PhysRevLett.121.050502":{"value":{"indexed":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T16:41:05Z","timestamp":1699893665332},"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":104,"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":"f6c2f6c4b157ce69649bb5b666a77be2f78489384f0e0a3024daaf99f92dc60c"},"expire":1732265128257},"doi:10.22331/q-2017-04-25-2":{"value":{"indexed":{"date-parts":[[2023,10,24]],"date-time":"2023-10-24T17:10:06Z","timestamp":1698167406500},"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":"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.","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":67,"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":[[2023,8,23]],"date-time":"2023-08-23T07:52:12Z","timestamp":1692777132000},"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 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Realizations"],"short-title":[],"issued":{"date-parts":[[2008,3,11]]},"ISBN":["9780429146510"],"references-count":0,"URL":"http://dx.doi.org/10.1201/9781420012293","relation":{},"published":{"date-parts":[[2008,3,11]]},"reference":[],"id":"doi:10.1201/9781420012293","_hash":"9f43ccb87bf8fc7e85c7e8bc853a7765ec5f03c847754805f9195ae4d6b6efb1"},"expire":1732265130383},"doi:10.1103/PRXQuantum.3.030319":{"value":{"indexed":{"date-parts":[[2023,10,12]],"date-time":"2023-10-12T10:30:24Z","timestamp":1697106624500},"reference-count":65,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2022,8,8]],"date-time":"2022-08-08T00:00:00Z","timestamp":1659916800000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"funder":[{"DOI":"10.13039/501100000023","name":"Government of Canada","doi-asserted-by":"publisher"},{"DOI":"10.13039/100011332","name":"Innovation, Science and Economic Development 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Quantum","published":{"date-parts":[[2022,8,8]]},"article-number":"030319","id":"doi:10.1103/PRXQuantum.3.030319","_hash":"fa3ea84b86257ee5526987ef7522be2429b19637b2a66b4168f754cc20f72897"},"expire":1732265131324},"doi:10.1038/s41586-022-04819-6":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T22:42:17Z","timestamp":1700606537161},"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":"AbstractSolid-state spin qubits is a promising platform for quantum computation and quantum networks1,2. Recent experiments have demonstrated high-quality control over multi-qubit systems3–8, elementary quantum algorithms8–11 and non-fault-tolerant error correction12–14. Large-scale systems will require using error-corrected logical qubits that are operated fault tolerantly, so that reliable computation becomes possible despite noisy operations15–18. Overcoming imperfections in this way remains an important outstanding challenge for quantum science15,19–27. 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 qubits28–30. 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.","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":52,"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":"20a5bb1e2780caed34392cd2b48063a80244d3f4831a231805ef58582afa8a90"},"expire":1732265132381},"doi:10.1093/nsr/nwab011":{"value":{"indexed":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T15:57:25Z","timestamp":1696607845718},"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":"Abstract\n 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.","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":19,"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 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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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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":"0f6548aeec52018d0ddc6061a5d7058bfd699981a0ea4b77a17ac97bb5fc7a2a"},"expire":1732265146865},"doi:10.22331/q-2017-10-03-31":{"value":{"indexed":{"date-parts":[[2023,8,10]],"date-time":"2023-08-10T02:27:14Z","timestamp":1691634434941},"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":"We present an infinite family of protocols to distill magic states for T-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 T-gate depth of the circuit scale linearly in the logarithm of the target error δ (up to log⁡log⁡1/δ). 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. 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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 \n \n\t\n\t \n\t 1\n\t \n\t\n\t×\n\t\n\t \n\t 2\n\t \n\t\n \n based on the Shor code state, the effectiveness of our loss-tolerant scheme and the loss tolerance of the encoded RGS are respectively verified. 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Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2014,2,2]],"date-time":"2014-02-02T00:00:00Z","timestamp":1391299200000},"content-version":"tdm","delay-in-days":0,"URL":"http://www.springer.com/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2014,3]]},"DOI":"10.1038/nnano.2014.2","type":"journal-article","created":{"date-parts":[[2014,2,1]],"date-time":"2014-02-01T04:54:19Z","timestamp":1391230459000},"page":"171-176","source":"Crossref","is-referenced-by-count":288,"title":"Universal control and error correction in multi-qubit spin registers in diamond","prefix":"10.1038","volume":"9","author":[{"given":"T. H.","family":"Taminiau","sequence":"first","affiliation":[]},{"given":"J.","family":"Cramer","sequence":"additional","affiliation":[]},{"given":"T.","family":"van der Sar","sequence":"additional","affiliation":[]},{"given":"V. V.","family":"Dobrovitski","sequence":"additional","affiliation":[]},{"given":"R.","family":"Hanson","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2014,2,2]]},"reference":[],"container-title":"Nature Nanotechnology","original-title":[],"language":"en","link":[{"URL":"http://www.nature.com/articles/nnano.2014.2.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/nnano.2014.2","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/nnano.2014.2.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T23:40:56Z","timestamp":1684453256000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/nnano.2014.2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,2,2]]},"references-count":32,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2014,3]]}},"alternative-id":["BFnnano20142"],"URL":"http://dx.doi.org/10.1038/nnano.2014.2","relation":{},"ISSN":["1748-3387","1748-3395"],"subject":["Electrical and Electronic Engineering","Condensed Matter Physics","General Materials Science","Biomedical Engineering","Atomic and Molecular Physics, and Optics","Bioengineering"],"container-title-short":"Nature Nanotech","published":{"date-parts":[[2014,2,2]]},"id":"doi:10.1038/nnano.2014.2","_hash":"77a09ca5fcf4eb79cf2b643b8d3977377b7c2c9684763cd7a7a74c157520a3d4"},"expire":1732265154855},"doi:10.1038/s42005-022-00875-6":{"value":{"indexed":{"date-parts":[[2023,11,20]],"date-time":"2023-11-20T16:00:10Z","timestamp":1700496010663},"reference-count":53,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T00:00:00Z","timestamp":1651017600000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T00:00:00Z","timestamp":1651017600000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/501100001691","name":"MEXT | Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["20H05661","20K2044120"]},{"DOI":"10.13039/501100002241","name":"MEXT | Japan Science and Technology Agency","doi-asserted-by":"publisher","award":["JPMJCR1773","JPMJMS2062"]},{"DOI":"10.13039/501100009105","name":"Ministry of Internal Affairs and Communications","doi-asserted-by":"publisher","award":["JPMI00316"]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"AbstractFault-tolerant quantum memory plays a key role in interfacing quantum computers with quantum networks to construct quantum computer networks. 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.","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":6,"title":"Quantum error correction of spin quantum memories in diamond under a zero magnetic field","prefix":"10.1038","volume":"5","author":[{"given":"Takaya","family":"Nakazato","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-4880-1654","authenticated-orcid":false,"given":"Raustin","family":"Reyes","sequence":"additional","affiliation":[]},{"given":"Nobuaki","family":"Imaike","sequence":"additional","affiliation":[]},{"given":"Kazuyasu","family":"Matsuda","sequence":"additional","affiliation":[]},{"given":"Kazuya","family":"Tsurumoto","sequence":"additional","affiliation":[]},{"given":"Yuhei","family":"Sekiguchi","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-3778-7236","authenticated-orcid":false,"given":"Hideo","family":"Kosaka","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,27]]},"reference":[],"container-title":"Communications 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Physics and Astronomy"],"container-title-short":"Commun Phys","published":{"date-parts":[[2022,4,27]]},"assertion":[{"value":"3 December 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 March 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 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":"102","id":"doi:10.1038/s42005-022-00875-6","_hash":"d3803a5b72b50f77296e99af358f73c5577c7f6be0fe22ef5997060f7f4724e7"},"expire":1732265155897},"doi:10.1038/nature12919":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T22:39:15Z","timestamp":1700606355262},"reference-count":39,"publisher":"Springer Science and Business Media 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register","prefix":"10.1038","volume":"506","author":[{"given":"G.","family":"Waldherr","sequence":"first","affiliation":[]},{"given":"Y.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"S.","family":"Zaiser","sequence":"additional","affiliation":[]},{"given":"M.","family":"Jamali","sequence":"additional","affiliation":[]},{"given":"T.","family":"Schulte-Herbrüggen","sequence":"additional","affiliation":[]},{"given":"H.","family":"Abe","sequence":"additional","affiliation":[]},{"given":"T.","family":"Ohshima","sequence":"additional","affiliation":[]},{"given":"J.","family":"Isoya","sequence":"additional","affiliation":[]},{"given":"J. F.","family":"Du","sequence":"additional","affiliation":[]},{"given":"P.","family":"Neumann","sequence":"additional","affiliation":[]},{"given":"J.","family":"Wrachtrup","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2014,2,12]]},"reference":[],"container-title":"Nature","original-title":[],"language":"en","link":[{"URL":"http://www.nature.com/articles/nature12919.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/nature12919","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/articles/nature12919.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T18:18:17Z","timestamp":1684433897000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/nature12919"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,2,12]]},"references-count":39,"journal-issue":{"issue":"7487","published-print":{"date-parts":[[2014,2,13]]}},"alternative-id":["BFnature12919"],"URL":"http://dx.doi.org/10.1038/nature12919","relation":{},"ISSN":["0028-0836","1476-4687"],"subject":["Multidisciplinary"],"container-title-short":"Nature","published":{"date-parts":[[2014,2,12]]},"assertion":[{"value":"2 September 2013","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 November 2013","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 February 2014","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 February 2014","order":4,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":5,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A new reference (31) has been added to the main-text reference list and all subsequent references have been renumbered.","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/nature12919","_hash":"77393642a755cc5d1d88604416f98c4e599f09677d4b5309bdad5ccc19bb866f"},"expire":1732265156890},"doi:10.1038/s41586-022-04986-6":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T22:40:08Z","timestamp":1700606408537},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"7924","license":[{"start":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T00:00:00Z","timestamp":1661299200000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T00:00:00Z","timestamp":1661299200000},"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":[[2022,8,25]]},"abstract":"AbstractFuture large-scale quantum computers will rely on quantum error correction (QEC) to protect the fragile quantum information during computation1,2. 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 computers3–5. Recent advances in silicon-based qubits have enabled the implementations of high-quality one-qubit and two-qubit systems6–8. However, the demonstration of QEC, which requires three or more coupled qubits1, and involves a three-qubit gate9–11 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.","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":30,"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":"0596baf0d34fa7b9497612ae8313e8c255e80c5b29fecbe0f0ab1fb5298acf2a"},"expire":1732265157881},"doi:10.1038/s41467-022-29906-0":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T19:02:11Z","timestamp":1700593331212},"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":"AbstractThe 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.","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":12,"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":"7a28eedb58e169835c97647dab4034f09121c927e8e739a1261b4e9ae7babf38"},"expire":1732265159060},"doi:10.1038/nature14270":{"value":{"indexed":{"date-parts":[[2023,11,20]],"date-time":"2023-11-20T15:29:08Z","timestamp":1700494148977},"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":647,"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 topological_codes module of Qiskit-Ignis, which is designed to provide the tools necessary to perform such tests. Specifically, we use the RepetitionCode and GraphDecoder 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 Rochester device is presented.","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":16,"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":"ae8e417086060b9fdf6a053d814c6336bd66d6003c2d460c56384b7c87a5d565"},"expire":1732265160815},"doi:10.1103/PhysRevA.97.052313":{"value":{"indexed":{"date-parts":[[2023,10,5]],"date-time":"2023-10-05T12:36:27Z","timestamp":1696509387569},"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":37,"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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Quantum error correction protects quantum states by encoding a logical quantum bit (qubit) in multiple physical qubits. To be compatible with universal fault-tolerant computations, it is essential that states remain encoded at all times and that errors are actively corrected. Here we demonstrate such active error correction on a continuously protected logical qubit using a diamond quantum processor. We encode the logical qubit in three long-lived nuclear spins, repeatedly detect phase errors by non-destructive measurements, and apply corrections by real-time feedback. The actively error-corrected qubit is robust against errors and encoded quantum superposition states are preserved beyond the natural dephasing time of the best physical qubit in the encoding. These results establish a powerful platform to investigate error correction under different types of noise and mark an important step towards fault-tolerant quantum information processing.","DOI":"10.1038/ncomms11526","type":"journal-article","created":{"date-parts":[[2016,5,5]],"date-time":"2016-05-05T11:03:40Z","timestamp":1462446220000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":158,"title":"Repeated quantum error correction on a continuously encoded qubit by real-time feedback","prefix":"10.1038","volume":"7","author":[{"given":"J.","family":"Cramer","sequence":"first","affiliation":[]},{"given":"N.","family":"Kalb","sequence":"additional","affiliation":[]},{"given":"M. A.","family":"Rol","sequence":"additional","affiliation":[]},{"given":"B.","family":"Hensen","sequence":"additional","affiliation":[]},{"given":"M. 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H.","family":"Taminiau","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2016,5,5]]},"reference":[],"container-title":"Nature Communications","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/ncomms11526.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/ncomms11526","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/ncomms11526.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,4]],"date-time":"2023-01-04T11:28:11Z","timestamp":1672831691000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/ncomms11526"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,5,5]]},"references-count":38,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,9,1]]}},"alternative-id":["BFncomms11526"],"URL":"http://dx.doi.org/10.1038/ncomms11526","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":[[2016,5,5]]},"assertion":[{"value":"22 December 2015","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 April 2016","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 May 2016","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":"11526","id":"doi:10.1038/ncomms11526","_hash":"bceaf54667213cc99ecde7605c3b932ee327c4f093e4899068dd27d8ecbca408"},"expire":1732265162980},"doi:10.1038/ncomms7983":{"value":{"indexed":{"date-parts":[[2023,11,20]],"date-time":"2023-11-20T15:34:34Z","timestamp":1700494474429},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2015,4,29]],"date-time":"2015-04-29T00:00:00Z","timestamp":1430265600000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2015,4,29]],"date-time":"2015-04-29T00:00:00Z","timestamp":1430265600000},"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":"AbstractQuantum data are susceptible to decoherence induced by the environment and to errors in the hardware processing it. A future fault-tolerant quantum computer will use quantum error correction to actively protect against both. In the smallest error correction codes, the information in one logical qubit is encoded in a two-dimensional subspace of a larger Hilbert space of multiple physical qubits. For each code, a set of non-demolition multi-qubit measurements, termed stabilizers, can discretize and signal physical qubit errors without collapsing the encoded information. Here using a five-qubit superconducting processor, we realize the two parity measurements comprising the stabilizers of the three-qubit repetition code protecting one logical qubit from physical bit-flip errors. While increased physical qubit coherence times and shorter quantum error correction blocks are required to actively safeguard the quantum information, this demonstration is a critical step towards larger codes based on multiple parity measurements.","DOI":"10.1038/ncomms7983","type":"journal-article","created":{"date-parts":[[2015,4,29]],"date-time":"2015-04-29T13:13:35Z","timestamp":1430313215000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":207,"title":"Detecting bit-flip errors in a logical qubit using stabilizer measurements","prefix":"10.1038","volume":"6","author":[{"given":"D.","family":"Ristè","sequence":"first","affiliation":[]},{"given":"S.","family":"Poletto","sequence":"additional","affiliation":[]},{"given":"M.-Z.","family":"Huang","sequence":"additional","affiliation":[]},{"given":"A.","family":"Bruno","sequence":"additional","affiliation":[]},{"given":"V.","family":"Vesterinen","sequence":"additional","affiliation":[]},{"given":"O.-P.","family":"Saira","sequence":"additional","affiliation":[]},{"given":"L.","family":"DiCarlo","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2015,4,29]]},"reference":[],"container-title":"Nature Communications","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/ncomms7983.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/ncomms7983","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/ncomms7983.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,5]],"date-time":"2023-01-05T11:42:59Z","timestamp":1672918979000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/ncomms7983"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,4,29]]},"references-count":31,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,11,3]]}},"alternative-id":["BFncomms7983"],"URL":"http://dx.doi.org/10.1038/ncomms7983","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":[[2015,4,29]]},"assertion":[{"value":"19 February 2015","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 March 2015","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 April 2015","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":"6983","id":"doi:10.1038/ncomms7983","_hash":"9d81de71803b4caf82fdc52d334ddacd25244f7993c6a5ed972edbb542792acd"},"expire":1732265163894},"doi:10.1038/nature10786":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T18:22:54Z","timestamp":1700590974667},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"7385","license":[{"start":{"date-parts":[[2012,2,1]],"date-time":"2012-02-01T00:00:00Z","timestamp":1328054400000},"content-version":"tdm","delay-in-days":0,"URL":"http://www.springer.com/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2012,2]]},"DOI":"10.1038/nature10786","type":"journal-article","created":{"date-parts":[[2012,1,31]],"date-time":"2012-01-31T14:02:15Z","timestamp":1328018535000},"page":"382-385","source":"Crossref","is-referenced-by-count":449,"title":"Realization of three-qubit quantum error correction with superconducting circuits","prefix":"10.1038","volume":"482","author":[{"given":"M. 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However, quantum sensors lose their sensitivity in the presence of noise. To protect them, the use of quantum error-correcting codes has been proposed. Trapped ions are an excellent technological platform for both quantum sensing and quantum error correction. Here we present a quantum error correction scheme that harnesses dissipation to stabilize a trapped-ion qubit. In our approach, always-on couplings to an engineered environment protect the qubit against spin-flips or phase-flips. Our dissipative error correction scheme operates in a continuous manner without the need to perform measurements or feedback operations. We show that the resulting enhanced coherence time translates into a significantly enhanced precision for quantum measurements. Our work constitutes a stepping stone towards the paradigm of self-correcting quantum information processing.","DOI":"10.1038/s41467-017-01895-5","type":"journal-article","created":{"date-parts":[[2017,11,22]],"date-time":"2017-11-22T14:29:39Z","timestamp":1511360979000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":76,"title":"Dissipative quantum error correction and application to quantum sensing with trapped ions","prefix":"10.1038","volume":"8","author":[{"ORCID":"http://orcid.org/0000-0002-5217-3064","authenticated-orcid":false,"given":"F.","family":"Reiter","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-1337-9163","authenticated-orcid":false,"given":"A. S.","family":"Sørensen","sequence":"additional","affiliation":[]},{"given":"P.","family":"Zoller","sequence":"additional","affiliation":[]},{"given":"C. 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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.","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":6,"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":"d8bb9e5048f741bde476007246da8b0addab0dc612da4753fea06647ec54e715"},"expire":1732265194808},"doi:10.22331/q-2021-12-20-605":{"value":{"indexed":{"date-parts":[[2023,11,16]],"date-time":"2023-11-16T15:35:20Z","timestamp":1700148920383},"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":"Recently, Hastings & 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 0.2&#x0025;&#x2212;0.3&#x0025; for the honeycomb code compared to a threshold of 0.5&#x0025;&#x2212;0.7&#x0025; 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 1.5&#x0025;&#x003C;p&#x003C;2.0&#x0025;, where p 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 10&#x2212;3, we project that the honeycomb code can reach the teraquop regime with only 600 physical qubits. 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We discuss small instances of the code.","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":14,"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":"393c7d35a888030994255b012376817219765000872ad52d0ff888c4434ed4d9"},"expire":1732265197815},"doi:10.26081/K6F65V":{"value":{"type":"article","id":"doi:10.26081/K6F65V","author":[{"family":"Brown","given":"Ben"}],"issued":{"date-parts":[[2022]]},"DOI":"10.26081/K6F65V","publisher":"Kavli Institute for Theoretical Physics","title":"Anyon condensation and the color code","URL":"https://online.kitp.ucsb.edu/online/dynisq-c22/brown/","reference":[],"_hash":"93e96e1f8c954b6b2bcfd00c23db64624f313f80b16e8ac1a894b53904f05bee"},"expire":1732265198683},"doi:10.1103/PhysRevResearch.2.033042":{"value":{"indexed":{"date-parts":[[2023,11,19]],"date-time":"2023-11-19T03:32:50Z","timestamp":1700364770400},"reference-count":34,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2020,7,9]],"date-time":"2020-07-09T00:00:00Z","timestamp":1594252800000},"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-1125565"]},{"DOI":"10.13039/100000936","name":"Gordon and Betty Moore Foundation","doi-asserted-by":"publisher","award":["GBMF-2644"]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevresearch.2.033042","type":"journal-article","created":{"date-parts":[[2020,7,9]],"date-time":"2020-07-09T18:14:26Z","timestamp":1594318466000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":26,"title":"Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel","prefix":"10.1103","volume":"2","author":[{"given":"Nicolas","family":"Delfosse","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-6041-9554","authenticated-orcid":true,"given":"Gilles","family":"Zémor","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2020,7,9]]},"reference":[],"container-title":"Physical Review Research","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevResearch.2.033042","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevResearch.2.033042/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,24]],"date-time":"2021-03-24T03:48:08Z","timestamp":1616557688000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevResearch.2.033042"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,9]]},"references-count":34,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,7]]}},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.033042","relation":{},"ISSN":["2643-1564"],"subject":["General Engineering"],"container-title-short":"Phys. 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Many schemes for fault-tolerant quantum information processing have been developed so far, one of which, called topological quantum computation, makes use of degrees of freedom that are inherently insensitive to local errors. However, this scheme is not so reliable against thermal errors. Other fault-tolerant schemes achieve better reliability through active error correction, but incur a substantial overhead cost. Thus, it is of practical importance and theoretical interest to design and assess fault-tolerant schemes that work well at finite temperature without active error correction.\n\nIn this thesis, a three-dimensional gapped lattice spin model is found which demonstrates for the first time that a reliable quantum memory at finite temperature is possible, at least to some extent. When quantum information is encoded into a highly entangled ground state of this model and subjected to thermal errors, the errors remain easily correctable for a long time without any active intervention, because a macroscopic energy barrier keeps the errors well localized. As a result, stored quantum information can be retrieved faithfully for a memory time which grows exponentially with the square of the inverse temperature. In contrast, for previously known types of topological quantum storage in three or fewer spatial dimensions the memory time scales exponentially with the inverse temperature, rather than its square.\n\nThis spin model exhibits a previously unexpected topological quantum order, in which ground states are locally indistinguishable, pointlike excitations are immobile, and the immobility is not affected by small perturbations of the Hamiltonian. The degeneracy of the ground state, though also insensitive to perturbations, is a complicated number-theoretic function of the system size, and the system bifurcates into multiple noninteracting copies of itself under real-space renormalization group transformations. The degeneracy, the excitations, and the renormalization group flow can be analyzed using a framework that exploits the spin model's symmetry and some associated free resolutions of modules over polynomial algebras.","DOI":"10.7907/GCYW-ZE58","publisher":"California Institute of Technology","title":"Lattice Quantum Codes and Exotic Topological Phases of Matter","URL":"https://resolver.caltech.edu/CaltechTHESIS:05292013-140541902","copyright":"No commercial reproduction, distribution, display or performance rights in this work are provided.","version":"Final","reference":[],"_hash":"7e9320ff258d493d1014350a30e5178046fac94201b8fdfaaa45d4989b91c470"},"expire":1732265201697},"doi:10.1007/s00220-013-1810-2":{"value":{"indexed":{"date-parts":[[2023,10,10]],"date-time":"2023-10-10T12:49:38Z","timestamp":1696942178409},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2013,10,10]],"date-time":"2013-10-10T00:00:00Z","timestamp":1381363200000},"content-version":"tdm","delay-in-days":0,"URL":"http://www.springer.com/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2013,12]]},"DOI":"10.1007/s00220-013-1810-2","type":"journal-article","created":{"date-parts":[[2013,10,9]],"date-time":"2013-10-09T06:33:03Z","timestamp":1381300383000},"page":"351-399","source":"Crossref","is-referenced-by-count":71,"title":"Commuting Pauli Hamiltonians as Maps between Free Modules","prefix":"10.1007","volume":"324","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2013,10,10]]},"reference":[],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","link":[{"URL":"http://link.springer.com/content/pdf/10.1007/s00220-013-1810-2.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http://link.springer.com/article/10.1007/s00220-013-1810-2/fulltext.html","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http://link.springer.com/content/pdf/10.1007/s00220-013-1810-2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T10:57:05Z","timestamp":1558695425000},"score":1,"resource":{"primary":{"URL":"http://link.springer.com/10.1007/s00220-013-1810-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,10,10]]},"references-count":39,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2013,12]]}},"alternative-id":["1810"],"URL":"http://dx.doi.org/10.1007/s00220-013-1810-2","relation":{},"ISSN":["0010-3616","1432-0916"],"subject":["Mathematical Physics","Statistical and Nonlinear Physics"],"container-title-short":"Commun. 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In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in D-dimensional hyperbolic space.","DOI":"10.22331/q-2022-05-13-711","type":"journal-article","created":{"date-parts":[[2022,5,13]],"date-time":"2022-05-13T12:50:42Z","timestamp":1652446242000},"page":"711","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":"Connectivity constrains quantum codes","prefix":"10.22331","volume":"6","author":[{"given":"Nouédyn","family":"Baspin","sequence":"first","affiliation":[{"name":"Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1"}]},{"given":"Anirudh","family":"Krishna","sequence":"additional","affiliation":[{"name":"Stanford University, Stanford, CA, USA, 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Phys.","published":{"date-parts":[[2016,1,25]]},"assertion":[{"value":"New Journal of Physics","name":"journal_title","label":"Journal title"},{"value":"paper","name":"article_type","label":"Article type"},{"value":"A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)","name":"article_title","label":"Article title"},{"value":"© 2016 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft","name":"copyright_information","label":"Copyright information"},{"value":"cc-by Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. 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":"2015-08-12","name":"date_received","label":"Date received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2015-11-24","name":"date_accepted","label":"Date accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2016-01-25","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/18/1/013050","_hash":"0859a650f912ddd41a6feed730d18de81b39177fcf9457e9d5a7b2b279948e5c"},"expire":1732265221879},"doi:10.1038/s41534-017-0019-1":{"value":{"indexed":{"date-parts":[[2023,11,22]],"date-time":"2023-11-22T13:58:59Z","timestamp":1700661539069},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2017,4,19]],"date-time":"2017-04-19T00:00:00Z","timestamp":1492560000000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2017,4,19]],"date-time":"2017-04-19T00:00:00Z","timestamp":1492560000000},"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":"AbstractPhotonic cat states stored in high-Q resonators show great promise for hardware efficient universal quantum computing. We propose an approach to efficiently prepare such cat states in a Kerr-nonlinear resonator by the use of a two-photon drive. Significantly, we show that this preparation is robust against single-photon loss. An outcome of this observation is that a two-photon drive can eliminate undesirable phase evolution induced by a Kerr nonlinearity. By exploiting the concept of transitionless quantum driving, we moreover demonstrate how non-adiabatic initialization of cat states is possible. Finally, we present a universal set of quantum logical gates that can be performed on the engineered eigenspace of such a two-photon driven resonator and discuss a possible realization using superconducting circuits. The robustness of the engineered subspace to higher-order circuit nonlinearities makes this implementation favorable for scalable quantum computation.","DOI":"10.1038/s41534-017-0019-1","type":"journal-article","created":{"date-parts":[[2017,4,13]],"date-time":"2017-04-13T13:23:14Z","timestamp":1492089794000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":172,"title":"Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving","prefix":"10.1038","volume":"3","author":[{"given":"Shruti","family":"Puri","sequence":"first","affiliation":[]},{"given":"Samuel","family":"Boutin","sequence":"additional","affiliation":[]},{"given":"Alexandre","family":"Blais","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,4,19]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41534-017-0019-1.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-017-0019-1","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-017-0019-1.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,23]],"date-time":"2022-12-23T01:44:11Z","timestamp":1671759851000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-017-0019-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,19]]},"references-count":43,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,12]]}},"alternative-id":["19"],"URL":"http://dx.doi.org/10.1038/s41534-017-0019-1","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":[[2017,4,19]]},"assertion":[{"value":"2 November 2016","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 March 2017","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 March 2017","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 April 2017","order":4,"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"}}],"article-number":"18","id":"doi:10.1038/s41534-017-0019-1","_hash":"e472969f32578a9aebdc749761858f49b2171f53490a63d457e5be6dcb47596f"},"expire":1732265222858},"doi:10.1098/rsta.2011.0485":{"value":{"indexed":{"date-parts":[[2023,8,18]],"date-time":"2023-08-18T23:10:25Z","timestamp":1692400225945},"reference-count":30,"publisher":"The Royal Society","issue":"1979","license":[{"start":{"date-parts":[[2012,11,28]],"date-time":"2012-11-28T00:00:00Z","timestamp":1354060800000},"content-version":"tdm","delay-in-days":0,"URL":"https://royalsociety.org/journals/ethics-policies/data-sharing-mining/"}],"content-domain":{"domain":["royalsocietypublishing.org"],"crossmark-restriction":true},"published-print":{"date-parts":[[2012,11,28]]},"abstract":"\n We provide a solution to the problem of determining whether a target pure state can be asymptotically prepared using dissipative Markovian dynamics under\n fixed\n locality constraints. Besides recovering existing results for a large class of physically relevant entangled states, our approach has the advantage of providing an explicit stabilization test solely based on the input state and constraints of the problem. Connections with the formalism of frustration-free parent Hamiltonians are discussed, as well as control implementations in terms of a switching output-feedback law.\n ","DOI":"10.1098/rsta.2011.0485","type":"journal-article","created":{"date-parts":[[2012,10,22]],"date-time":"2012-10-22T07:32:58Z","timestamp":1350891178000},"page":"5259-5269","update-policy":"http://dx.doi.org/10.1098/crossmark-policy","source":"Crossref","is-referenced-by-count":49,"title":"Stabilizing entangled states with quasi-local quantum dynamical semigroups","prefix":"10.1098","volume":"370","author":[{"given":"Francesco","family":"Ticozzi","sequence":"first","affiliation":[{"name":"Dipartimento di Ingegneria dell'Informazione, Università di Padova, via Gradenigo 6/B, 35131 Padova, Italy"},{"name":"Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA"}]},{"given":"Lorenza","family":"Viola","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA"}]}],"member":"175","published-online":{"date-parts":[[2012,11,28]]},"reference":[],"container-title":"Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","link":[{"URL":"https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2011.0485","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://royalsocietypublishing.org/doi/full-xml/10.1098/rsta.2011.0485","content-type":"application/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2011.0485","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,20]],"date-time":"2021-02-20T23:54:14Z","timestamp":1613865254000},"score":1,"resource":{"primary":{"URL":"https://royalsocietypublishing.org/doi/10.1098/rsta.2011.0485"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,11,28]]},"references-count":30,"journal-issue":{"issue":"1979","published-print":{"date-parts":[[2012,11,28]]}},"alternative-id":["10.1098/rsta.2011.0485"],"URL":"http://dx.doi.org/10.1098/rsta.2011.0485","relation":{},"ISSN":["1364-503X","1471-2962"],"subject":["General Physics and Astronomy","General Engineering","General Mathematics"],"container-title-short":"Phil. Trans. R. Soc. 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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. 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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.","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":23,"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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(miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2020,7,2]]},"article-number":"288","id":"doi:10.22331/q-2020-07-02-288","_hash":"40a5979ff3a10f463658847bb7b976f70bb8958b3d187020da4c050b0bef9d20"},"expire":1732265267883},"doi:10.1103/PhysRevLett.126.150503":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T20:37:03Z","timestamp":1700599023596},"reference-count":63,"publisher":"American Physical Society (APS)","issue":"15","license":[{"start":{"date-parts":[[2021,4,15]],"date-time":"2021-04-15T00:00:00Z","timestamp":1618444800000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"funder":[{"DOI":"10.13039/100000893","name":"Simons Foundation","doi-asserted-by":"publisher"},{"DOI":"10.13039/501100000023","name":"Government of Canada","doi-asserted-by":"publisher"},{"DOI":"10.13039/100011332","name":"Innovation, Science and Economic 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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. We also present a type of covariant codes which nearly saturates these lower bounds.","DOI":"10.22331/q-2021-08-09-521","type":"journal-article","created":{"date-parts":[[2021,8,9]],"date-time":"2021-08-09T11:21:59Z","timestamp":1628508119000},"page":"521","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":19,"title":"New perspectives on covariant quantum error correction","prefix":"10.22331","volume":"5","author":[{"given":"Sisi","family":"Zhou","sequence":"first","affiliation":[{"name":"Department of Physics, Yale University, New Haven, Connecticut 06511, USA"},{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Illinois 60637, USA"}]},{"given":"Zi-Wen","family":"Liu","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada"}]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Illinois 60637, USA"}]}],"member":"9598","published-online":{"date-parts":[[2021,8,9]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2021-08-09-521/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,1,6]],"date-time":"2023-01-06T23:31:04Z","timestamp":1673047864000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2021-08-09-521/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,9]]},"references-count":75,"URL":"http://dx.doi.org/10.22331/q-2021-08-09-521","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2021,8,9]]},"article-number":"521","id":"doi:10.22331/q-2021-08-09-521","_hash":"66042970da09f32ffae04932212f5a41e3402769aacf7eba0bd91e573c6c2291"},"expire":1732265269880},"doi:10.1109/TIT.2018.2873764":{"value":{"indexed":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T17:20:31Z","timestamp":1701105631307},"reference-count":91,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","license":[{"start":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T00:00:00Z","timestamp":1554076800000},"content-version":"vor","delay-in-days":0,"URL":"https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html"},{"start":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T00:00:00Z","timestamp":1554076800000},"content-version":"am","delay-in-days":0,"URL":"https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html"},{"start":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T00:00:00Z","timestamp":1554076800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https://doi.org/10.15223/policy-029"},{"start":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T00:00:00Z","timestamp":1554076800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https://doi.org/10.15223/policy-037"}],"funder":[{"name":"ARL-CDQI","award":["W911NF-15-2-0067"]},{"name":"ARO","award":["W911NF-14-1-0011","W911NF-14-1-0563","W911NF-18-1-0020","W911NF-18-1-0212"]},{"name":"ARO MURI","award":["W911NF-16-1-0349"]},{"name":"AFOSR MURI","award":["FA9550-14-1-0052","FA9550-15-1-0015"]},{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["EFMA-1640959"]},{"DOI":"10.13039/100000879","name":"Alfred P. 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Hence determining the quantum capacity of these channels is an outstanding open problem for quantum computation and communication. Here we derive several upper bounds on the quantum capacity of qubit and bosonic thermal attenuators. We introduce an extended version of such channels which is degradable and hence has a single-letter quantum capacity, bounding that of the original thermal attenuators. Another bound for bosonic attenuators is given by the bottleneck inequality applied to a particular channel decomposition. With respect to previously known bounds we report better results in a broad range of attenuation and noise: we can now approximate the quantum capacity up to a negligible uncertainty for most practical applications, e.g., for low thermal noise.","DOI":"10.1038/s41467-018-06848-0","type":"journal-article","created":{"date-parts":[[2018,10,12]],"date-time":"2018-10-12T14:05:36Z","timestamp":1539353136000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":"Narrow bounds for the quantum capacity of thermal attenuators","prefix":"10.1038","volume":"9","author":[{"ORCID":"http://orcid.org/0000-0002-8972-2936","authenticated-orcid":false,"given":"Matteo","family":"Rosati","sequence":"first","affiliation":[]},{"given":"Andrea","family":"Mari","sequence":"additional","affiliation":[]},{"given":"Vittorio","family":"Giovannetti","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,10,18]]},"reference":[],"container-title":"Nature Communications","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41467-018-06848-0","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-018-06848-0.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41467-018-06848-0.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T19:18:00Z","timestamp":1671563880000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41467-018-06848-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,18]]},"references-count":53,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,12]]}},"alternative-id":["6848"],"URL":"http://dx.doi.org/10.1038/s41467-018-06848-0","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":[[2018,10,18]]},"assertion":[{"value":"5 June 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 September 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 October 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 January 2019","order":4,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":5,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The original version of this Article contained an error in Equation (40). The numerator of the fraction inside the logarithm was missing an overall minus sign. 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We present general results that we illustrate through examples taken from different classes of codes, including scaled self-dual GKP codes and the concatenated surface-GKP code.","DOI":"10.22331/q-2022-02-10-648","type":"journal-article","created":{"date-parts":[[2022,2,10]],"date-time":"2022-02-10T11:00:14Z","timestamp":1644490814000},"page":"648","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":7,"title":"Gottesman-Kitaev-Preskill codes: A lattice perspective","prefix":"10.22331","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0001-6120-9930","authenticated-orcid":false,"given":"Jonathan","family":"Conrad","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"},{"name":"Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, 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EEC-1941583, OMA-2137642"]},{"name":"NTT Research","award":["NTT Research"]},{"DOI":"10.13039/100000008","name":"Packard Foundation","doi-asserted-by":"crossref","award":["2020-71479"]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"abstract":"Quantum error correction has recently been shown to benefit greatly from specific physical encodings of the code qubits. 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.","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":8,"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, 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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. 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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 distancedthrough 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 ofO(d/log⁡d). 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For one of these problems, namely[Formula: see text]where dpq is the Euclidean distance between points P and Q and m is the number of points, we discuss the results for m ≤ 16 and 1 ≤ n ≤ ∞. For the cases m = 5, 11, 13–16 we find hitherto undiscovered solutions. Our solutions for m = 5 and 11 correct earlier results in the literature. We also sharpen the existing literature results for m = 7 and 10. 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Comput.","published":{"date-parts":[[1998,6]]},"id":"doi:10.1137/S0097539796302531","_hash":"8a6be0a2edcf4f436caf0be798e0555367bbed2a0bb2e9ea9c0b56939a7f739c"},"expire":1732265728881},"doi:10.1145/502090.502098":{"value":{"indexed":{"date-parts":[[2023,11,28]],"date-time":"2023-11-28T05:27:04Z","timestamp":1701149224835},"reference-count":32,"publisher":"Association for Computing Machinery (ACM)","issue":"4","content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"published-print":{"date-parts":[[2001,7]]},"abstract":"\n We prove optimal, up to an arbitrary ε > 0, inapproximability results for Max-E\n k\n -Sat for\n k\n ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime\n p\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 ","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","update-policy":"http://dx.doi.org/10.1145/crossmark-policy","source":"Crossref","is-referenced-by-count":773,"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","published-online":{"date-parts":[[2001,7]]},"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":[[2023,1,2]],"date-time":"2023-01-02T21:40:24Z","timestamp":1672695624000},"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":"d6089c0143c32c32141b4f17359d02b92c6bf47aba112f2538c6af052bef9870"},"expire":1732265730911},"doi:10.1145/3051093":{"value":{"indexed":{"date-parts":[[2023,8,29]],"date-time":"2023-08-29T22:53:55Z","timestamp":1693349635987},"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":390,"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":["dl.acm.org"],"crossmark-restriction":true},"published-print":{"date-parts":[[2017,4,30]]},"abstract":"\n Locally correctable codes (LCCs) and locally testable codes (LTCs) are error-correcting codes that admit\n local\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 \n \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 n\n , constant rate (which can even be taken arbitrarily close to 1), and constant relative distance, whose query complexity is exp(Õ(√log\n n\n )) (for LCCs) and (log\n n\n )\n \n O\n (log log\n n\n )\n \n (for LTCs).\n \n \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 o\n (\n n\n ) query complexity. Our results on LCCs also immediately give locally decodable codes with the same parameters.\n ","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","update-policy":"http://dx.doi.org/10.1145/crossmark-policy","source":"Crossref","is-referenced-by-count":20,"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","published-online":{"date-parts":[[2017,5,25]]},"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":[[2022,12,31]],"date-time":"2022-12-31T09:47:48Z","timestamp":1672480068000},"score":1,"resource":{"primary":{"URL":"https://dl.acm.org/doi/10.1145/3051093"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,30]]},"references-count":69,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2017,4,30]]}},"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":[[2017,4,30]]},"assertion":[{"value":"2016-05-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2017-02-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2017-05-25","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"id":"doi:10.1145/3051093","_hash":"deaca14d109a2689ae98627787190a7748fc6ba32efe87c52fef6408d5986575"},"expire":1732265731887},"doi:10.1145/1162349.1162351":{"value":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T18:22:46Z","timestamp":1700590966966},"reference-count":32,"publisher":"Association for Computing Machinery (ACM)","issue":"4","content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"published-print":{"date-parts":[[2006,7]]},"abstract":"\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 k\n information bits are encoded by a codeword of length\n k\n ⋅ exp(Õ(√(log\n k\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 n\n ⋅ exp(Õ(√(log\n n\n ))) and verification in performed by a constant number (i.e., 19) of queries, as opposed to previous results that used proof length\n n\n \n (1 +\n O\n (1/\n q\n ))\n \n for verification by\n q\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 ","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","update-policy":"http://dx.doi.org/10.1145/crossmark-policy","source":"Crossref","is-referenced-by-count":84,"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","published-online":{"date-parts":[[2006,7]]},"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":[[2022,12,28]],"date-time":"2022-12-28T19:56:08Z","timestamp":1672257368000},"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 largest travel time differences for any solar phenomena are observed.Conclusions.With sufficient modeling effort, these should lead to better understanding of sunspot structure.","DOI":"10.1051/0004-6361/201732424","type":"journal-article","created":{"date-parts":[[2018,1,31]],"date-time":"2018-01-31T08:48:16Z","timestamp":1517388496000},"page":"A73","source":"Crossref","is-referenced-by-count":4,"title":"Probing sunspots with two-skip time–distance helioseismology","prefix":"10.1051","volume":"613","author":[{"suffix":"Jr.","given":"Thomas L.","family":"Duvall","sequence":"first","affiliation":[]},{"given":"Paul S.","family":"Cally","sequence":"additional","affiliation":[]},{"given":"Damien","family":"Przybylski","sequence":"additional","affiliation":[]},{"given":"Kaori","family":"Nagashima","sequence":"additional","affiliation":[]},{"given":"Laurent","family":"Gizon","sequence":"additional","affiliation":[]}],"member":"250","published-online":{"date-parts":[[2018,6,4]]},"reference":[],"container-title":"Astronomy & Astrophysics","original-title":[],"link":[{"URL":"https://www.aanda.org/10.1051/0004-6361/201732424/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,13]],"date-time":"2022-08-13T15:39:52Z","timestamp":1660405192000},"score":1,"resource":{"primary":{"URL":"https://www.aanda.org/10.1051/0004-6361/201732424"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5]]},"references-count":36,"alternative-id":["aa32424-17"],"URL":"http://dx.doi.org/10.1051/0004-6361/201732424","relation":{},"ISSN":["0004-6361","1432-0746"],"subject":["Space and Planetary Science","Astronomy and Astrophysics"],"container-title-short":"A&A","published":{"date-parts":[[2018,5]]},"id":"doi:10.1051/0004-6361/201732424","_hash":"684444e104b819e95f029ee0306a38de104ba1b472b459709f5f4ccfe41824c5"},"expire":1732286485695},"doi:10.26421/QIC14.9-10-1":{"value":{"indexed":{"date-parts":[[2023,8,23]],"date-time":"2023-08-23T23:32:32Z","timestamp":1692833552065},"reference-count":0,"publisher":"Rinton Press","issue":"9&10","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2014,7]]},"abstract":"We present a three-dimensional generalization of a renormalization group decoding algorithm for topological codes with Abelian anyonic excitations that we introduced for two dimensions in \\cite{DP09a,DP10a}. 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Provided with input magic states, our protocol operates on a two-dimensional square grid by measurements of ZZ on horizontal pairs of qubits, XX on vertical pairs, and Z,X on single qubits.","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":1,"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":"793809ddb352ada0f94f1227933c3fb6d00c3b1552b28a55faff64e78430a3d9"},"expire":1734264356137},"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":"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.","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":1734264356931},"doi:10.1103/PhysRevA.57.127":{"value":{"indexed":{"date-parts":[[2023,12,20]],"date-time":"2023-12-20T17:00:50Z","timestamp":1703091650237},"reference-count":13,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[1998,1,1]],"date-time":"1998-01-01T00:00:00Z","timestamp":883612800000},"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.57.127","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T01:39:22Z","timestamp":1027733962000},"page":"127-137","source":"Crossref","is-referenced-by-count":490,"title":"Theory of fault-tolerant quantum computation","prefix":"10.1103","volume":"57","author":[{"given":"Daniel","family":"Gottesman","sequence":"first","affiliation":[]}],"member":"16","published-online":{"date-parts":[[1998,1,1]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.57.127","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.57.127/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,15]],"date-time":"2017-06-15T08:21:12Z","timestamp":1497514872000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.57.127"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,1,1]]},"references-count":13,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1998,1]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.57.127","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. 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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.","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":70,"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 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circuits","prefix":"10.1103","volume":"65","author":[{"given":"Barbara M.","family":"Terhal","sequence":"first","affiliation":[]},{"given":"David P.","family":"DiVincenzo","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2002,3,1]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.65.032325","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.65.032325/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,15]],"date-time":"2017-06-15T07:35:44Z","timestamp":1497512144000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.65.032325"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,3,1]]},"references-count":9,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2002,3]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.65.032325","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. Rev. A","published":{"date-parts":[[2002,3,1]]},"article-number":"032325","id":"doi:10.1103/PhysRevA.65.032325","_hash":"f04ea05d72e6da785d55e8b60fafc003f485f437b3db33087ca04c3b62fe421c"},"expire":1734264360845},"doi:10.22331/q-2017-04-25-4":{"value":{"indexed":{"date-parts":[[2023,8,10]],"date-time":"2023-08-10T11:50:21Z","timestamp":1691668221215},"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":"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 n, the number of encoded qubits k, the code distance d, the accuracy parameter &#x03B4; that quantifies how well the erasure channel can be reversed, and the locality parameter &#x2113; that specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracy &#x03B4; that is exponentially small in &#x2113;, which is the case for perturbations of local commuting projector codes, our bound reads kd2D&#x2212;1&#x2264;O(n(log&#x2061;n)2DD&#x2212;1) for codes on D-dimensional lattices of Euclidean metric. We also find that the code distance of any local approximate code cannot exceed O(&#x2113;n(D&#x2212;1)/D) if &#x03B4;&#x2264;O(&#x2113;n&#x2212;1/D). As a corollary of our formulation of correctability in terms of logical operator avoidance, we show that the code distance d and the size d&#x007E; of a minimal region that can support all approximate logical operators satisfies d&#x007E;d1D&#x2212;1&#x2264;O(n&#x2113;DD&#x2212;1), where the logical operators are accurate up to O((n&#x03B4;/d)1/2) 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.","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":19,"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. J. Watson Research Center, Yorktown Heights, New York, USA"},{"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":[[2017,4,25]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2017-04-25-4/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,2,7]],"date-time":"2022-02-07T13:13:17Z","timestamp":1644239597000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2017-04-25-4/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,25]]},"references-count":56,"URL":"http://dx.doi.org/10.22331/q-2017-04-25-4","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":"4","id":"doi:10.22331/q-2017-04-25-4","_hash":"315ea08cc0ae082500e19cb236880c49a8c9be9d5a76a24c63593d47ea4c277f"},"expire":1734264361956},"doi:10.1098/rsta.1995.0106":{"value":{"indexed":{"date-parts":[[2023,11,4]],"date-time":"2023-11-04T01:56:56Z","timestamp":1699063016140},"reference-count":45,"publisher":"The Royal Society","issue":"1703","license":[{"start":{"date-parts":[[1995,12,15]],"date-time":"1995-12-15T00:00:00Z","timestamp":818985600000},"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":[[1995,12,15]]},"abstract":"\n Technologies differ in their explicit utilization of quantum mechanical behaviour. 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 localization\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 ","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":102,"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. 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Specifically, we investigate the quantum phase-flip repetition code as a quantum memory and theoretically demonstrate that it can preserve arbitrary quantum information longer than the lifetime of a single idle qubit in a dephasing-time-limited system, e.g. in semiconductor qubits. Our circuit-based analytical calculations show the efficiency of the phase-flip code as a quantum memory in the presence of relaxation, dephasing, and faulty quantum gates. Moreover, we identify the optimal repetition number of quantum error correction cycles required to reach the break-even point by considering the gate error probabilities of current platforms for quantum computing. Our results provide guidelines for developing quantum memories in semiconductor quantum devices.","DOI":"10.1088/1367-2630/acfba5","type":"journal-article","created":{"date-parts":[[2023,9,20]],"date-time":"2023-09-20T22:28:00Z","timestamp":1695248880000},"page":"103004","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":"Break-even point of the phase-flip error correcting code","prefix":"10.1088","volume":"25","author":[{"given":"Áron","family":"Rozgonyi","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4178-5759","authenticated-orcid":true,"given":"Gábor","family":"Széchenyi","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2023,10,5]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,5]],"date-time":"2023-10-05T11:57:00Z","timestamp":1696507020000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/acfba5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,1]]},"references-count":40,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10,5]]},"published-print":{"date-parts":[[2023,10,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/acfba5","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. Phys.","published":{"date-parts":[[2023,10,1]]},"assertion":[{"value":"Break-even point of the phase-flip error correcting code","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":"© 2023 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":"2023-05-15","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2023-09-20","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2023-10-05","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/acfba5","_hash":"cd36bd2f1a52af022ead4eef16300d136cc60dfad25c6d98466cd54418b0bcc9"},"expire":1734778369509},"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":"Abstract\n \n 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(n\n 1−ν ) for any ν ∈ (0, 1) and Ω(log n) 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 d-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.","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":1734778370851},"doi:10.1017/fms.2023.98":{"value":{"indexed":{"date-parts":[[2023,11,29]],"date-time":"2023-11-29T01:01:49Z","timestamp":1701219709765},"reference-count":40,"publisher":"Cambridge University Press (CUP)","license":[{"start":{"date-parts":[[2023,11,28]],"date-time":"2023-11-28T00:00:00Z","timestamp":1701129600000},"content-version":"unspecified","delay-in-days":331,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"published-print":{"date-parts":[[2023]]},"abstract":"Abstract\n\t We show that every ergodic Davies generator associated to any 2D Kitaev’s quantum double model has a nonvanishing spectral gap in the thermodynamic limit. This validates rigorously the extended belief that those models are useless as self-correcting quantum memories, even in the non-abelian case. The proof uses recent ideas and results regarding the characterization of the spectral gap for parent Hamiltonians associated to Projected Entangled Pair States in terms of a bulk-boundary correspondence.","DOI":"10.1017/fms.2023.98","type":"journal-article","created":{"date-parts":[[2023,11,28]],"date-time":"2023-11-28T10:01:08Z","timestamp":1701165668000},"update-policy":"http://dx.doi.org/10.1017/policypage","source":"Crossref","is-referenced-by-count":0,"title":"Thermalization in Kitaev’s quantum double models via tensor network techniques","prefix":"10.1017","volume":"11","author":[{"ORCID":"http://orcid.org/0000-0003-1709-1220","authenticated-orcid":false,"given":"Angelo","family":"Lucia","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-2990-791X","authenticated-orcid":false,"given":"David","family":"Pérez-García","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-8600-7083","authenticated-orcid":false,"given":"Antonio","family":"Pérez-Hernández","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,11,28]]},"reference":[],"container-title":"Forum of Mathematics, Sigma","original-title":[],"language":"en","link":[{"URL":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S2050509423000981","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,28]],"date-time":"2023-11-28T10:01:13Z","timestamp":1701165673000},"score":1,"resource":{"primary":{"URL":"https://www.cambridge.org/core/product/identifier/S2050509423000981/type/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023]]},"references-count":40,"alternative-id":["S2050509423000981"],"URL":"http://dx.doi.org/10.1017/fms.2023.98","relation":{},"ISSN":["2050-5094"],"subject":["Computational Mathematics","Discrete Mathematics and Combinatorics","Geometry and Topology","Mathematical Physics","Statistics and Probability","Algebra and Number Theory","Theoretical Computer Science","Analysis"],"container-title-short":"Forum of Mathematics, Sigma","published":{"date-parts":[[2023]]},"assertion":[{"value":"© The Author(s), 2023. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}],"article-number":"e107","id":"doi:10.1017/fms.2023.98","_hash":"78b79372db2887a939619d6094efb4ec696bd1300af70ed95bc09e50548dc6ff"},"expire":1734778371835},"doi:10.1007/JHEP01(2022)170":{"value":{"indexed":{"date-parts":[[2023,10,28]],"date-time":"2023-10-28T23:25:55Z","timestamp":1698535555058},"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":"Abstract\n 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 N and a large gap in the spectrum of operators in the emergence of complementary recovery.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.","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":8,"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. Phys.","published":{"date-parts":[[2022,1]]},"assertion":[{"value":"4 July 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 December 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 January 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"170","id":"doi:10.1007/JHEP01(2022)170","_hash":"04e87b4a3419aa06427adab3e265d4bae557a83fdf0df9e461e31c190807b756"},"expire":1734778372747},"doi:10.1038/s41534-023-00788-4":{"value":{"indexed":{"date-parts":[[2023,11,22]],"date-time":"2023-11-22T00:25:46Z","timestamp":1700612746143},"reference-count":97,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T00:00:00Z","timestamp":1700524800000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T00:00:00Z","timestamp":1700524800000},"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":"AbstractQuantum error correction (QEC) is a key concept in quantum computation as well as many areas of physics. There are fundamental tensions between continuous symmetries and QEC. One vital situation is unfolded by the Eastin–Knill theorem, which forbids the existence of QEC codes that admit transversal continuous symmetry actions (transformations). Here, we systematically study the competition between continuous symmetries and QEC in a quantitative manner. We first define a series of meaningful measures of approximate symmetries motivated from different perspectives, and then establish a series of trade-off bounds between them and QEC accuracy utilizing multiple different methods. Remarkably, the results allow us to derive general quantitative limitations of transversally implementable logical gates, an important topic in fault-tolerant quantum computation. As concrete examples, we showcase two explicit types of quantum codes, obtained from quantum Reed–Muller codes and thermodynamic codes, respectively, that nearly saturate our bounds. Finally, we discuss several potential applications of our results in physics.","DOI":"10.1038/s41534-023-00788-4","type":"journal-article","created":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T20:05:28Z","timestamp":1700597128000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":"Approximate symmetries and quantum error correction","prefix":"10.1038","volume":"9","author":[{"ORCID":"http://orcid.org/0000-0002-3402-9763","authenticated-orcid":false,"given":"Zi-Wen","family":"Liu","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0003-4618-8590","authenticated-orcid":false,"given":"Sisi","family":"Zhou","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,11,21]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41534-023-00788-4.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-023-00788-4","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-023-00788-4.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T20:13:02Z","timestamp":1700597582000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-023-00788-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,21]]},"references-count":97,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,12]]}},"alternative-id":["788"],"URL":"http://dx.doi.org/10.1038/s41534-023-00788-4","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":[[2023,11,21]]},"assertion":[{"value":"25 April 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 November 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 November 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":"119","id":"doi:10.1038/s41534-023-00788-4","_hash":"243f4756f0b35aa6fe7e92b50f8b2fdd5f527c3169cbe4c42c6f83cbf294b897"},"expire":1734778373968},"doi:10.1103/PRXQuantum.4.040334":{"value":{"indexed":{"date-parts":[[2023,12,2]],"date-time":"2023-12-02T00:50:42Z","timestamp":1701478242123},"reference-count":73,"publisher":"American Physical Society (APS)","issue":"4","license":[{"start":{"date-parts":[[2023,12,1]],"date-time":"2023-12-01T00:00:00Z","timestamp":1701388800000},"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.4.040334","type":"journal-article","created":{"date-parts":[[2023,12,1]],"date-time":"2023-12-01T16:53:38Z","timestamp":1701449618000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":"Closest Lattice Point Decoding for 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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.","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":25,"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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Similarly to Reed-Muller codes, multiplicity codes have a local nature that allows for local correction and local testing. Recently, [Karliner et al., 2022] showed that the plane test, which tests the degree of the codeword on a random plane, is a good local tester for small enough degrees. In this work we simplify and extend the analysis of local testing for multiplicity codes, giving a more general and tight analysis. In particular, we show that multiplicity codes MRM_p(m, d, s) over prime fields with arbitrary d are locally testable by an appropriate k-flat test, which tests the degree of the codeword on a random k-dimensional affine subspace. The relationship between the degree parameter d and the required dimension k is shown to be nearly optimal, and improves on [Karliner et al., 2022] in the case of planes.\r\nOur analysis relies on a generalization of the technique of canonincal monomials introduced in [Haramaty et al., 2013]. Generalizing canonical monomials to the multiplicity case requires substantially different proofs which exploit the algebraic structure of multiplicity codes.","DOI":"10.4230/LIPICS.APPROX/RANDOM.2022.11","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","title":"Improved Local Testing for Multiplicity Codes","URL":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.11","copyright":"Creative Commons Attribution 4.0 International license","reference":[],"_hash":"1d2e9cb3e09c75a51554f6cf8cb7f6fd8ae638928f4d39c4e6d494b315d51bd9"},"expire":1740762681168},"doi:10.4230/LIPIcs.CCC.2022.14":{"value":{"type":"","id":"doi:10.4230/LIPIcs.CCC.2022.14","categories":["local testing","multiplicity codes","Reed Muller codes","Theory of computation → Error-correcting codes"],"language":"en","author":[{"family":"Karliner","given":"Dan"},{"family":"Salama","given":"Roie"},{"family":"Ta-Shma","given":"Amnon"}],"contributor":[{"family":"Lovett","given":"Shachar"}],"issued":{"date-parts":[[2022]]},"abstract":"Multiplicity codes are a generalization of RS and RM codes where for each evaluation point we output the evaluation of a low-degree polynomial and all of its directional derivatives up to order s. Multi-variate multiplicity codes are locally decodable with the natural local decoding algorithm that reads values on a random line and corrects to the closest uni-variate multiplicity code. However, it was not known whether multiplicity codes are locally testable, and this question has been posed since the introduction of these codes with no progress up to date. In fact, it has been also open whether multiplicity codes can be characterized by local constraints, i.e., if there exists a probabilistic algorithm that queries few symbols of a word c, accepts every c in the code with probability 1, and rejects every c not in the code with nonzero probability. \r\nWe begin by giving a simple example showing the line test does not give local characterization when d > q. Surprisingly, we then show the plane test is a local characterization when s < q and d < qs-1 for prime q. In addition, we show the s-dimensional test is a local tester for multiplicity codes, when s < q. Combining the two results, we show our main result that the plane test is a local tester for multiplicity codes of degree d < qs-1, with constant rejection probability for constant q, s.\r\nOur technique is new. We represent the given input as a possibly very high-degree polynomial, and we show that for some choice of plane, the restriction of the polynomial to the plane is a high-degree bi-variate polynomial. The argument has to work modulo the appropriate kernels, and for that we use Grobner theory, the Combinatorial Nullstellensatz theorem and its generalization to multiplicities. Even given that, the argument is delicate and requires choosing a non-standard monomial order for the argument to work.","DOI":"10.4230/LIPICS.CCC.2022.14","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","title":"The Plane Test Is a Local Tester for Multiplicity Codes","URL":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.14","copyright":"Creative Commons Attribution 4.0 International license","reference":[],"_hash":"d44d705a16b7ec66a9f8ab93dd34ed727cd96235fbad724daa94c59cd64a6129"},"expire":1740762682270},"doi:10.1145/2629416":{"value":{"indexed":{"date-parts":[[2024,3,2]],"date-time":"2024-03-02T12:21:49Z","timestamp":1709382109732},"reference-count":49,"publisher":"Association for Computing Machinery (ACM)","issue":"5","license":[{"start":{"date-parts":[[2014,9,8]],"date-time":"2014-09-08T00:00:00Z","timestamp":1410134400000},"content-version":"vor","delay-in-days":0,"URL":"http://www.acm.org/publications/policies/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"published-print":{"date-parts":[[2014,9,8]]},"abstract":"\n Locally decodable codes are error-correcting codes that admit efficient decoding algorithms; any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword. The tradeoff between the rate of a code and the locality/efficiency of its decoding algorithms has been well studied, and it has widely been suspected that nontrivial locality must come at the price of low rate. A particular setting of potential interest in practice is codes of constant rate. For such codes, decoding algorithms with locality\n O\n (\n k∈\n ) were known only for codes of rate\n ∈\n Ω(1/\n ∈\n ), where\n k\n is the length of the message. Furthermore, for codes of rate > 1/2, no nontrivial locality had been achieved.\n \n \n In this article, we construct a new family of locally decodable codes that have very efficient local decoding algorithms, and at the same time have rate approaching 1. We show that for every\n ∈\n > 0 and\n α\n > 0, for infinitely many\n k\n , there exists a code\n C\n which encodes messages of length\n k\n with rate 1 −\n α\n , and is locally decodable from a constant fraction of errors using\n O\n (\n k∈\n ) queries and time.\n \n These codes, which we call multiplicity codes, are based on evaluating multivariate polynomials and their derivatives. Multiplicity codes extend traditional multivariate polynomial codes; they inherit the local-decodability of these codes, and at the same time achieve better tradeoffs and flexibility in the rate and minimum distance.","DOI":"10.1145/2629416","type":"journal-article","created":{"date-parts":[[2014,9,9]],"date-time":"2014-09-09T14:39:29Z","timestamp":1410273569000},"page":"1-20","update-policy":"http://dx.doi.org/10.1145/crossmark-policy","source":"Crossref","is-referenced-by-count":31,"title":"High-rate codes with sublinear-time decoding","prefix":"10.1145","volume":"61","author":[{"given":"Swastik","family":"Kopparty","sequence":"first","affiliation":[{"name":"Institute for Advanced Study"}]},{"given":"Shubhangi","family":"Saraf","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology"}]},{"given":"Sergey","family":"Yekhanin","sequence":"additional","affiliation":[{"name":"Microsoft Research Silicon Valley"}]}],"member":"320","published-online":{"date-parts":[[2014,9,8]]},"reference":[],"container-title":"Journal of the ACM","original-title":[],"language":"en","link":[{"URL":"https://dl.acm.org/doi/pdf/10.1145/2629416","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,30]],"date-time":"2022-12-30T19:51:11Z","timestamp":1672429871000},"score":1,"resource":{"primary":{"URL":"https://dl.acm.org/doi/10.1145/2629416"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,9,8]]},"references-count":49,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2014,9,8]]}},"alternative-id":["10.1145/2629416"],"URL":"http://dx.doi.org/10.1145/2629416","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 construction generates hybrid classical-quantum stabilizer codes and we formulate a theorem that fully characterizes the Pauli errors that are correctable for a given code, generalizing the fundamental theorems for the QEC and OQEC stabilizer formalisms. We discover hybrid versions of the Bacon-Shor subsystem codes motivated by the formalism, and we apply the theorem to derive a result that gives the distance of such codes. We show how some recent hybrid subspace code constructions are captured by the formalism, and we also indicate how it extends to qudits.","DOI":"10.22331/q-2024-02-21-1261","type":"journal-article","created":{"date-parts":[[2024,2,21]],"date-time":"2024-02-21T13:07:47Z","timestamp":1708520867000},"page":"1261","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Stabilizer Formalism for Operator Algebra Quantum Error Correction","prefix":"10.22331","volume":"8","author":[{"given":"Guillaume","family":"Dauphinais","sequence":"first","affiliation":[{"name":"Xanadu, Toronto, ON M5G 2C8, Canada"}]},{"given":"David W.","family":"Kribs","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON M5G 2C8, Canada"},{"name":"Department of Mathematics & Statistics, University of Guelph, Guelph, ON N1G 2W1, Canada"}]},{"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON M5G 2C8, Canada"},{"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":[[2024,2,21]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2024-02-21-1261/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2024,2,21]],"date-time":"2024-02-21T13:08:01Z","timestamp":1708520881000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2024-02-21-1261/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,21]]},"references-count":70,"URL":"http://dx.doi.org/10.22331/q-2024-02-21-1261","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2024,2,21]]},"article-number":"1261","id":"doi:10.22331/q-2024-02-21-1261","_hash":"90c5842ddca1dfda9a8f7d9a055e4cb5d2a8415c3933979c04b90053d1c10897"},"expire":1740764053527},"doi:10.1103/PhysRevA.102.062402":{"value":{"indexed":{"date-parts":[[2024,2,5]],"date-time":"2024-02-05T15:47:03Z","timestamp":1707148023271},"reference-count":71,"publisher":"American Physical Society (APS)","issue":"6","license":[{"start":{"date-parts":[[2020,12,2]],"date-time":"2020-12-02T00:00:00Z","timestamp":1606867200000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"funder":[{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1820939"]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physreva.102.062402","type":"journal-article","created":{"date-parts":[[2020,12,3]],"date-time":"2020-12-03T15:15:52Z","timestamp":1607008552000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":6,"title":"Minimal distances for certain quantum product codes and tensor products of chain complexes","prefix":"10.1103","volume":"102","author":[{"ORCID":"http://orcid.org/0000-0002-1871-5034","authenticated-orcid":true,"given":"Weilei","family":"Zeng","sequence":"first","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2020,12,2]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevA.102.062402","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.102.062402/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,12,3]],"date-time":"2020-12-03T15:17:57Z","timestamp":1607008677000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.102.062402"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,2]]},"references-count":71,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2020,12]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.102.062402","relation":{},"ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. 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The data and ancilla qubits of the small stellated dodecahedron code can be located on the edges respectively vertices of a small stellated dodecahedron, making this code suitable for three-dimensional connectivity. This code encodes eight logical qubits into 30 physical qubits (plus 22 ancilla qubits for parity check measurements) in contrast with one logical qubit into nine physical qubits (plus eight ancilla qubits) for the surface code. We develop fault-tolerant parity check circuits and a decoder for this code, allowing us to numerically assess the circuit-based pseudo-threshold.\n This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.","DOI":"10.1098/rsta.2017.0323","type":"journal-article","created":{"date-parts":[[2018,5,28]],"date-time":"2018-05-28T18:10:15Z","timestamp":1527531015000},"page":"20170323","update-policy":"http://dx.doi.org/10.1098/crossmark-policy","source":"Crossref","is-referenced-by-count":13,"title":"The small stellated dodecahedron code and friends","prefix":"10.1098","volume":"376","author":[{"given":"J.","family":"Conrad","sequence":"first","affiliation":[{"name":"JARA Institute for Quantum Information, RWTH Aachen University, Aachen 52056, Germany"}]},{"ORCID":"http://orcid.org/0000-0003-3239-5783","authenticated-orcid":false,"given":"C.","family":"Chamberland","sequence":"additional","affiliation":[{"name":"Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1"}]},{"given":"N. P.","family":"Breuckmann","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, London WC1E 6BT, UK"}]},{"given":"B. M.","family":"Terhal","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands"},{"name":"Institute for Theoretical Nanoelectronics, Forschungszentrum Juelich, 52425 Juelich, Germany"}]}],"member":"175","published-online":{"date-parts":[[2018,5,28]]},"reference":[],"container-title":"Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","link":[{"URL":"https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2017.0323","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://royalsocietypublishing.org/doi/full-xml/10.1098/rsta.2017.0323","content-type":"application/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2017.0323","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,18]],"date-time":"2021-02-18T19:24:22Z","timestamp":1613676262000},"score":1,"resource":{"primary":{"URL":"https://royalsocietypublishing.org/doi/10.1098/rsta.2017.0323"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,28]]},"references-count":31,"journal-issue":{"issue":"2123","published-print":{"date-parts":[[2018,7,13]]}},"alternative-id":["10.1098/rsta.2017.0323"],"URL":"http://dx.doi.org/10.1098/rsta.2017.0323","relation":{},"ISSN":["1364-503X","1471-2962"],"subject":["General Physics and Astronomy","General Engineering","General Mathematics"],"container-title-short":"Phil. 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However, determining the most suited encoding for unknown error channels or specific laboratory setups is highly challenging. Here, we present a reinforcement learning framework for optimizing and fault-tolerantly adapting quantum error correction codes. We consider a reinforcement learning agent tasked with modifying a family of surface code quantum memories until a desired logical error rate is reached. Using efficient simulations with about 70 data qubits with arbitrary connectivity, we demonstrate that such a reinforcement learning agent can determine near-optimal solutions, in terms of the number of data qubits, for various error models of interest. Moreover, we show that agents trained on one setting are able to successfully transfer their experience to different settings. 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Conventionally, protocols for fault-tolerant quantum computation demand excessive space overheads by using many physical qubits for each logical qubit. A more recent protocol using quantum analogues of low-density parity-check codes needs only a constant space overhead that does not grow with the number of logical qubits. However, the overhead in the processing time required to implement this protocol grows polynomially with the number of computational steps. To address these problems, here we introduce an alternative approach to constant-space-overhead fault-tolerant quantum computing using a concatenation of multiple small-size quantum codes rather than a single large-size quantum low-density parity-check code. We develop techniques for concatenating different quantum Hamming codes with growing size. As a result, we construct a low-overhead protocol to achieve constant space overhead and only quasi-polylogarithmic time overhead simultaneously. Our protocol is fault tolerant even if a decoder has a non-constant runtime, unlike the existing constant-space-overhead protocol. 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The robustness of magic was recently proposed as a well-behaved magic monotone for multi-qubit states and quantifies the simulation overhead of circuits composed of Clifford + Tgates, or circuits using other gates from the Clifford hierarchy. Here we present a general theory of the ‘non-stabilizerness’ of quantum operations rather than states, which are useful for classical simulation of more general circuits. We introduce two magic monotones, called channel robustness and magic capacity, which are well-defined for generaln-qubit channels and treat all stabilizer-preserving CPTP maps as free operations. We present two complementary Monte Carlo-type classical simulation algorithms with sample complexity given by these quantities and provide examples of channels where the complexity of our algorithms is exponentially better than previously known simulators. We present additional techniques that ease the difficulty of calculating our monotones for special classes of channels.","DOI":"10.1098/rspa.2019.0251","type":"journal-article","created":{"date-parts":[[2019,7,31]],"date-time":"2019-07-31T07:06:20Z","timestamp":1564556780000},"page":"20190251","update-policy":"http://dx.doi.org/10.1098/crossmark-policy","source":"Crossref","is-referenced-by-count":45,"title":"Quantifying magic for multi-qubit operations","prefix":"10.1098","volume":"475","author":[{"ORCID":"http://orcid.org/0000-0002-6059-4125","authenticated-orcid":true,"given":"James R.","family":"Seddon","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University College London, London, UK"}]},{"given":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of Sheffield, Sheffield, UK"}]}],"member":"175","published-online":{"date-parts":[[2019,7,31]]},"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.2019.0251","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://royalsocietypublishing.org/doi/full-xml/10.1098/rspa.2019.0251","content-type":"application/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2019.0251","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,24]],"date-time":"2022-09-24T22:15:47Z","timestamp":1664057747000},"score":1,"resource":{"primary":{"URL":"https://royalsocietypublishing.org/doi/10.1098/rspa.2019.0251"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7]]},"references-count":56,"journal-issue":{"issue":"2227","published-print":{"date-parts":[[2019,7]]}},"alternative-id":["10.1098/rspa.2019.0251"],"URL":"http://dx.doi.org/10.1098/rspa.2019.0251","relation":{},"ISSN":["1364-5021","1471-2946"],"subject":["General Physics and Astronomy","General Engineering","General Mathematics"],"container-title-short":"Proc. 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Our work is motivated by themagic state modelof fault-tolerant quantum computation. In this model, all unitaries belong to the Clifford group. Non-Clifford operations are effected by injecting non-stabiliser states, which are referred to asmagic statesin this context. Therobustness of magicmeasures the complexity of simulating such a circuit using a classical Monte Carlo algorithm. It is closely related to the degree negativity that slows down Monte Carlo simulations through the infamoussign problem. Surprisingly, the robustness of magic issub- multiplicative. This implies that the classical simulation overhead scales subexponentially with the number of injected magic states - better than a naive analysis would suggest. However, determining the robustness ofncopies of a magic state is difficult, as its definition involves a convex optimisation problem in a 4n-dimensional space. In this paper, we make use of inherent symmetries to reduce the problem tondimensions. The total run-time of our algorithm, while still exponential inn, is super-polynomially faster than previously published methods. We provide a computer implementation and give the robustness of up to 10 copies of the most commonly used magic states. Guided by the exact results, we find a finite hierarchy of approximate solutions where each level can be evaluated in polynomial time and yields rigorous upper bounds to the robustness. Technically, we use symmetries of the stabiliser polytope to connect the robustness of magic to the geometry of a low-dimensional convex polytope generated by certainsigned quantum weight enumerators. As a by-product, we characterised the automorphism group of the stabiliser polytope, and, more generally, of projections onto complex projective 3-designs.","DOI":"10.22331/q-2019-04-08-132","type":"journal-article","created":{"date-parts":[[2019,4,8]],"date-time":"2019-04-08T16:26:35Z","timestamp":1554740795000},"page":"132","source":"Crossref","is-referenced-by-count":36,"title":"Robustness of Magic and Symmetries of the Stabiliser Polytope","prefix":"10.22331","volume":"3","author":[{"ORCID":"http://orcid.org/0000-0002-1334-7885","authenticated-orcid":false,"given":"Markus","family":"Heinrich","sequence":"first","affiliation":[{"name":"Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany"}]},{"given":"David","family":"Gross","sequence":"additional","affiliation":[{"name":"Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2019,4,8]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2019-04-08-132/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,9,15]],"date-time":"2022-09-15T13:40:54Z","timestamp":1663249254000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2019-04-08-132/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,8]]},"references-count":42,"URL":"http://dx.doi.org/10.22331/q-2019-04-08-132","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2019,4,8]]},"article-number":"132","id":"doi:10.22331/q-2019-04-08-132","_hash":"139f3eec56ec930d1ba448c3e4a861bde151f1d1973d88c68e5e0b0305df1849"},"expire":1740764071748},"doi:10.1103/PhysRevA.59.141":{"value":{"indexed":{"date-parts":[[2024,3,1]],"date-time":"2024-03-01T14:31:29Z","timestamp":1709303489400},"reference-count":19,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[1999,1,1]],"date-time":"1999-01-01T00:00:00Z","timestamp":915148800000},"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.59.141","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T01:41:14Z","timestamp":1027734074000},"page":"141-155","source":"Crossref","is-referenced-by-count":431,"title":"Robustness of entanglement","prefix":"10.1103","volume":"59","author":[{"given":"Guifré","family":"Vidal","sequence":"first","affiliation":[]},{"given":"Rolf","family":"Tarrach","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[1999,1,1]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevA.59.141","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.59.141/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,15]],"date-time":"2017-06-15T08:22:32Z","timestamp":1497514952000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.59.141"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,1,1]]},"references-count":19,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1999,1]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.59.141","relation":{},"ISSN":["1050-2947","1094-1622"],"subject":["Atomic and Molecular Physics, and Optics"],"container-title-short":"Phys. 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Diagonal gates such as the transversal T gate play an important role in implementing a universal set of quantum operations. This paper introduces a framework that describes the process of preparing a code state, applying a diagonal physical gate, measuring a code syndrome, and applying a Pauli correction that may depend on the measured syndrome (the average logical channel induced by an arbitrary diagonal gate). It focuses on CSS codes, and describes the interaction of code states and physical gates in terms of generator coefficients determined by the induced logical operator. The interaction of code states and diagonal gates depends very strongly on the signs of Z-stabilizers in the CSS code, and the proposed generator coefficient framework explicitly includes this degree of freedom. The paper derives necessary and sufficient conditions for an arbitrary diagonal gate to preserve the code space of a stabilizer code, and provides an explicit expression of the induced logical operator. When the diagonal gate is a quadratic form diagonal gate (introduced by Rengaswamy et al.), the conditions can be expressed in terms of divisibility of weights in the two classical codes that determine the CSS code. These codes find application in magic state distillation and elsewhere. When all the signs are positive, the paper characterizes all possible CSS codes, invariant under transversal Z-rotation through &#x03C0;/2l, that are constructed from classical Reed-Muller codes by deriving the necessary and sufficient constraints on l. The generator coefficient framework extends to arbitrary stabilizer codes but there is nothing to be gained by considering the more general class of non-degenerate stabilizer codes.","DOI":"10.22331/q-2022-09-08-802","type":"journal-article","created":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T15:16:45Z","timestamp":1662650205000},"page":"802","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":"Designing the Quantum Channels Induced by Diagonal Gates","prefix":"10.22331","volume":"6","author":[{"ORCID":"http://orcid.org/0000-0002-2699-3966","authenticated-orcid":false,"given":"Jingzhen","family":"Hu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Duke University, Durham, NC 27708, USA"}]},{"ORCID":"http://orcid.org/0000-0002-5073-9431","authenticated-orcid":false,"given":"Qingzhong","family":"Liang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Duke 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Science, University College London, London, U.K."}]}],"member":"263","reference":[],"container-title":"IEEE Transactions on Information Theory","original-title":[],"link":[{"URL":"http://xplorestaging.ieee.org/ielx7/18/9861208/09763547.pdf?arnumber=9763547","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,22]],"date-time":"2024-01-22T22:49:39Z","timestamp":1705963779000},"score":1,"resource":{"primary":{"URL":"https://ieeexplore.ieee.org/document/9763547/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9]]},"references-count":58,"journal-issue":{"issue":"9"},"URL":"http://dx.doi.org/10.1109/TIT.2022.3170846","relation":{},"ISSN":["0018-9448","1557-9654"],"subject":["Library and Information Sciences","Computer Science Applications","Information Systems"],"container-title-short":"IEEE Trans. Inform. Theory","published":{"date-parts":[[2022,9]]},"id":"doi:10.1109/TIT.2022.3170846","_hash":"57d704dc1978de81b7dcbe8e203ac4a9dc1854ed4fba86ab44b2374cf75dc312"},"expire":1740764080417},"doi:10.22331/q-2018-06-07-71":{"value":{"indexed":{"date-parts":[[2024,1,22]],"date-time":"2024-01-22T23:53:08Z","timestamp":1705967588990},"reference-count":31,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2018,6,7]],"date-time":"2018-06-07T00:00:00Z","timestamp":1528329600000},"content-version":"unspecified","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"We present several different codes and protocols to distillT, controlled-S, and Toffoli (orCCZ) gates. One construction is based on codes that generalize the triorthogonal codes, allowing any of these gates to be induced at the logical level by transversalT. We present a randomized construction of generalized triorthogonal codes obtaining an asymptotic distillation efficiencyγ→1. We also present a Reed-Muller based construction of these codes which obtains a worseγbut performs well at small sizes. Additionally, we present protocols based on checking the stabilizers ofCCZmagic states at the logical level by transversal gates applied to codes; these protocols generalize the protocols of. Several examples, including a Reed-Muller code forT-to-Toffoli distillation, punctured Reed-Muller codes forT-gate distillation, and some of the check based protocols, require a lower ratio of input gates to output gates than other known protocols at the given order of error correction for the given code size. In particular, we find a512T-gate to10Toffoli gate code with distance8as well as triorthogonal codes with parameters[[887,137,5]],[[912,112,6]],[[937,87,7]]with very low prefactors in front of the leading order error terms in those codes.","DOI":"10.22331/q-2018-06-07-71","type":"journal-article","created":{"date-parts":[[2018,6,7]],"date-time":"2018-06-07T16:37:15Z","timestamp":1528389435000},"page":"71","source":"Crossref","is-referenced-by-count":37,"title":"Codes and Protocols for DistillingT, controlled-S, and Toffoli Gates","prefix":"10.22331","volume":"2","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Quantum Architectures and Computation Group, 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 Group, Microsoft Research, Redmond, WA 98052, USA"}]}],"member":"9598","published-online":{"date-parts":[[2018,6,7]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","deposited":{"date-parts":[[2022,8,25]],"date-time":"2022-08-25T06:14:10Z","timestamp":1661408050000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2018-06-07-71/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,7]]},"references-count":31,"URL":"http://dx.doi.org/10.22331/q-2018-06-07-71","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2018,6,7]]},"article-number":"71","id":"doi:10.22331/q-2018-06-07-71","_hash":"203849329702b7f60c0729af4f945da288580a543efc4560f89121ee343180b0"},"expire":1740764082728},"doi:10.1007/JHEP12(2022)124":{"value":{"indexed":{"date-parts":[[2024,2,14]],"date-time":"2024-02-14T11:07:36Z","timestamp":1707908856983},"reference-count":75,"publisher":"Springer Science and Business Media LLC","issue":"12","license":[{"start":{"date-parts":[[2022,12,21]],"date-time":"2022-12-21T00:00:00Z","timestamp":1671580800000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,12,21]],"date-time":"2022-12-21T00:00:00Z","timestamp":1671580800000},"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":"Abstract\n Holography has taught us that spacetime is emergent and its properties depend on the entanglement structure of the dual theory. In this paper, we describe how changes in the entanglement due to a local projective measurement (LPM) on a subregion A of the boundary theory modify the bulk dual spacetime. We find that LPMs destroy portions of the bulk geometry, yielding post-measurement bulk spacetimes dual to the complementary unmeasured region Ac that are cut off by end-of-the-world branes. Using a bulk calculation in AdS3 and tensor network models of holography (in particular, the HaPPY code and random tensor networks), we show that the portions of the bulk geometry that are preserved after the measurement depend on the size of A and the state we project onto. The post-measurement bulk dual to Ac includes regions that were originally part of the entanglement wedge of A prior to measurement. This suggests that LPMs performed on a boundary subregion A teleport part of the bulk information originally encoded in A into the complementary region Ac. In semiclassical holography an arbitrary amount of bulk information can be teleported in this way, while in tensor network models the teleported information is upper-bounded by the amount of entanglement shared between A and Ac due to finite-N effects. When A is the union of two disjoint subregions, the measurement triggers an entangled/disentangled phase transition between the remaining two unmeasured subregions, corresponding to a connected/disconnected phase transition in the bulk description. Our results shed new light on the effects of measurement on the entanglement structure of holographic theories and give insight on how bulk information can be manipulated from the boundary theory. They could also represent a first step towards a holographic description of measurement-induced phase transitions.","DOI":"10.1007/jhep12(2022)124","type":"journal-article","created":{"date-parts":[[2022,12,22]],"date-time":"2022-12-22T07:02:51Z","timestamp":1671692571000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":"Holographic measurement and bulk teleportation","prefix":"10.1007","volume":"2022","author":[{"ORCID":"http://orcid.org/0000-0002-0950-653X","authenticated-orcid":false,"given":"Stefano","family":"Antonini","sequence":"first","affiliation":[]},{"given":"Gregory","family":"Bentsen","sequence":"additional","affiliation":[]},{"given":"ChunJun","family":"Cao","sequence":"additional","affiliation":[]},{"given":"Jonathan","family":"Harper","sequence":"additional","affiliation":[]},{"given":"Shao-Kai","family":"Jian","sequence":"additional","affiliation":[]},{"given":"Brian","family":"Swingle","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,12,21]]},"reference":[],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","link":[{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP12(2022)124.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/article/10.1007/JHEP12(2022)124/fulltext.html","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://link.springer.com/content/pdf/10.1007/JHEP12(2022)124.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,25]],"date-time":"2023-03-25T22:22:12Z","timestamp":1679782932000},"score":1,"resource":{"primary":{"URL":"https://link.springer.com/10.1007/JHEP12(2022)124"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,21]]},"references-count":75,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["19861"],"URL":"http://dx.doi.org/10.1007/JHEP12(2022)124","relation":{},"ISSN":["1029-8479"],"subject":["Nuclear and High Energy Physics"],"container-title-short":"J. High Energ. Phys.","published":{"date-parts":[[2022,12,21]]},"assertion":[{"value":"5 October 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 December 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 December 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"124","id":"doi:10.1007/JHEP12(2022)124","_hash":"226b88362499abdfc6d8296ab416732f84109e5a0aaff3d16e7ce3c4c79d95ec"},"expire":1740764083855},"doi:10.1103/PhysRevB.99.115123":{"value":{"indexed":{"date-parts":[[2024,2,23]],"date-time":"2024-02-23T15:47:06Z","timestamp":1708703226513},"reference-count":32,"publisher":"American Physical Society (APS)","issue":"11","license":[{"start":{"date-parts":[[2019,3,18]],"date-time":"2019-03-18T00:00:00Z","timestamp":1552867200000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"},{"start":{"date-parts":[[2020,3,17]],"date-time":"2020-03-17T00:00:00Z","timestamp":1584403200000},"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":["DMR-1654340"]},{"DOI":"10.13039/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher"},{"DOI":"10.13039/501100003579","name":"University of Toronto","doi-asserted-by":"publisher"}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physrevb.99.115123","type":"journal-article","created":{"date-parts":[[2019,3,18]],"date-time":"2019-03-18T13:31:53Z","timestamp":1552915913000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":38,"title":"Foliated fracton order in the checkerboard model","prefix":"10.1103","volume":"99","author":[{"given":"Wilbur","family":"Shirley","sequence":"first","affiliation":[]},{"given":"Kevin","family":"Slagle","sequence":"additional","affiliation":[]},{"given":"Xie","family":"Chen","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2019,3,18]]},"reference":[],"container-title":"Physical Review B","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/accepted/10.1103/PhysRevB.99.115123","content-type":"application/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"https://link.aps.org/article/10.1103/PhysRevB.99.115123","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevB.99.115123/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,3,18]],"date-time":"2019-03-18T13:31:55Z","timestamp":1552915915000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevB.99.115123"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,18]]},"references-count":32,"journal-issue":{"issue":"11","published-print":{"date-parts":[[2019,3]]}},"URL":"http://dx.doi.org/10.1103/PhysRevB.99.115123","relation":{},"ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","published":{"date-parts":[[2019,3,18]]},"article-number":"115123","id":"doi:10.1103/PhysRevB.99.115123","_hash":"1bcac9c34e7072dfe059d1fe3bdd68068bf4c34806bf8a623cc6ea442f4714f9"},"expire":1740764084847},"doi:10.21468/SciPostPhys.6.1.015":{"value":{"indexed":{"date-parts":[[2024,3,3]],"date-time":"2024-03-03T14:47:01Z","timestamp":1709477221590},"reference-count":48,"publisher":"Stichting SciPost","issue":"1","license":[{"start":{"date-parts":[[2019,1,31]],"date-time":"2019-01-31T00:00:00Z","timestamp":1548892800000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/100000879","name":"Alfred P. Sloan Foundation","doi-asserted-by":"publisher"},{"DOI":"10.13039/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMR-1654340"]},{"DOI":"10.13039/501100000038","name":"Natural Sciences and Engineering Research Council","doi-asserted-by":"publisher"}],"content-domain":{"domain":["scipost.org"],"crossmark-restriction":false},"abstract":"Fracton models exhibit a variety of exotic properties and lie beyond\nthe conventional framework of gapped topological order. In , we\ngeneralized the notion of gapped phase to one of foliated\nfracton phase by allowing the addition of layers of gapped\ntwo-dimensional resources in the adiabatic evolution between gapped\nthree-dimensional models. Moreover, we showed that the X-cube model is a\nfixed point of one such phase. In this paper, according to this\ndefinition, we look for universal properties of such phases which remain\ninvariant throughout the entire phase. We propose multi-partite\nentanglement quantities, generalizing the proposal of topological\nentanglement entropy designed for conventional topological phases. 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In the neural network state ansatz, the complex amplitude function of a quantum state is computed by a neural network. The resulting multipartite entanglement structure captured by this ansatz has proven rich enough to describe the ground states and unitary dynamics of various physical systems of interest. In the present paper, we initiate the study of neural network states in quantum information-processing tasks. We demonstrate that neural network states are capable of efficiently representing quantum codes for quantum information transmission and quantum error correction, supplying further evidence for the usefulness of neural network states to describe multipartite entanglement. In particular, we show the following main results: (a) neural network states yield quantum codes with a high coherent information for two important quantum channels, the generalized amplitude damping channel and the dephrasure channel. These codes outperform all other known codes for these channels, and cannot be found using a direct parametrization of the quantum state. (b) For the depolarizing channel, the neural network state ansatz reliably finds the best known codes given by repetition codes. (c) Neural network states can be used to represent absolutely maximally entangled states, a special type of quantum error-correcting codes. In all three cases, the neural network state ansatz provides an efficient and versatile means as a variational parametrization of these highly entangled states.","DOI":"10.1088/1367-2630/ab6cdd","type":"journal-article","created":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:00:34Z","timestamp":1579215634000},"page":"023005","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":32,"title":"Quantum codes from neural networks","prefix":"10.1088","volume":"22","author":[{"ORCID":"http://orcid.org/0000-0003-3189-9162","authenticated-orcid":false,"given":"Johannes","family":"Bausch","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-1073-9795","authenticated-orcid":false,"given":"Felix","family":"Leditzky","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2020,2,4]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab6cdd","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab6cdd/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab6cdd/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab6cdd","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab6cdd","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab6cdd/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab6cdd/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab6cdd/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab6cdd/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab6cdd/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab6cdd","content-type":"text/html","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab6cdd/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,24]],"date-time":"2021-11-24T22:45:30Z","timestamp":1637793930000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/ab6cdd"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,1]]},"references-count":82,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2,4]]},"published-print":{"date-parts":[[2020,2,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/ab6cdd","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":"2019-11-16","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-01-16","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-02-04","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/1367-2630/ab6cdd","_hash":"6dfda558a629a86086439e0ea98e276f787ccbe9928ef2abd6d6f630b9f18097"},"expire":1740764102380},"doi:10.1103/PhysRevX.8.031084":{"value":{"indexed":{"date-parts":[[2024,3,2]],"date-time":"2024-03-02T13:01:13Z","timestamp":1709384473106},"reference-count":56,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2018,9,27]],"date-time":"2018-09-27T00:00:00Z","timestamp":1538006400000},"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.8.031084","type":"journal-article","created":{"date-parts":[[2018,9,27]],"date-time":"2018-09-27T14:24:10Z","timestamp":1538058250000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":135,"title":"Reinforcement Learning with Neural Networks for Quantum Feedback","prefix":"10.1103","volume":"8","author":[{"given":"Thomas","family":"Fösel","sequence":"first","affiliation":[]},{"given":"Petru","family":"Tighineanu","sequence":"additional","affiliation":[]},{"given":"Talitha","family":"Weiss","sequence":"additional","affiliation":[]},{"given":"Florian","family":"Marquardt","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2018,9,27]]},"reference":[],"container-title":"Physical Review X","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevX.8.031084","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevX.8.031084/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2018,9,27]],"date-time":"2018-09-27T14:24:13Z","timestamp":1538058253000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevX.8.031084"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,27]]},"references-count":56,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2018,9]]}},"URL":"http://dx.doi.org/10.1103/PhysRevX.8.031084","relation":{},"ISSN":["2160-3308"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. 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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).","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/entities/document/10.4230/LIPIcs.ITCS.2022.68","copyright":"Creative Commons Attribution 4.0 International license","reference":[],"_hash":"54871e54aed216dbc58b69c3467b2bc25a5b10bdd6d73aac6870abe3d7f1a42c"},"expire":1740764107580},"doi:10.1103/PhysRevA.100.022304":{"value":{"indexed":{"date-parts":[[2024,3,1]],"date-time":"2024-03-01T02:29:17Z","timestamp":1709260157899},"reference-count":17,"publisher":"American Physical Society (APS)","issue":"2","license":[{"start":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T00:00:00Z","timestamp":1565136000000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"},{"start":{"date-parts":[[2020,8,6]],"date-time":"2020-08-06T00:00:00Z","timestamp":1596672000000},"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":["1718494"]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physreva.100.022304","type":"journal-article","created":{"date-parts":[[2019,8,8]],"date-time":"2019-08-08T14:56:02Z","timestamp":1565276162000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":12,"title":"Unifying the Clifford hierarchy via symmetric matrices over rings","prefix":"10.1103","volume":"100","author":[{"ORCID":"http://orcid.org/0000-0002-2369-3159","authenticated-orcid":true,"given":"Narayanan","family":"Rengaswamy","sequence":"first","affiliation":[]},{"given":"Robert","family":"Calderbank","sequence":"additional","affiliation":[]},{"given":"Henry D.","family":"Pfister","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2019,8,7]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/accepted/10.1103/PhysRevA.100.022304","content-type":"application/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"https://link.aps.org/article/10.1103/PhysRevA.100.022304","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.100.022304/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,23]],"date-time":"2019-09-23T07:43:46Z","timestamp":1569224626000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.100.022304"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,7]]},"references-count":17,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,8]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.100.022304","relation":{},"ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. 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A","published":{"date-parts":[[2019,8,7]]},"article-number":"022304","id":"doi:10.1103/PhysRevA.100.022304","_hash":"595cf4d5ed1aff6c591695fc4a6381218b2d9655e6f0cb6056b929bdf7051978"},"expire":1740764108533},"doi:10.1103/PhysRevLett.93.130502":{"value":{"indexed":{"date-parts":[[2024,3,2]],"date-time":"2024-03-02T17:19:14Z","timestamp":1709399954864},"reference-count":15,"publisher":"American Physical Society (APS)","issue":"13","license":[{"start":{"date-parts":[[2004,9,20]],"date-time":"2004-09-20T00:00:00Z","timestamp":1095638400000},"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.93.130502","type":"journal-article","created":{"date-parts":[[2004,9,20]],"date-time":"2004-09-20T22:23:46Z","timestamp":1095719026000},"source":"Crossref","is-referenced-by-count":197,"title":"Quantum Circuits for General Multiqubit Gates","prefix":"10.1103","volume":"93","author":[{"given":"Mikko","family":"Möttönen","sequence":"first","affiliation":[]},{"given":"Juha J.","family":"Vartiainen","sequence":"additional","affiliation":[]},{"given":"Ville","family":"Bergholm","sequence":"additional","affiliation":[]},{"given":"Martti M.","family":"Salomaa","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2004,9,20]]},"reference":[],"container-title":"Physical Review Letters","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevLett.93.130502","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.93.130502/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,25]],"date-time":"2021-06-25T20:46:25Z","timestamp":1624653985000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevLett.93.130502"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,9,20]]},"references-count":15,"journal-issue":{"issue":"13","published-print":{"date-parts":[[2004,9]]}},"URL":"http://dx.doi.org/10.1103/PhysRevLett.93.130502","relation":{},"ISSN":["0031-9007","1079-7114"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. Rev. Lett.","published":{"date-parts":[[2004,9,20]]},"article-number":"130502","id":"doi:10.1103/PhysRevLett.93.130502","_hash":"a8719efc1c82241cdbb921f9252587d0b5ef90f33f6722d99038804482b596d8"},"expire":1740764109210},"doi:10.1088/2058-9565/ac5d20":{"value":{"indexed":{"date-parts":[[2024,3,2]],"date-time":"2024-03-02T17:19:40Z","timestamp":1709399980513},"reference-count":43,"publisher":"IOP Publishing","issue":"4","license":[{"start":{"date-parts":[[2022,7,21]],"date-time":"2022-07-21T00:00:00Z","timestamp":1658361600000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0/"},{"start":{"date-parts":[[2022,7,21]],"date-time":"2022-07-21T00:00:00Z","timestamp":1658361600000},"content-version":"tdm","delay-in-days":0,"URL":"https://iopscience.iop.org/info/page/text-and-data-mining"}],"funder":[{"DOI":"10.13039/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA2386-18-1-4028"]},{"DOI":"10.13039/501100003246","name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek","doi-asserted-by":"publisher","award":["Rubicon Fellowship"]}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"published-print":{"date-parts":[[2022,10,1]]},"abstract":"Abstract\n We introduce an enhanced technique for strong classical simulation of quantum circuits which combines the ‘sum-of-stabilisers’ method with an automated simplification strategy based on the ZX-calculus. Recently it was shown that quantum circuits can be classically simulated by expressing the non-stabiliser gates in a circuit as magic state injections and decomposing them in chunks of 2–6 states at a time, obtaining sums of (efficiently-simulable) stabiliser states with many fewer terms than the naive approach. We adapt these techniques from the original setting of Clifford circuits with magic state injection to generic ZX-diagrams and show that, by interleaving this ‘chunked’ decomposition with a ZX-calculus-based simplification strategy, we can obtain stabiliser decompositions that are many orders of magnitude smaller than existing approaches. We illustrate this technique to perform exact norm calculations (and hence strong simulation) on the outputs of random 50- and 100-qubit Clifford + T circuits with up to 70 T-gates as well as a family of hidden shift circuits previously considered by Bravyi and Gosset with over 1000 T-gates.","DOI":"10.1088/2058-9565/ac5d20","type":"journal-article","created":{"date-parts":[[2022,3,11]],"date-time":"2022-03-11T22:46:36Z","timestamp":1647038796000},"page":"044001","update-policy":"http://dx.doi.org/10.1088/crossmark-policy","source":"Crossref","is-referenced-by-count":11,"title":"Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions","prefix":"10.1088","volume":"7","author":[{"ORCID":"http://orcid.org/0000-0002-6090-9684","authenticated-orcid":true,"given":"Aleks","family":"Kissinger","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-5405-8959","authenticated-orcid":true,"given":"John","family":"van de Wetering","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2022,7,21]]},"reference":[],"container-title":"Quantum Science and Technology","original-title":[],"link":[{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ac5d20","content-type":"text/html","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ac5d20/pdf","content-type":"application/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ac5d20","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ac5d20/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ac5d20/pdf","content-type":"application/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ac5d20/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ac5d20/pdf","content-type":"application/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ac5d20/pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,7,21]],"date-time":"2022-07-21T07:32:56Z","timestamp":1658388776000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/2058-9565/ac5d20"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,21]]},"references-count":43,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,7,21]]},"published-print":{"date-parts":[[2022,10,1]]}},"URL":"http://dx.doi.org/10.1088/2058-9565/ac5d20","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":[[2022,7,21]]},"assertion":[{"value":"Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions","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":"© 2022 The Author(s). Published by IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2021-11-16","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2022-03-11","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2022-07-21","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}],"id":"doi:10.1088/2058-9565/ac5d20","_hash":"1d1b57342590cdbf0bfe7e3fc6a54eb3cb28ae7c98a62a761161dde507f13f17"},"expire":1740764110636},"doi:10.4204/EPTCS.318.9":{"value":{"indexed":{"date-parts":[[2024,3,2]],"date-time":"2024-03-02T17:18:35Z","timestamp":1709399915123},"reference-count":13,"publisher":"Open Publishing Association","content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.4204/eptcs.318.9","type":"journal-article","created":{"date-parts":[[2020,4,30]],"date-time":"2020-04-30T12:54:56Z","timestamp":1588251296000},"page":"131-149","source":"Crossref","is-referenced-by-count":12,"title":"Techniques to Reduce π/4-Parity-Phase Circuits, Motivated by the ZX Calculus","prefix":"10.4204","volume":"318","author":[{"given":"Niel","family":"de Beaudrap","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Oxford"}]},{"given":"Xiaoning","family":"Bian","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Dalhousie University"}]},{"given":"Quanlong","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Oxford; Cambridge Quantum Computing Ltd."}]}],"member":"2720","published-online":{"date-parts":[[2020,5,1]]},"reference":[],"container-title":"Electronic Proceedings in Theoretical Computer Science","original-title":[],"language":"en","deposited":{"date-parts":[[2020,5,14]],"date-time":"2020-05-14T03:09:52Z","timestamp":1589425792000},"score":1,"resource":{"primary":{"URL":"http://arxiv.org/abs/1911.09039v2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,1]]},"references-count":13,"URL":"http://dx.doi.org/10.4204/EPTCS.318.9","relation":{},"ISSN":["2075-2180"],"subject":["Psychiatry and Mental health"],"container-title-short":"Electron. Proc. Theor. Comput. Sci.","published":{"date-parts":[[2020,5,1]]},"id":"doi:10.4204/EPTCS.318.9","_hash":"b563f9060be0446d01afb7708b8599069191c27395e91b9c9191e3af1b479054"},"expire":1740764111590},"doi:10.1103/PhysRevA.102.022406":{"value":{"indexed":{"date-parts":[[2024,3,2]],"date-time":"2024-03-02T17:21:06Z","timestamp":1709400066381},"reference-count":29,"publisher":"American Physical Society (APS)","issue":"2","license":[{"start":{"date-parts":[[2020,8,11]],"date-time":"2020-08-11T00:00:00Z","timestamp":1597104000000},"content-version":"vor","delay-in-days":0,"URL":"https://link.aps.org/licenses/aps-default-license"}],"funder":[{"DOI":"10.13039/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA2386-18-1-4028"]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"DOI":"10.1103/physreva.102.022406","type":"journal-article","created":{"date-parts":[[2020,8,11]],"date-time":"2020-08-11T16:21:58Z","timestamp":1597162918000},"update-policy":"http://dx.doi.org/10.1103/crossmark-policy","source":"Crossref","is-referenced-by-count":56,"title":"Reducing the number of non-Clifford gates in quantum circuits","prefix":"10.1103","volume":"102","author":[{"ORCID":"http://orcid.org/0000-0002-6090-9684","authenticated-orcid":true,"given":"Aleks","family":"Kissinger","sequence":"first","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-5405-8959","authenticated-orcid":true,"given":"John","family":"van de Wetering","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2020,8,11]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"https://link.aps.org/article/10.1103/PhysRevA.102.022406","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.102.022406/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,8,11]],"date-time":"2020-08-11T16:24:13Z","timestamp":1597163053000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.102.022406"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,11]]},"references-count":29,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,8]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.102.022406","relation":{},"ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","published":{"date-parts":[[2020,8,11]]},"article-number":"022406","id":"doi:10.1103/PhysRevA.102.022406","_hash":"e66aae849ddb1f39341c6fc32a23d617cd4d4d386137bb1c48f9d9d83f9e14f4"},"expire":1740764112411},"doi:10.1038/s41534-022-00651-y":{"value":{"indexed":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T08:15:46Z","timestamp":1699863346427},"reference-count":53,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T00:00:00Z","timestamp":1669766400000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T00:00:00Z","timestamp":1669766400000},"content-version":"vor","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"}],"funder":[{"DOI":"10.13039/501100000023","name":"Government of Canada","doi-asserted-by":"publisher"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"abstract":"AbstractWe design an algorithm to determine the (minimum) T-count of any n-qubit (n ≥ 1) unitary W of size 2n × 2n, over the Clifford+T gate set. The space and time complexity of our algorithm are $$O\\left({2}^{2n}\\right)$$\n \n O\n \n \n \n \n 2\n \n \n 2\n n\n \n \n \n \n \n and $$O\\left({2}^{2n{{{{\\mathcal{T}}}}}_{\\epsilon }(W)+4n}\\right)$$\n \n O\n \n \n \n \n 2\n \n \n 2\n n\n \n \n T\n \n \n ϵ\n \n \n \n (\n \n W\n \n )\n \n +\n 4\n n\n \n \n \n \n \n , respectively. $${{{{\\mathcal{T}}}}}_{\\epsilon }(W)$$\n \n \n \n T\n \n \n ϵ\n \n \n \n (\n \n W\n \n )\n \n \n (ϵ-T-count) is the (minimum) T-count of an exactly implementable unitary U ($${{{\\mathcal{T}}}}(U)$$\n \n T\n \n (\n \n U\n \n )\n \n \n ), such that d(U,W) ≤ ϵ and $${{{\\mathcal{T}}}}(U)\\le {{{\\mathcal{T}}}}({U}^{{\\prime} })$$\n \n T\n \n (\n \n U\n \n )\n \n ≤\n T\n \n (\n \n \n \n U\n \n \n ′\n \n \n \n )\n \n \n where $${U}^{{\\prime} }$$\n \n \n U\n \n \n ′\n \n \n is any exactly implementable unitary with $$d({U}^{{\\prime} },W)\\le \\epsilon$$\n \n d\n \n (\n \n \n \n U\n \n \n ′\n \n \n ,\n W\n \n )\n \n ≤\n ϵ\n \n . d(. , .) is the global phase invariant distance. Our algorithm can also be used to determine the (minimum) T-depth as well as the minimum non-Clifford-gate count or depth required to implement any multi-qubit unitary with a finite universal gate set like Clifford+CS, Clifford+V, etc. For small enough ϵ, we can synthesize the optimal circuits.","DOI":"10.1038/s41534-022-00651-y","type":"journal-article","created":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T12:03:54Z","timestamp":1669809834000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":"T-count and T-depth of any multi-qubit unitary","prefix":"10.1038","volume":"8","author":[{"given":"Vlad","family":"Gheorghiu","sequence":"first","affiliation":[]},{"given":"Michele","family":"Mosca","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-6463-9100","authenticated-orcid":false,"given":"Priyanka","family":"Mukhopadhyay","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,11,30]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41534-022-00651-y.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-022-00651-y","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-022-00651-y.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T12:06:02Z","timestamp":1669809962000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-022-00651-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,30]]},"references-count":53,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["651"],"URL":"http://dx.doi.org/10.1038/s41534-022-00651-y","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":[[2022,11,30]]},"assertion":[{"value":"6 March 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 November 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 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":"141","id":"doi:10.1038/s41534-022-00651-y","_hash":"cc47cb29fb74b01559346aa1986ff0ca5b391e3ea7237b12450db0e9c7709c60"},"expire":1740764113276},"doi:10.1038/s41534-018-0072-4":{"value":{"indexed":{"date-parts":[[2024,3,2]],"date-time":"2024-03-02T17:19:22Z","timestamp":1709399962961},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2018,5,10]],"date-time":"2018-05-10T00:00:00Z","timestamp":1525910400000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2018,5,10]],"date-time":"2018-05-10T00:00:00Z","timestamp":1525910400000},"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":"AbstractWe develop and implement automated methods for optimizing quantum circuits of the size and type expected in quantum computations that outperform classical computers. We show how to handle continuous gate parameters and report a collection of fast algorithms capable of optimizing large-scale quantum circuits. For the suite of benchmarks considered, we obtain substantial reductions in gate counts. In particular, we provide better optimization in significantly less time than previous approaches, while making minimal structural changes so as to preserve the basic layout of the underlying quantum algorithms. Our results help bridge the gap between the computations that can be run on existing hardware and those that are expected to outperform classical computers.","DOI":"10.1038/s41534-018-0072-4","type":"journal-article","created":{"date-parts":[[2018,5,4]],"date-time":"2018-05-04T12:47:44Z","timestamp":1525438064000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":114,"title":"Automated optimization of large quantum circuits with continuous parameters","prefix":"10.1038","volume":"4","author":[{"given":"Yunseong","family":"Nam","sequence":"first","affiliation":[]},{"given":"Neil J.","family":"Ross","sequence":"additional","affiliation":[]},{"given":"Yuan","family":"Su","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0002-9903-837X","authenticated-orcid":false,"given":"Andrew M.","family":"Childs","sequence":"additional","affiliation":[]},{"ORCID":"http://orcid.org/0000-0001-7381-4556","authenticated-orcid":false,"given":"Dmitri","family":"Maslov","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,5,10]]},"reference":[],"container-title":"npj Quantum Information","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/s41534-018-0072-4.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-018-0072-4","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/s41534-018-0072-4.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T22:53:36Z","timestamp":1671576816000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/s41534-018-0072-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,10]]},"references-count":41,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,12]]}},"alternative-id":["72"],"URL":"http://dx.doi.org/10.1038/s41534-018-0072-4","relation":{},"ISSN":["2056-6387"],"subject":["Computational 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Systems","original-title":[],"link":[{"URL":"http://xplorestaging.ieee.org/ielx7/43/6899718/06899791.pdf?arnumber=6899791","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,12]],"date-time":"2022-01-12T16:52:07Z","timestamp":1642006327000},"score":1,"resource":{"primary":{"URL":"https://ieeexplore.ieee.org/document/6899791"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,10]]},"references-count":39,"journal-issue":{"issue":"10"},"URL":"http://dx.doi.org/10.1109/TCAD.2014.2341953","relation":{},"ISSN":["0278-0070","1937-4151"],"subject":["Electrical and Electronic Engineering","Computer Graphics and Computer-Aided Design","Software"],"container-title-short":"IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.","published":{"date-parts":[[2014,10]]},"id":"doi:10.1109/TCAD.2014.2341953","_hash":"bf3f197d999f74cb2613c619febfea637182377236c10a09c0e518ff339d3659"},"expire":1740764115215},"doi:10.1038/ncomms15043":{"value":{"indexed":{"date-parts":[[2024,3,4]],"date-time":"2024-03-04T03:31:00Z","timestamp":1709523060049},"reference-count":69,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2017,4,26]],"date-time":"2017-04-26T00:00:00Z","timestamp":1493164800000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2017,4,26]],"date-time":"2017-04-26T00:00:00Z","timestamp":1493164800000},"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":"AbstractQuantum communications promises reliable transmission of quantum information, efficient distribution of entanglement and generation of completely secure keys. For all these tasks, we need to determine the optimal point-to-point rates that are achievable by two remote parties at the ends of a quantum channel, without restrictions on their local operations and classical communication, which can be unlimited and two-way. These two-way assisted capacities represent the ultimate rates that are reachable without quantum repeaters. Here, by constructing an upper bound based on the relative entropy of entanglement and devising a dimension-independent technique dubbed ‘teleportation stretching’, we establish these capacities for many fundamental channels, namely bosonic lossy channels, quantum-limited amplifiers, dephasing and erasure channels in arbitrary dimension. In particular, we exactly determine the fundamental rate-loss tradeoff affecting any protocol of quantum key distribution. Our findings set the limits of point-to-point quantum communications and provide precise and general benchmarks for quantum repeaters.","DOI":"10.1038/ncomms15043","type":"journal-article","created":{"date-parts":[[2017,4,26]],"date-time":"2017-04-26T11:15:20Z","timestamp":1493205320000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":818,"title":"Fundamental limits of repeaterless quantum communications","prefix":"10.1038","volume":"8","author":[{"ORCID":"http://orcid.org/0000-0001-6165-5615","authenticated-orcid":false,"given":"Stefano","family":"Pirandola","sequence":"first","affiliation":[]},{"given":"Riccardo","family":"Laurenza","sequence":"additional","affiliation":[]},{"given":"Carlo","family":"Ottaviani","sequence":"additional","affiliation":[]},{"given":"Leonardo","family":"Banchi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,4,26]]},"reference":[],"container-title":"Nature Communications","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/ncomms15043.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/ncomms15043","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http://www.nature.com/doifinder/10.1038/ncomms15043","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https://www.nature.com/articles/ncomms15043.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,23]],"date-time":"2022-12-23T00:29:57Z","timestamp":1671755397000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/ncomms15043"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,26]]},"references-count":69,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,4,28]]}},"alternative-id":["BFncomms15043"],"URL":"http://dx.doi.org/10.1038/ncomms15043","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":[[2017,4,26]]},"assertion":[{"value":"15 March 2016","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 February 2017","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 April 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":"15043","id":"doi:10.1038/ncomms15043","_hash":"a5d1bef1812c97288d17e1f8878fb7374e404e349da642ef56a16d105695e960"},"expire":1740764116310},"doi:10.1063/1.1498491":{"value":{"indexed":{"date-parts":[[2024,2,28]],"date-time":"2024-02-28T18:18:35Z","timestamp":1709144315425},"reference-count":7,"publisher":"AIP Publishing","issue":"9","content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"published-print":{"date-parts":[[2002,9,1]]},"abstract":"A conjecture arising naturally in the investigation of additivity of classical information capacity of quantum channels states that the maximal purity of outputs from a quantum channel, as measured by the p-norm, should be multiplicative with respect to the tensor product of channels. We disprove this conjecture for p&gt;4.79. The same example (with p=∞) also disproves a conjecture for the multiplicativity of the injective norm of Hilbert space tensor products.","DOI":"10.1063/1.1498491","type":"journal-article","created":{"date-parts":[[2002,9,18]],"date-time":"2002-09-18T17:44:03Z","timestamp":1032371043000},"page":"4353-4357","update-policy":"http://dx.doi.org/10.1063/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":118,"title":"Counterexample to an additivity conjecture for output purity of quantum channels","prefix":"10.1063","volume":"43","author":[{"given":"R. F.","family":"Werner","sequence":"first","affiliation":[{"name":"Institut für Mathematische Physik, TU Braunschweig, Mendelssohnstr.3, 38106 Braunschweig, Germany"}]},{"given":"A. S.","family":"Holevo","sequence":"additional","affiliation":[{"name":"Steklov Mathematical Institute, Gubkina 8, 117966 Moscow, Russia"}]}],"member":"317","reference":[],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","link":[{"URL":"https://pubs.aip.org/aip/jmp/article-pdf/43/9/4353/19182938/4353_1_online.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https://pubs.aip.org/aip/jmp/article-pdf/43/9/4353/19182938/4353_1_online.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,6]],"date-time":"2024-02-06T05:53:54Z","timestamp":1707198834000},"score":1,"resource":{"primary":{"URL":"https://pubs.aip.org/jmp/article/43/9/4353/230894/Counterexample-to-an-additivity-conjecture-for"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,9,1]]},"references-count":7,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2002,9,1]]}},"URL":"http://dx.doi.org/10.1063/1.1498491","relation":{},"ISSN":["0022-2488","1089-7658"],"subject":["Mathematical Physics","Statistical and Nonlinear Physics"],"published-other":{"date-parts":[[2002,9]]},"published":{"date-parts":[[2002,9,1]]},"id":"doi:10.1063/1.1498491","_hash":"eac76cfcfb39293b8779f172938240a7a813f098eb34f1c9e2fe1587faec81cb"},"expire":1740764117207},"doi:10.1103/PhysRevA.95.012304":{"value":{"indexed":{"date-parts":[[2024,3,1]],"date-time":"2024-03-01T01:17:49Z","timestamp":1709255869542},"reference-count":18,"publisher":"American Physical Society (APS)","issue":"1","license":[{"start":{"date-parts":[[2017,1,4]],"date-time":"2017-01-04T00:00:00Z","timestamp":1483488000000},"content-version":"vor","delay-in-days":0,"URL":"http://link.aps.org/licenses/aps-default-license"},{"start":{"date-parts":[[2018,1,4]],"date-time":"2018-01-04T00:00:00Z","timestamp":1515024000000},"content-version":"am","delay-in-days":365,"URL":"http://link.aps.org/licenses/aps-default-accepted-manuscript-license"}],"funder":[{"DOI":"10.13039/100000185","name":"Defense Advanced Research Projects Agency","doi-asserted-by":"publisher"},{"DOI":"10.13039/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-14-1-0052"]},{"DOI":"10.13039/100006602","name":"Air Force Research Laboratory","doi-asserted-by":"publisher","award":["FA8750-14-2-0120"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1103/physreva.95.012304","type":"journal-article","created":{"date-parts":[[2017,1,4]],"date-time":"2017-01-04T22:28:57Z","timestamp":1483568937000},"source":"Crossref","is-referenced-by-count":93,"title":"Rate-distance tradeoff and resource costs for all-optical quantum repeaters","prefix":"10.1103","volume":"95","author":[{"given":"Mihir","family":"Pant","sequence":"first","affiliation":[]},{"given":"Hari","family":"Krovi","sequence":"additional","affiliation":[]},{"given":"Dirk","family":"Englund","sequence":"additional","affiliation":[]},{"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2017,1,4]]},"reference":[],"container-title":"Physical Review A","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/accepted/10.1103/PhysRevA.95.012304","content-type":"application/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"http://link.aps.org/article/10.1103/PhysRevA.95.012304","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.95.012304/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,4,6]],"date-time":"2017-04-06T14:56:04Z","timestamp":1491490564000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevA.95.012304"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1,4]]},"references-count":18,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,1]]}},"URL":"http://dx.doi.org/10.1103/PhysRevA.95.012304","relation":{},"ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. 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A","published":{"date-parts":[[2017,1,4]]},"article-number":"012304","id":"doi:10.1103/PhysRevA.95.012304","_hash":"faa8697d0789e3a059b907494c6178f3cb2cc6ac5d3ecc83743ec600a1a151aa"},"expire":1740764118343},"doi:10.1038/ncomms7787":{"value":{"indexed":{"date-parts":[[2024,3,3]],"date-time":"2024-03-03T14:28:43Z","timestamp":1709476123287},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2015,4,15]],"date-time":"2015-04-15T00:00:00Z","timestamp":1429056000000},"content-version":"tdm","delay-in-days":0,"URL":"https://creativecommons.org/licenses/by/4.0"},{"start":{"date-parts":[[2015,4,15]],"date-time":"2015-04-15T00:00:00Z","timestamp":1429056000000},"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":"AbstractQuantum communication holds promise for unconditionally secure transmission of secret messages and faithful transfer of unknown quantum states. Photons appear to be the medium of choice for quantum communication. Owing to photon losses, robust quantum communication over long lossy channels requires quantum repeaters. It is widely believed that a necessary and highly demanding requirement for quantum repeaters is the existence of matter quantum memories. Here we show that such a requirement is, in fact, unnecessary by introducing the concept of all-photonic quantum repeaters based on flying qubits. In particular, we present a protocol based on photonic cluster-state machine guns and a loss-tolerant measurement equipped with local high-speed active feedforwards. We show that, with such all-photonic quantum repeaters, the communication efficiency scales polynomially with the channel distance. Our result paves a new route towards quantum repeaters with efficient single-photon sources rather than matter quantum memories.","DOI":"10.1038/ncomms7787","type":"journal-article","created":{"date-parts":[[2015,4,15]],"date-time":"2015-04-15T12:16:15Z","timestamp":1429100175000},"update-policy":"http://dx.doi.org/10.1007/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":336,"title":"All-photonic quantum repeaters","prefix":"10.1038","volume":"6","author":[{"given":"Koji","family":"Azuma","sequence":"first","affiliation":[]},{"given":"Kiyoshi","family":"Tamaki","sequence":"additional","affiliation":[]},{"given":"Hoi-Kwong","family":"Lo","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2015,4,15]]},"reference":[],"container-title":"Nature Communications","original-title":[],"language":"en","link":[{"URL":"https://www.nature.com/articles/ncomms7787.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/ncomms7787","content-type":"text/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https://www.nature.com/articles/ncomms7787.pdf","content-type":"application/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,5]],"date-time":"2023-01-05T11:49:24Z","timestamp":1672919364000},"score":1,"resource":{"primary":{"URL":"https://www.nature.com/articles/ncomms7787"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,4,15]]},"references-count":51,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,11,3]]}},"alternative-id":["BFncomms7787"],"URL":"http://dx.doi.org/10.1038/ncomms7787","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":[[2015,4,15]]},"assertion":[{"value":"17 October 2014","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 February 2015","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 April 2015","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":"6787","id":"doi:10.1038/ncomms7787","_hash":"6181d824bc869720ed559d9809cfb18c7028f8f496094f324a5f480b314ced79"},"expire":1740764119772},"doi:10.1103/PhysRevLett.97.120501":{"value":{"indexed":{"date-parts":[[2024,1,29]],"date-time":"2024-01-29T12:21:34Z","timestamp":1706530894167},"reference-count":17,"publisher":"American Physical Society (APS)","issue":"12","license":[{"start":{"date-parts":[[2006,9,20]],"date-time":"2006-09-20T00:00:00Z","timestamp":1158710400000},"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.97.120501","type":"journal-article","created":{"date-parts":[[2006,9,20]],"date-time":"2006-09-20T18:33:47Z","timestamp":1158777227000},"source":"Crossref","is-referenced-by-count":145,"title":"Loss Tolerance in One-Way Quantum Computation via Counterfactual Error Correction","prefix":"10.1103","volume":"97","author":[{"given":"Michael","family":"Varnava","sequence":"first","affiliation":[]},{"given":"Daniel E.","family":"Browne","sequence":"additional","affiliation":[]},{"given":"Terry","family":"Rudolph","sequence":"additional","affiliation":[]}],"member":"16","published-online":{"date-parts":[[2006,9,20]]},"reference":[],"container-title":"Physical Review Letters","original-title":[],"language":"en","link":[{"URL":"http://link.aps.org/article/10.1103/PhysRevLett.97.120501","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.97.120501/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,4,5]],"date-time":"2017-04-05T22:45:27Z","timestamp":1491432327000},"score":1,"resource":{"primary":{"URL":"https://link.aps.org/doi/10.1103/PhysRevLett.97.120501"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,9,20]]},"references-count":17,"journal-issue":{"issue":"12","published-print":{"date-parts":[[2006,9]]}},"URL":"http://dx.doi.org/10.1103/PhysRevLett.97.120501","relation":{},"ISSN":["0031-9007","1079-7114"],"subject":["General Physics and Astronomy"],"container-title-short":"Phys. 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However, the hardware path from nearest-neighbor-connection-based topological codes to long-range-interaction-demanding QLDPC codes is likely a challenging one. Given the practical difficulty in building a monolithic architecture for quantum systems, such as computers, based on optimal QLDPC codes, it is worth considering a distributed implementation of such codes over a network of interconnected medium-sized quantum processors. In such a setting, all syndrome measurements and logical operations must be performed through the use of high-fidelity shared entangled states between the processing nodes. Since probabilistic many-to-1 distillation schemes for purifying entanglement are inefficient, we investigate quantum error correction based entanglement purification in this work. Specifically, we employ QLDPC codes to distill GHZ states, as the resulting high-fidelity logical GHZ states can interact directly with the code used to perform distributed quantum computing (DQC), e.g. for fault-tolerant Steane syndrome extraction. This protocol is applicable beyond the application of DQC since entanglement distribution and purification is a quintessential task of any quantum network. We use the min-sum algorithm (MSA) based iterative decoder with a sequential schedule for distilling 3-qubit GHZ states using a rate 0.118 family of lifted product QLDPC codes and obtain an input fidelity threshold of &#x2248;0.7974 under i.i.d. single-qubit depolarizing noise. This represents the best threshold for a yield of 0.118 for any GHZ purification protocol. Our results apply to larger size GHZ states as well, where we extend our technical result about a measurement property of 3-qubit GHZ states to construct a scalable GHZ purification protocol. ","DOI":"10.22331/q-2024-01-24-1233","type":"journal-article","created":{"date-parts":[[2024,1,29]],"date-time":"2024-01-29T16:33:16Z","timestamp":1706545996000},"page":"1233","update-policy":"http://dx.doi.org/10.22331/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":"Entanglement Purification with Quantum LDPC Codes and Iterative Decoding","prefix":"10.22331","volume":"8","author":[{"ORCID":"http://orcid.org/0000-0002-2369-3159","authenticated-orcid":false,"given":"Narayanan","family":"Rengaswamy","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]},{"ORCID":"http://orcid.org/0000-0002-1024-8099","authenticated-orcid":false,"given":"Nithin","family":"Raveendran","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]},{"ORCID":"http://orcid.org/0000-0002-9022-3595","authenticated-orcid":false,"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"}]},{"ORCID":"http://orcid.org/0000-0003-2365-4106","authenticated-orcid":false,"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":[[2024,1,24]]},"reference":[],"container-title":"Quantum","original-title":[],"language":"en","link":[{"URL":"https://quantum-journal.org/papers/q-2024-01-24-1233/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2024,1,29]],"date-time":"2024-01-29T16:33:24Z","timestamp":1706546004000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2024-01-24-1233/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,24]]},"references-count":49,"URL":"http://dx.doi.org/10.22331/q-2024-01-24-1233","relation":{},"ISSN":["2521-327X"],"subject":["Physics and Astronomy (miscellaneous)","Atomic and Molecular Physics, and Optics"],"container-title-short":"Quantum","published":{"date-parts":[[2024,1,24]]},"article-number":"1233","id":"doi:10.22331/q-2024-01-24-1233","_hash":"3ce7c334391134ad0e731af5c28477045c23f8ad0329a98f998415771a43c274"},"expire":1740764129858},"doi:10.1088/1367-2630/12/2/025013":{"value":{"indexed":{"date-parts":[[2023,9,13]],"date-time":"2023-09-13T20:47:22Z","timestamp":1694638042717},"reference-count":25,"publisher":"IOP Publishing","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.1088/1367-2630/12/2/025013","type":"journal-article","created":{"date-parts":[[2010,2,26]],"date-time":"2010-02-26T23:13:54Z","timestamp":1267226034000},"page":"025013","source":"Crossref","is-referenced-by-count":36,"title":"Thermodynamic stability criteria for a quantum memory based on stabilizer and subsystem codes","prefix":"10.1088","volume":"12","author":[{"given":"Stefano","family":"Chesi","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Loss","sequence":"additional","affiliation":[]},{"given":"Sergey","family":"Bravyi","sequence":"additional","affiliation":[]},{"given":"Barbara M","family":"Terhal","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2010,2,26]]},"reference":[],"container-title":"New Journal of Physics","original-title":[],"deposited":{"date-parts":[[2021,10,24]],"date-time":"2021-10-24T00:17:43Z","timestamp":1635034663000},"score":1,"resource":{"primary":{"URL":"https://iopscience.iop.org/article/10.1088/1367-2630/12/2/025013"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,2,26]]},"references-count":25,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2010,2,1]]}},"URL":"http://dx.doi.org/10.1088/1367-2630/12/2/025013","relation":{},"ISSN":["1367-2630"],"subject":["General Physics and Astronomy"],"container-title-short":"New J. 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Phys. A: Math. Theor.","published":{"date-parts":[[2009,1,14]]},"id":"doi:10.1088/1751-8113/42/6/065303","_hash":"fa548503469b1c9829bcc320281f1308cd02318660c51a47cd2f6315ed5f70df"},"expire":1740764131807},"doi:10.22331/q-2022-10-06-828":{"value":{"indexed":{"date-parts":[[2024,3,3]],"date-time":"2024-03-03T14:51:09Z","timestamp":1709477469833},"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":"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., ((n,2n&#x2212;6,3))2 for n from 7 to 14. We also found new ((6,2,3))2 and ((7,2,3))2 codes that are not equivalent to any stabilizer code, and extensive numerical evidence with VarQEC suggests that a ((7,3,3))2 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.","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":9,"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 Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China"}]}],"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-2022-10-06-828/pdf/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,10,6]],"date-time":"2022-10-06T13:23:50Z","timestamp":1665062630000},"score":1,"resource":{"primary":{"URL":"https://quantum-journal.org/papers/q-2022-10-06-828/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,6]]},"references-count":202,"URL":"http://dx.doi.org/10.22331/q-2022-10-06-828","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":"828","id":"doi:10.22331/q-2022-10-06-828","_hash":"e2fd152f9cf35850a1b8041bd5601ea27c6b4e02d70e56624109a9c8cc95eb54"},"expire":1740764132881},"doi:10.1155/2013/198710":{"value":{"indexed":{"date-parts":[[2024,2,26]],"date-time":"2024-02-26T15:24:22Z","timestamp":1708961062678},"reference-count":137,"publisher":"Hindawi Limited","license":[{"start":{"date-parts":[[2013,3,27]],"date-time":"2013-03-27T00:00:00Z","timestamp":1364342400000},"content-version":"unspecified","delay-in-days":0,"URL":"http://creativecommons.org/licenses/by/3.0/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2013,3,27]]},"abstract":"We review the progress in the last 20–30 years, during which we discovered that there are many new phases of matter that are beyond the traditional Landau symmetry breaking theory. We discuss new “topological” phenomena, such as topological degeneracy that reveals the existence of those new phases—topologically ordered phases. Just like zero viscosity defines the superfluid order, the new “topological” phenomena define the topological order at macroscopic level. More recently, we found that at the microscopical level, topological order is due to long-range quantum entanglements. Long-range quantum entanglements lead to many amazing emergent phenomena, such as fractional charges and fractional statistics. Long-range quantum entanglements can even provide a unified origin of light and electrons; light is a fluctuation of long-range entanglements, and electrons are defects in long-range entanglements.","DOI":"10.1155/2013/198710","type":"journal-article","created":{"date-parts":[[2013,3,27]],"date-time":"2013-03-27T21:01:39Z","timestamp":1364418099000},"page":"1-20","source":"Crossref","is-referenced-by-count":86,"title":"Topological Order: From Long-Range Entangled Quantum Matter to a Unified Origin of Light and Electrons","prefix":"10.1155","volume":"2013","author":[{"given":"Xiao-Gang","family":"Wen","sequence":"first","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada N2L 2Y5"},{"name":"Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA"},{"name":"Institute for Advanced Study, Tsinghua University, Beijing 100084, China"}]}],"member":"98","reference":[],"container-title":"ISRN Condensed Matter 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Condensed Matter Physics","published":{"date-parts":[[2013,3,27]]},"id":"doi:10.1155/2013/198710","_hash":"8002909cf6a93be2b2cb3855de2b341667306a2cca09f95702953ee6cc3ff042"},"expire":1740764133853},"doi:10.1016/j.pquantelec.2023.100496":{"value":{"indexed":{"date-parts":[[2024,2,29]],"date-time":"2024-02-29T00:23:42Z","timestamp":1709166222363},"reference-count":297,"publisher":"Elsevier 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in bosonic quantum error correction with Gottesman–Kitaev–Preskill Codes: Theory, engineering and applications","prefix":"10.1016","volume":"93","author":[{"given":"Anthony J.","family":"Brady","sequence":"first","affiliation":[]},{"given":"Alec","family":"Eickbusch","sequence":"additional","affiliation":[]},{"given":"Shraddha","family":"Singh","sequence":"additional","affiliation":[]},{"given":"Jing","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Quntao","family":"Zhuang","sequence":"additional","affiliation":[]}],"member":"78","reference":[],"container-title":"Progress in Quantum 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All rights reserved.","name":"copyright","label":"Copyright"}],"article-number":"100496","id":"doi:10.1016/j.pquantelec.2023.100496","_hash":"528e0f06ab7ffa6496612140eeddb1ae76188cf3973a4c3dc9e700e416831873"},"expire":1740764134930},"doi:10.1126/science.adk7560":{"value":{"indexed":{"date-parts":[[2024,2,23]],"date-time":"2024-02-23T15:53:44Z","timestamp":1708703624875},"reference-count":43,"publisher":"American Association for the Advancement of Science (AAAS)","issue":"6680","content-domain":{"domain":["www.science.org"],"crossmark-restriction":true},"published-print":{"date-parts":[[2024,1,19]]},"abstract":"To harness the potential of a quantum computer, quantum information must be protected against error by encoding it into a logical state that is suitable for quantum error correction. The Gottesman-Kitaev-Preskill (GKP) qubit is a promising candidate because the required multiqubit operations are readily available at optical frequency. To date, however, GKP qubits have been demonstrated only at mechanical and microwave frequencies. We realized a GKP state in propagating light at telecommunication wavelength and verified it through homodyne measurements without loss corrections. The generation is based on interference of cat states, followed by homodyne measurements. Our final states exhibit nonclassicality and non-Gaussianity, including the trident shape of faint instances of GKP states. Improvements toward brighter, multipeaked GKP qubits will be the basis for quantum computation with light.","DOI":"10.1126/science.adk7560","type":"journal-article","created":{"date-parts":[[2024,1,18]],"date-time":"2024-01-18T18:59:12Z","timestamp":1705604352000},"page":"289-293","update-policy":"http://dx.doi.org/10.34133/aaas_crossmark","source":"Crossref","is-referenced-by-count":3,"title":"Logical states for fault-tolerant quantum computation with propagating light","prefix":"10.1126","volume":"383","author":[{"ORCID":"http://orcid.org/0000-0001-5805-5029","authenticated-orcid":true,"given":"Shunya","family":"Konno","sequence":"first","affiliation":[{"name":"Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan."}]},{"ORCID":"http://orcid.org/0000-0002-3722-3676","authenticated-orcid":true,"given":"Warit","family":"Asavanant","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan."},{"name":"Optical Quantum Computing Research Team, RIKEN Center for Quantum Computing, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan."}]},{"ORCID":"http://orcid.org/0000-0002-2382-4593","authenticated-orcid":true,"given":"Fumiya","family":"Hanamura","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan."}]},{"ORCID":"http://orcid.org/0009-0005-3516-5643","authenticated-orcid":true,"given":"Hironari","family":"Nagayoshi","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan."}]},{"ORCID":"http://orcid.org/0000-0002-8231-7241","authenticated-orcid":true,"given":"Kosuke","family":"Fukui","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, 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In this paper, we consider the simplest scenario where a large superadditive capacity is known: a pure-loss channel with a coherent-state binary phase-shift keyed (BPSK) modulation. The two BPSK states can be mapped conceptually to two non-orthogonal states of a qubit, described by an inner product that is a function of the mean photon number per pulse. Using this map, we derive an explicit construction of the quantum circuit of a joint-detection receiver based on a recent idea of “belief-propagation with quantum messages” (BPQM). We quantify its performance improvement over the Dolinar receiver that performs optimal pulse-by-pulse detection, which represents the best “classical” approach. We analyze the scheme rigorously and show that it achieves the quantum limit of minimum average error probability in discriminating 8 (BPSK) codewords of a length-5 binary linear code with a tree factor graph. 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In the polynomial ideal framework, a code is specified by some ideals in a polynomial ring, messages are polynomials and their encoding is the residue modulo the ideals. We present an alternate way of viewing this class of codes in terms of linear operators, and show that this alternate view makes their algorithmic list-decodability amenable to analysis.\r\nOur framework leads to a new class of codes that we call affine Folded Reed-Solomon codes (which are themselves a special case of the broader class we explore). These codes are common generalizations of the well-studied Folded Reed-Solomon codes and Univariate Multiplicity codes, while also capturing the less-studied Additive Folded Reed-Solomon codes as well as a large family of codes that were not previously known/studied. \r\nMore significantly our framework also captures the algorithmic list-decodability of the constituent codes. Specifically, we present a unified view of the decoding algorithm for ideal-theoretic codes and show that the decodability reduces to the analysis of the distance of some related codes. We show that good bounds on this distance lead to capacity-achieving performance of the underlying code, providing a unifying explanation of known capacity-achieving results. 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Kotelnikov, “The theory of optimum noise immunity,” PhD Thesis, Molotov Energy Institute, Moscow, Jan. 1947}","_hash":"7e85a0a05b41f411b5615fe53e7c9c08d765807745f01fdab75a8babc7f08225"},"expire":1709664184432}} \ No newline at end of file diff --git a/jscomponents/codegraph/index.js b/jscomponents/codegraph/index.js index 7f7f458..0eb8554 100644 --- a/jscomponents/codegraph/index.js +++ b/jscomponents/codegraph/index.js @@ -117,10 +117,12 @@ export class EczCodeGraph }, rootPositioning: { - rootAbstractCodesXSpacing: 180, - rootAbstractCodesYPosition: 100, - domainNodesXSpacing: 180, - domainNodesYPosition: 0, + rootAbstractCodesXSpacing: 750, + rootAbstractCodesYPosition: 0, + rootAbstractCodesYPositionSingleOffset: 150, + domainNodesXSpacing: 750, + domainNodesYPosition: 250, + domainNodesYPositionSingleOffset: 150, }, }, graphOptions ?? {}, @@ -208,7 +210,9 @@ export class EczCodeGraph let domainColorIndexCounter = 0; let domainColorIndexByDomainId = {}; - for (const [domainId, domain] of Object.entries(this.eczoodb.objects.domain)) { + let domainsListWithIds = [ ... Object.entries(this.eczoodb.objects.domain) ]; + + for (const [domainId, domain] of domainsListWithIds) { //debug(`Adding domain =`, domain); @@ -452,17 +456,35 @@ export class EczCodeGraph const { rootAbstractCodesXSpacing, rootAbstractCodesYPosition, + rootAbstractCodesYPositionSingleOffset, domainNodesXSpacing, domainNodesYPosition, + domainNodesYPositionSingleOffset, } = this.graphOptions.rootPositioning; let graphRootNodesPrelayoutHints = {}; let domainIds = Object.keys(this.eczoodb.objects.domain); + + debug(`Domains before custom ordering: ${domainIds}`); + // hard-code some constraints on the order: + // put 'classical_domain' first, and 'quantum_domain' last + const customDomainIdsOrder = { + classical_domain: -100, + quantum_domain: 100, + }; + domainIds.sort( + (aId, bId) => (customDomainIdsOrder[aId] ?? 0) - (customDomainIdsOrder[bId] ?? 0) + ); + debug(`Domains after custom ordering: ${domainIds}`); + for (const [j, domainId] of domainIds.entries()) { const nodeId = this.getNodeIdDomain(domainId); graphRootNodesPrelayoutHints[nodeId] = { - position: {x: (j - (domainIds.length-1)/2) * rootAbstractCodesXSpacing, - y: rootAbstractCodesYPosition}, + position: { + x: (j - (domainIds.length-1)/2) * domainNodesXSpacing, + y: domainNodesYPosition + + Math.min(j, domainIds.length-1-j) * domainNodesYPositionSingleOffset + }, radiusOffset: 50, direction: Math.PI - Math.PI * (j+0.5) / domainIds.length, angularSpread: Math.PI / domainIds.length, @@ -473,8 +495,11 @@ export class EczCodeGraph //debug(`rootCodeNodeIds = `, rootCodeNodeIds); for (const [j, codeNodeId] of rootCodeNodeIds.entries()) { graphRootNodesPrelayoutHints[codeNodeId] = { - position: {x: (j - (rootCodeNodeIds.length-1)/2) * domainNodesXSpacing, - y: domainNodesYPosition}, + position: { + x: (j - (rootCodeNodeIds.length-1)/2) * rootAbstractCodesXSpacing, + y: rootAbstractCodesYPosition + - Math.min(j, rootCodeNodeIds.length-1-j) * rootAbstractCodesYPositionSingleOffset + }, radiusOffset: 50, direction: Math.PI + Math.PI * (j+0.5) / rootCodeNodeIds.length, angularSpread: Math.PI / rootCodeNodeIds.length,