Free spectrahedra are dimension free solution sets to linear matrix inequalities of the form L-A(X) = I-d circle times I-n+A(1)circle times X-1+A(2)circle times X-2+center dot center dot center dot+A(g)circle times X-g >= 0, where the Ai and Xi are symmetric matrices and the Xi have any size n x n. Free spectrahedra are ubiquitous in systems engineering, operator algebras, and the theory of matrix convex sets. Matrix and free extreme points of free spectrahedra are particularly important. We present theoretical, algorithmic, and experimental results illuminating basic properties of extreme points. For example, though many authors have studied matrix and free extreme points, it has until now been unknown if these two types of extreme points are actually different. This paper settles that issue.We also present and analyze several algorithms. Namely, we perfect an algorithm for computing an expansion of an element of a free spectrahedron in terms of free extreme points. We also give algorithms for testing if a point is matrix extreme and for computing matrix extreme points that are not free extreme.

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.