Quantum Statistical Learning via Quantum Wasserstein Natural Gradient

Abstract

In this article, we introduce a new approach towards the statistical learning problem argminρ(θ)∈PθWQ2(ρ⋆,ρ(θ))$$\mathrm{argmin}{\rho (\theta ) \in {\mathcal {P}}{\theta }} W_{Q}^2 (\rho _{\star },\rho (\theta ))$$ to approximate a target quantum state ρ⋆$$\rho _{\star }$$ by a set of parametrized quantum states ρ(θ)$$\rho (\theta )$$ in a quantum L2$$L^2$$-Wasserstein metric. We solve this estimation problem by considering Wasserstein natural gradient flows for density operators on finite-dimensional C∗$$C^*$$ 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.