Quantum Statistical Learning via Quantum Wasserstein Natural Gradient

In this article, we introduce a new approach towards the statistical learning problem $\operatorname{argmin}_{\rho(\theta) \in \mathcal P_{\theta}} W_{Q}^2 (\rho_{\star},\rho(\theta))$ to approximate target quantum state $\rho_{\star}$ by set of parametrized states $\rho(\theta)$ in $L^2$-Wasserstein metric. We solve estimation considering Wasserstein natural gradient flows for density operators on finite-dimensional $C^*$ algebras. For continuous parametric models operators, pull back metric such that parameter space becomes Riemannian manifold with information matrix. Using analogue Benamou-Brenier formula, derive flow space. also discuss certain continuous-variable studying transport associated Wigner probability distributions.