Quantum statistical learning via Quantum Wasserstein natural gradient

Abstract

In this article, we introduce a new approach towards the statistical learning problem ${\mathrm{argmin}}_{\rho (\theta )\in {\mathcal{P}}_{\theta }}{W}_Q^2({\rho }_{\star },\rho (\theta ))$ to approximate a target quantum state ${\rho }_{\star }$ by a set of parametrized quantum states $\rho (\theta )$ in a quantum $L^2$-Wasserstein metric. We solve this estimation problem by considering Wasserstein natural gradient flows for density operators on finite-dimensional $C^{\ast }$ 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.

References in corpus (15)

• Continuous-variable optical quantum state tomography
• Quantum state tomography via compressed sensing
• Experimental Quantum State Tomography of Optical Fields and Ultrafast Statistical Sampling
• Unknown Quantum States and Operations, a Bayesian View
• Many-body quantum state tomography with neural networks
• Optimal, reliable estimation of quantum states
• On the Mean Field and Classical Limits of Quantum Mechanics
• Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems
• Quantum optimal transport with quantum channels
• Quantum tomography as a tool for the characterization of optical devices
• Quantum Wasserstein Generative Adversarial Networks
• An entropic gradient structure for Lindblad equations and couplings of quantum systems to macroscopic models
• Transport information geometry I: Riemannian calculus on probability simplex
• Convergence rates for the quantum central limit theorem
• Parametric Fokker-Planck equation

Cited by in corpus (3)

• The Quantum Wasserstein Distance of Order 1
• Variational quantum algorithm for unconstrained black box binary optimization: Application to feature selection
• A dual formula for the noncommutative transport distance