SDP Relaxations for Non-Commutative Polynomial Optimization

Abstract

We consider the problem of minimizing an arbitrary hermitian polynomial p(X) in non-commutative variables X = (X 1, …, X N ), where the polynomial p(X) is evaluated over all states and bounded operators (X 1, …, X n ) satisfying a finite set of polynomial constraints. Problems of this type appear frequently in areas as diverse as quantum chemistry, condensed matter physics, and quantum information science; finding numerical tools to attack them is thus essential. In this chapter, we describe a hierarchy of semidefinite programming relaxations of this generic problem, which converges to the optimal solution in the asymptotic limit. Furthermore, we derive sufficient optimality conditions for each step of the hierarchy. Our method is related to recent results in non-commutative algebraic geometry and can be seen as a generalization to the non-commutative setting of well-known semidefinite programming hierarchies that have been introduced in scalar (i.e. commutative) polynomial optimization. After presenting our results, we discuss at the end of the chapter some open questions and possible directions for future research.