Quantum statistical learning via Quantum Wasserstein natural gradient

TL;DR

This paper introduces a quantum Wasserstein natural gradient approach for statistical learning to approximate quantum states, leveraging a Riemannian structure on the parameter space and extending to continuous-variable states.

Contribution

It develops a quantum Wasserstein natural gradient framework for quantum state estimation, combining optimal transport with quantum information geometry.

Findings

Derivation of a quantum Wasserstein natural gradient flow for density operators.

Extension of the framework to continuous-variable quantum states via Wigner distributions.

Establishment of a Riemannian structure on the parameter space for quantum models.

Abstract

In this article, we introduce a new approach towards the statistical learning problem $\operatorname{argmin}_{\rho(\theta) \in \mathcal P_{\theta}} W_{Q}^2 (\rho_{\star},\rho(\theta))$ 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 $C^*$ 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.