Quantum Statistical Learning via Quantum Wasserstein Natural Gradient

Abstract

Abstract: In this article, we introduce a new approach towards the statistical learning problem argminρ(θ)∈PθWQ2(ρ⋆, ρ(θ)) to approximate a target quantum state ρ⋆ by a set of parametrized quantum states ρ(θ) in a quantum L2-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.