Approximation algorithms for noncommutative CSPs

TL;DR

This paper introduces approximation algorithms for noncommutative CSPs, including a 0.864-approximation for NC-Max-3-Cut, extending to various classical and quantum problems with new theoretical concepts.

Contribution

It presents the first approximation algorithm for NC-Max-3-Cut and introduces new concepts like approximate isometry and $ ext{ extasterisk} $-anticommutation.

Findings

Developed a 0.864-approximation algorithm for NC-Max-3-Cut

Extended the approach to a broader class of CSPs

Introduced new theoretical concepts of independent interest

Abstract

Noncommutative constraint satisfaction problems (NC-CSPs) are higher-dimensional operator extensions of classical CSPs. Despite their significance in quantum information, their approximability remains largely unexplored. A notable example of a noncommutative CSP that is not solvable in polynomial time is NC-Max-$3$-Cut. We present a $0.864$-approximation algorithm for this problem. Our approach extends to a broader class of both classical and noncommutative CSPs. We introduce three key concepts: approximate isometry, relative distribution, and $\ast$-anticommutation, which may be of independent interest.