  The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing
approximation algorithms for local Hamiltonian problems. In this paper we
attack this problem using the algebraic structure of QMC, in particular the
relationship between the quantum max cut Hamiltonian and the representation
theory of the symmetric group.
  The first major contribution of this paper is an extension of non-commutative
Sum of Squares (ncSoS) optimization techniques to give a new hierarchy of
relaxations to Quantum Max Cut. The hierarchy we present is based on
optimizations over polynomials in the qubit swap operators. This is in contrast
to the "standard" quantum Lasserre Hierarchy, which is based on polynomials
expressed in terms of the Pauli matrices. To prove correctness of this
hierarchy, we exploit a finite presentation of the algebra generated by the
qubit swap operators. This presentation allows for the use of computer
algebraic techniques to manipulate and simplify polynomials written in terms of
the swap operators, and may be of independent interest. Surprisingly, we find
that level-2 of this new hierarchy is numerically exact (up to tolerance
10^(-7)) on all QMC instances with uniform edge weights on graphs with at most
8 vertices.
  The second major contribution of this paper is a polynomial-time algorithm
that computes (in exact arithmetic) the maximum eigenvalue of the QMC
Hamiltonian for certain graphs, including graphs that can be "decomposed" as a
signed combination of cliques. A special case of the latter are complete
bipartite graphs with uniform edge-weights, for which exact solutions are known
from the work of Lieb and Mattis. Our methods, which use representation theory
of the symmetric group, can be seen as a generalization of the Lieb-Mattis
result.
