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[Draft] Demo: The KAK theorem #1227
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{ | ||
"title": "The KAK theorem", | ||
"authors": [ | ||
{ | ||
"username": "dwierichs" | ||
} | ||
], | ||
"dateOfPublication": "2024-12-05T00:00:00+00:00", | ||
"dateOfLastModification": "2024-12-05T00:00:00+00:00", | ||
"categories": [ | ||
"Quantum Computing", | ||
"Algorithms" | ||
], | ||
"tags": [], | ||
"previewImages": [ | ||
{ | ||
"type": "thumbnail", | ||
"uri": "/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_kak_theorem.png" | ||
}, | ||
{ | ||
"type": "large_thumbnail", | ||
"uri": "/_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_kak_theorem.png" | ||
} | ||
], | ||
"seoDescription": "Learn about the KAK theorem and how it powers circuit decompositions.", | ||
"doi": "", | ||
"canonicalURL": "/qml/demos/tutorial_kak_theorem", | ||
"references": [ | ||
], | ||
"basedOnPapers": [], | ||
"referencedByPapers": [], | ||
"relatedContent": [ | ||
{ | ||
"type": "demonstration", | ||
"id": "tutorial_liealgebra", | ||
"weight": 1.0 | ||
} | ||
] | ||
} |
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r"""The KAK theorem | ||
=================== | ||
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The KAK theorem is a beautiful mathematical result from Lie theory, with | ||
particular relevance for quantum computing research. It can be seen as a | ||
generalization of the singular value decomposition, and therefore falls | ||
under the large umbrella of matrix factorizations. This allows us to | ||
use it for quantum circuit decompositions. However, it can also | ||
be understood from a more abstract point of view, as we will see. | ||
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In this demo, we will discuss so-called symmetric spaces, which arise from | ||
subgroups of Lie groups. For this, we will focus on the algebraic level | ||
and introduce Cartan involutions/decompositions, horizontal | ||
and vertical subspaces, as well as horizontal Cartan subalgebras. | ||
With these tools in our hands, we will then learn about the KAK theorem | ||
itself. | ||
We conclude with a famous application of the theorem to circuit decomposition | ||
by Khaneja and Glaser [#khaneja_glaser]_, which provides a circuit | ||
template for arbitrary unitaries on any number of qubits, and proved for | ||
the first time that single and two-qubit gates are sufficient to implement them. | ||
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While this demo is of more mathematical nature than others, we will include | ||
hands-on examples throughout. | ||
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.. figure:: ../_static/demo_thumbnails/opengraph_demo_thumbnails/OGthumbnail_kak_theorem.png | ||
:align: center | ||
:width: 60% | ||
:target: javascript:void(0) | ||
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.. note:: | ||
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In the following we will assume a basic understanding of vector spaces, | ||
linear maps, and Lie algebras. For the former two, we recommend a look | ||
at your favourite linear algebra material, for the latter see our | ||
:doc:`introduction to (dynamical) Lie algebras </demos/tutorial_liealgebra/>`. | ||
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Introduction | ||
------------ | ||
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Basic mathematical objects | ||
-------------------------- | ||
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Introduce the mathematical objects that will play together to yield | ||
the KAK theorem. | ||
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(Semi-)simple Lie algebras | ||
~~~~~~~~~~~~~~~~~~~~~~~~~~ | ||
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- Introduce the notion of a Lie algebra very briefly, refer to existing demo(s). | ||
- Focus on vector space notion being clear. | ||
- [optional] Briefly say what a simple/semisimple Lie algebra is. | ||
- [optional] In particular mention that the adjoint representation is faithful for semisimple algebras. | ||
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Group and algebra interaction | ||
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ | ||
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- Exponential map | ||
- adjoint action of group on algebra | ||
- adjoint action of algebra on algebra -> adjoint representation | ||
- adjoint identity (-> g-sim demo) | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Not sure how much you plan on talking about these points but could be kept brief |
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Subalgebras and Cartan pairs | ||
~~~~~~~~~~~~~~~~~~~~~~~~~~~~ | ||
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- Introduce the notion of a subalgebra. | ||
- Explain that there can be vector subspaces that are not subalgebras. | ||
- Define Cartan pairs via commutation relations | ||
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Cartan subalgebras | ||
~~~~~~~~~~~~~~~~~~ | ||
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- Define Cartan subalgebras of :math:`m`. | ||
- Dimension of Cartan subalgebras | ||
- Transition between Cartan subalgebras via :math:`K` | ||
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Involutions | ||
~~~~~~~~~~~ | ||
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- Explain linear maps on (matrix) algebras (-> homomorphism) | ||
- Define involutions. | ||
- Involutions define Cartan pairs (:math:`k = +1 | m = -1` eigenspaces) | ||
- Cartan pairs define involutions :math:`\theta = \Pi_{\mathfrak{k}} - \Pi_{\mathfrak{m}}` | ||
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KAK theorem | ||
~~~~~~~~~~~ | ||
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- KP decomposition | ||
- KAK decomposition | ||
- [optional] implication: KaK on algebra level | ||
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Two-qubit KAK decomposition | ||
--------------------------- | ||
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- Algebra/subalgebra :math:`\mathfrak{g} =\mathfrak{su}(4) | \mathfrak{k} =\mathfrak{su}(2) \oplus \mathfrak{su}(2)` | ||
- Involution: EvenOdd | ||
- CSA: :math:`\mathfrak{a} = \langle\{XX, YY, ZZ\}\rangle_{i\mathbb{R}}` | ||
- KAK decomposition :math:`U= (A\otimes B) \exp(i(\eta_x XX+\eta_y YY +\eta_z ZZ)) (C\otimes D)`. | ||
- [optional] Mention Cartan coordinates | ||
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Khaneja-Glaser decomposition | ||
---------------------------- | ||
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- Important first recursive decomposition showing universality of single- and two-qubit operations | ||
- Used for practical decompositions, replaced by other, similar decompositions by now | ||
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A recursive decomposition | ||
~~~~~~~~~~~~~~~~~~~~~~~~~ | ||
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- Show recursion on qubit count | ||
- display resulting decomposition structure | ||
- Mention that a two-qubit interaction is enough to get the CSA elements | ||
- Universality | ||
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The recursion step in detail | ||
~~~~~~~~~~~~~~~~~~~~~~~~~~~~ | ||
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- Two substeps in each recursion step: | ||
- Algebra/subalgebra :math:`\mathfrak{g}=\mathfrak{su}(2^n) | \mathfrak{k} = \mathfrak{su}(2^{n-1}) \oplus \mathfrak{su}(2^{n-1})` | ||
- Involution TBD | ||
- CSA TBD | ||
- Algebra/subalgebra :math:`\mathfrak{g}=\mathfrak{su}(2^{n-1}) \oplus \mathfrak{su}(2^{n-1}) | \mathfrak{k} = \mathfrak{su}(2^{n-1})` | ||
- Involution TBD | ||
- CSA TBD | ||
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Overview of resulting decomposition | ||
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ | ||
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- Count blocks | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. If things had to be scratched, Perhaps this could be made a separate demo (though I am very interested in the contents here so maybe we can already talk about it) There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Yeah, maybe we could actually have this demo focus on KAK theorem, including only the 2-qubit application. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Yeah something like that, or even more generally a demo on recursive decomposition, but I think that would depend on our findings this quarter and how much we want to invest (seems like a lot of work to do all of this 😅 ) |
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- [optional] CNOT count | ||
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Conclusion | ||
---------- | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Hi @Qottmann,
Thanks! 🙏 There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. It's a lot of ground to cover with quite involved topics. Do you reckon it is realistic to put this all in a demo? This seems like review paper level thoroughness (which makes me think, should we write a review paper on this stuff for physicists?) I actually wouldnt want to scratch anything because I'd be curious to read all of this myself, I am just wondering if you think it's realistic for you to cover all these things in reasonable time (your own time effort) and in a presentation that is not too long (time effort of a reader)? There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. It is a very fair question. Taking some of this offline.
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In this demo we learned about the KAK theorem and how it uses a Cartan | ||
decomposition of a Lie algebra to decompose its Lie group. | ||
A famous immediate application of this result is the circuit decomposition, or | ||
parametrization, for arbitrary qubit numbers by Khaneja and Glaser. It also allowed | ||
us to prove universality of single and two-qubit unitaries for quantum computation. | ||
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If you are interested in other applications of Lie theory in the field of | ||
quantum computing, you are in luck! It has been a handy tool throughout the last | ||
decades, e.g., for the simulation and compression of quantum circuits, # TODO: REFS | ||
in quantum optimal control, and for trainability analyses. For Lie algebraic | ||
classical simulation of quantum circuits, check the | ||
:doc:`g-sim </demos/tutorial_liesim/>` and | ||
:doc:`(g+P)-sim </demos/tutorial_liesim_extension/>` demos, and stay posted for | ||
a brand new demo on compiling Hamiltonian simulation circuits with the KAK theorem! | ||
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The props | ||
--------- | ||
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Adjoint representation | ||
~~~~~~~~~~~~~~~~~~~~~~ | ||
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""" | ||
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import pennylane as qml | ||
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###################################################################### | ||
# | ||
# References | ||
# ---------- | ||
# | ||
# .. [#khaneja_glaser] | ||
# | ||
# Navin Khaneja, Steffen Glaser | ||
# "Cartan decomposition of SU(2^n), constructive controllability of spin systems and universal quantum computing" | ||
# `arXiv:quant-ph/0010100 <https://arxiv.org/abs/quant-ph/0010100>`__, 2000 | ||
# | ||
# About the author | ||
# ---------------- |
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oh good to know! I wasnt aware!
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The precise criterion is that the center is trivial, which is a bit more general. But a center would be an Abelian ideal, which semisimple Lie algebras don't have, so semisimple implies faithful ad.rep.