Convergent relaxations of polynomial optimization problems with non-commuting variables

TL;DR

This paper develops a hierarchy of semidefinite programming relaxations for polynomial optimization problems involving non-commuting variables, with applications in quantum theory, ensuring convergence to the global optimum.

Contribution

It introduces a novel hierarchy of relaxations for non-commuting polynomial optimization problems, including a criterion for optimality and methods to extract solutions.

Findings

Hierarchy of relaxations converges to the global optimum

A criterion to detect when the global optimum is reached

Method to extract global optimizer from SDP solution

Abstract

We consider optimization problems with polynomial inequality constraints in non-commuting variables. These non-commuting variables are viewed as bounded operators on a Hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints. Such problems arise naturally in quantum theory and quantum information science. To solve them, we introduce a hierarchy of semidefinite programming relaxations which generates a monotone sequence of lower bounds that converges to the optimal solution. We also introduce a criterion to detect whether the global optimum is reached at a given relaxation step and show how to extract a global optimizer from the solution of the corresponding semidefinite programming problem.