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Quantum Statistical Learning via Quantum Wasserstein Natural Gradient

Published version
Peer-reviewed

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Authors

Becker, S 
Li, W 

Abstract

jats:titleAbstract</jats:title>jats:pIn this article, we introduce a new approach towards the statistical learning problem jats:inline-formulajats:alternativesjats: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"> mml:mrow mml:msub mml:miargmin</mml:mi> mml:mrow mml:miρ</mml:mi> mml:mrow mml:mo(</mml:mo> mml:miθ</mml:mi> mml:mo)</mml:mo> </mml:mrow> mml:mo∈</mml:mo> mml:msub mml:miP</mml:mi> mml:miθ</mml:mi> </mml:msub> </mml:mrow> </mml:msub> mml:msubsup mml:miW</mml:mi> mml:mrow mml:miQ</mml:mi> </mml:mrow> mml:mn2</mml:mn> </mml:msubsup> mml:mrow mml:mo(</mml:mo> mml:msub mml:miρ</mml:mi> mml:mo⋆</mml:mo> </mml:msub> mml:mo,</mml:mo> mml:miρ</mml:mi> mml:mrow mml:mo(</mml:mo> mml:miθ</mml:mi> mml:mo)</mml:mo> </mml:mrow> mml:mo)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> to approximate a target quantum state jats:inline-formulajats:alternativesjats:tex-math$$\rho _{\star }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:miρ</mml:mi> mml:mo⋆</mml:mo> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> by a set of parametrized quantum states jats:inline-formulajats:alternativesjats:tex-math$$\rho (\theta )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miρ</mml:mi> mml:mo(</mml:mo> mml:miθ</mml:mi> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> in a quantum jats:inline-formulajats:alternativesjats:tex-math$$L^2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:miL</mml:mi> mml:mn2</mml:mn> </mml:msup> </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-formulajats:alternativesjats:tex-math$$C^*$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:miC</mml:mi> mml:mo∗</mml:mo> </mml:msup> </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>

Description

Keywords

Quantum transport information geometry, Quantum state estimation, Quantum Wasserstein information matrix, Quantum Wasserstein natural gradient, Quantum Schrodinger bridge problem

Journal Title

Journal of Statistical Physics

Conference Name

Journal ISSN

0022-4715
1572-9613

Volume Title

182

Publisher

Springer Science and Business Media LLC
Sponsorship
Engineering and Physical Sciences Research Council (EP/L016516/1)