Quantum Statistical Learning via Quantum Wasserstein Natural Gradient

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"> <mml:mrow> <mml:msub> <mml:mi>argmin</mml:mi> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>θ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>θ</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mo>⋆</mml:mo> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>ρ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>θ</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-formula><jats:alternatives><jats:tex-math>$$\rho _{\star }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mo>⋆</mml:mo> </mml:msub> </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"> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>θ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> in a quantum <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</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-formula><jats:alternatives><jats:tex-math>$$C^*$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</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>