Approximating the order 2 quantum Wasserstein distance using the moment-SOS hierarchy

TL;DR

This paper develops a method to approximate the order 2 quantum Wasserstein distance using the moment-SOS hierarchy, enabling computational solutions for quantum optimal transport problems.

Contribution

It formulates the quantum Wasserstein distance as an infinite-dimensional linear program and applies the moment-SOS hierarchy for practical approximation.

Findings

The hierarchy provides converging lower bounds to the quantum Wasserstein distance.

Numerical experiments demonstrate the effectiveness of the approach.

The method bridges quantum optimal transport and computational moment problems.

Abstract

Optimal transport theory has recently been extended to quantum settings, where the density matrices generalize the probability measures. In this paper, we study the computational aspects of the order 2 quantum Wasserstein distance, formulating it as an infinite dimensional linear program in the space of positive Borel measures supported on products of two unit spheres. This formulation is recognized as an instance of the Generalized Moment Problem, which enables us to use the moment-sums of squares hierarchy to provide a sequence of lower bounds converging to the distance. We illustrate our approach with numerical experiments.