Quantum Statistical Learning via Quantum Wasserstein Natural Gradient
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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>
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1572-9613