Algorithm 950: Ncpol2sdpa---Sparse Semidefinite Programming Relaxations for Polynomial Optimization Problems of Noncommuting Variables

TL;DR

This paper introduces Ncpol2sdpa, a Python tool for efficiently generating sparse semidefinite programming relaxations for polynomial optimization problems involving noncommuting variables, with applications in quantum physics.

Contribution

It develops a scalable implementation that exploits problem sparsity and enables solving larger noncommuting polynomial optimization problems using SDPA.

Findings

Efficient relaxation generation for noncommuting variables.

Reduction in complexity through sparsity exploitation.

Application to quantum physics problems like ground state energy.

Abstract

A hierarchy of semidefinite programming (SDP) relaxations approximates the global optimum of polynomial optimization problems of noncommuting variables. Generating the relaxation, however, is a computationally demanding task, and only problems of commuting variables have efficient generators. We develop an implementation for problems of noncommuting problems that creates the relaxation to be solved by SDPA -- a high-performance solver that runs in a distributed environment. We further exploit the inherent sparsity of optimization problems in quantum physics to reduce the complexity of the resulting relaxations. Constrained problems with a relaxation of order two may contain up to a hundred variables. The implementation is available in Python. The tool helps solve problems such as finding the ground state energy or testing quantum correlations.