Barrier relaxations of the classical and quantum optimal transport problems

TL;DR

This paper explores barrier relaxation methods, specifically interior point methods, for classical and quantum optimal transport problems, highlighting their efficiency especially in the quantum case compared to traditional algorithms.

Contribution

It demonstrates that interior point methods serve as barrier relaxations for classical and quantum MPOTP, offering an alternative to Sinkhorn algorithms with promising efficiency in quantum scenarios.

Findings

IPM for classical MPOTP duals is slower than Sinkhorn algorithms.

IPM for quantum MPOTP duals shows high efficiency.

Barrier relaxation approach unifies classical and quantum optimal transport methods.

Abstract

In the last fifteen years a significant progress was achieved by considering an entropic relaxation of the classical multi-partite optimal transport problem (MPOTP). The entropic relaxation gives rise to the rescaling problem of a given tensor. This rescaling can be achieved fast with the Sinkhorn type algorithms. Recently, it was shown that a similar approach works for the quantum MPOTP. However, the analog of the rescaling Sinkhorn algorithm is much more complicated than in the classical MPOTP. In this paper we show that the interior point method (IPM) for the primary and dual problems of classical and quantum MPOTP problems can be considered as barrier relaxations of the optimal transport problems (OTP). It is well known that the dual of the OTP are advantageous as it has much less variables than the primary problem. The IPM for the dual problem of the classical MPOTP are not as fast as the Sinkhorn type algorithm. However, IPM method for the dual of the quantum MPOTP seems to work quite efficiently.