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Understanding and approximating extremal energy states of local Hamiltoniansis a central problem in quantum physics and complexity theory. Recent work hasfocused on developing approximation algorithms for local Hamiltonians, and inparticular the ``Quantum Max Cut'' (QMax-Cut) problem, which is closely relatedto the antiferromagnetic Heisenberg model. In this work, we introduce a familyof semidefinite programming (SDP) relaxations based on theNavascues-Pironio-Acin (NPA) hierarchy which is tailored for QMaxCut by takinginto account its SU(2) symmetry. We show that the hierarchy converges to theoptimal QMaxCut value at a finite level, which is based on a newcharacterization of the algebra of SWAP operators. We give several analyticproofs and computational results showing exactness/inexactness of our hierarchyat the lowest level on several important families of graphs.  We also discuss relationships between SDP approaches for QMaxCut andfrustration-freeness in condensed matter physics and numerically demonstratethat the SDP-solvability practically becomes an efficiently-computablegeneralization of frustration-freeness. Furthermore, by numerical demonstrationwe show the potential of SDP algorithms to perform as an approximate method tocompute physical quantities and capture physical features of someHeisenberg-type statistical mechanics models even away from thefrustration-free regions.

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