Quantum statistical learning via Quantum Wasserstein natural gradient

Abstract

In this article, we introduce a new approach towards the statistical learning problem ${\mathrm{argmin}}_{\ddot{I}(\hat{I}¸)\in {\mathcal{P}}_{\hat{I}}¸}{W}_Q^2(\ddot{I}{}_{\star },\ddot{I}(\hat{I}¸))$ to approximate a target quantum state $\ddot{I}{}_{\star }$ by a set of parametrized quantum states $\ddot{I}(\hat{I}¸)$ 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.