Quantum Statistical Learning via Quantum Wasserstein Natural Gradient
Authors
Becker, S
Li, W
Publication Date
2021-01-07Journal Title
Journal of Statistical Physics
ISSN
0022-4715
Publisher
Springer Science and Business Media LLC
Volume
182
Issue
1
Language
en
Type
Article
This Version
VoR
Metadata
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Becker, S., & Li, W. (2021). Quantum Statistical Learning via Quantum Wasserstein Natural Gradient. Journal of Statistical Physics, 182 (1) https://doi.org/10.1007/s10955-020-02682-1
Abstract
<jats:title>Abstract</jats:title><jats:p>In this article, we introduce a new approach towards the statistical learning problem <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathrm{argmin}_{\rho (\theta ) \in {\mathcal {P}}_{\theta }} W_{Q}^2 (\rho _{\star },\rho (\theta ))$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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</mml:math></jats:alternatives></jats:inline-formula> to approximate a target quantum state <jats:inline-formula><jats:alternatives><jats:tex-math>$$\rho _{\star }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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</mml:math></jats:alternatives></jats:inline-formula> by a set of parametrized quantum states <jats:inline-formula><jats:alternatives><jats:tex-math>$$\rho (\theta )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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</mml:math></jats:alternatives></jats:inline-formula>-Wasserstein metric. We solve this estimation problem by considering Wasserstein natural gradient flows for density operators on finite-dimensional <jats:inline-formula><jats:alternatives><jats:tex-math>$$C^*$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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</mml:math></jats:alternatives></jats:inline-formula> algebras. For continuous parametric models of density operators, we pull back the quantum Wasserstein metric such that the parameter space becomes a Riemannian manifold with quantum Wasserstein information matrix. Using a quantum analogue of the Benamou–Brenier formula, we derive a natural gradient flow on the parameter space. We also discuss certain continuous-variable quantum states by studying the transport of the associated Wigner probability distributions.</jats:p>
Keywords
Article, Quantum transport information geometry, Quantum state estimation, Quantum Wasserstein information matrix, Quantum Wasserstein natural gradient, Quantum Schrödinger bridge problem
Sponsorship
Engineering and Physical Sciences Research Council (EP/L016516/1)
Identifiers
s10955-020-02682-1, 2682
External DOI: https://doi.org/10.1007/s10955-020-02682-1
This record's URL: https://www.repository.cam.ac.uk/handle/1810/316643
Rights
Attribution 4.0 International (CC BY 4.0)
Licence URL: https://creativecommons.org/licenses/by/4.0/
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