Quantum Statistical Learning via Quantum Wasserstein Natural Gradient

Abstract

In this article, we introduce a new approach towards the statistical learning problem $$\mathrm{argmin}_{\rho (\theta ) \in {\mathcal {P}}_{\theta }} W_{Q}^2 (\rho _{\star },\rho (\theta ))$$ argmin ρ ( θ ) ∈ P θ W Q 2 ( ρ ⋆ , ρ ( θ ) ) to approximate a target quantum state $$\rho _{\star }$$ ρ ⋆ by a set of parametrized quantum states $$\rho (\theta )$$ ρ ( θ ) in a quantum $$L^2$$ 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.

A dual formula for the noncommutative transport distance

“…This result yields a noncommutative version of the dual formula obtained independently by Erbar, Maas and the author [EMW19] and Gangbo, Li and Mou [GLM19] for the Wasserstein-like transport distance on graphs. In fact, we prove a dual formula that is not only valid for the metric W, but also for the entropic regularization recently introduced by Becker-Li [BL21].…”

“…noncommutative transport distance from [CM17] and its entropic regularization introduced in [BL21]. Our notation mostly follows [CM17;CM20].…”

“…For ε = 0, this is the noncommutative transport distance W introduced in [CM17] (as distance function associated with a Riemannian metric on S(A) + ), and for ε > 0, this is the entropic regularization of W introduced in [BL21].…”

A Dual Formula for the Noncommutative Transport Distance

“…More precisely, we prove a dual formula that is a noncommutative analog of the expression of the classical -Wasserstein distance in terms of subsolutions of the Hamilton–Jacobi equation [ 5 , 24 ] This result yields a noncommutative version of the dual formula obtained independently by Erbar et al [ 15 ] and Gangb et al [ 16 ] for the Wasserstein-like transport distance on graphs. In fact, we prove a dual formula that is not only valid for the metric , but also for the entropic regularization recently introduced by Becker–Li [ 3 ]. When the generator is again of the simple form discussed above, the entropic regularization is a metric obtained when replacing the constraint in the definition of by With the notation introduced in the next section, the main result of this article reads as follows.…”

“…Moreover, stands for the set of all Hamilton–Jacobi–Bellmann subsolutions, a suitable noncommutative variant of solutions of the differential inequality Other metrics similar to also occur in the literature, most notably the one called the “anticommutator case” in [ 3 , 10 , 11 ]. In [ 9 , 30 ], a class of such metrics was studied in a systematic way, and our main theorem applies in fact to this wider class of metrics.…”