Quantum statistical learning via Quantum Wasserstein natural gradient

In this article, we introduce a new approach towards the statistical learning problem operatornameargmin_ρ(θ) ∈ mathcal Pθ WQ2 (ρstar,ρ(θ)) to approximate a target quantum state ρstar 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.