Quantum statistical learning via Quantum Wasserstein natural gradient

Abstract: In this article, we introduce a new approach towards the statistical learning problem $$\operatorname{argmin}<em>{\rho(\theta) \in \mathcal P</em>{\theta}} W_{Q}² (\rho_{\star},\rho(\theta))$$ to approximate a target quantum state $\rho_{\star}$ by a set of parametrized quantum states $\rho(\theta)$ in a quantum $L<sup>2$-Wasserstein metric. We solve this estimation problem by considering Wasserstein natural gradient flows for density operators on finite-dimensional $C<sup>*$ 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.