An SU(2)-symmetric Semidefinite Programming Hierarchy for Quantum Max Cut

TL;DR

This paper introduces an SU(2)-symmetric SDP hierarchy for Quantum Max Cut, converging at a finite level, with applications to quantum physics and complexity theory.

Contribution

It develops a new SU(2)-symmetric SDP hierarchy tailored for QMaxCut, proving its finite convergence and exploring its physical and computational implications.

Findings

Hierarchy converges to the optimal QMaxCut value at a finite level.

Exactness or inexactness of the hierarchy at the lowest level is demonstrated on various graphs.

SDP approaches relate to frustration-freeness and can approximate physical quantities in Heisenberg models.

Abstract

Understanding and approximating extremal energy states of local Hamiltonians is a central problem in quantum physics and complexity theory. Recent work has focused on developing approximation algorithms for local Hamiltonians, and in particular the ``Quantum Max Cut'' (QMax-Cut) problem, which is closely related to the antiferromagnetic Heisenberg model. In this work, we introduce a family of semidefinite programming (SDP) relaxations based on the Navascues-Pironio-Acin (NPA) hierarchy which is tailored for QMaxCut by taking into account its SU(2) symmetry. We show that the hierarchy converges to the optimal QMaxCut value at a finite level, which is based on a new characterization of the algebra of SWAP operators. We give several analytic proofs and computational results showing exactness/inexactness of our hierarchy at the lowest level on several important families of graphs. We also discuss relationships between SDP approaches for QMaxCut and frustration-freeness in condensed matter physics and numerically demonstrate that the SDP-solvability practically becomes an efficiently-computable generalization of frustration-freeness. Furthermore, by numerical demonstration we show the potential of SDP algorithms to perform as an approximate method to compute physical quantities and capture physical features of some Heisenberg-type statistical mechanics models even away from the frustration-free regions.