Faster quantum and classical SDP approximations for quadratic binary optimization

TL;DR

This paper introduces a quantum algorithm that accelerates solving semidefinite programming relaxations for binary quadratic optimization, achieving nearly quadratic speedups over existing classical methods, with applications to spin glasses and MaxCut problems.

Contribution

It presents the first quantum speedup for the canonical SDP relaxation in binary quadratic optimization, combining quantum Gibbs sampling with matrix exponent updates, and also offers a faster classical de-quantized solver.

Findings

Quantum solver achieves nearly quadratic speedup for generic instances.

Applicable to spin glasses and MaxCut on Erdős-Rényi graphs.

Provides an efficient randomized rounding procedure.

Abstract

We give a quantum speedup for solving the canonical semidefinite programming relaxation for binary quadratic optimization. This class of relaxations for combinatorial optimization has so far eluded quantum speedups. Our methods combine ideas from quantum Gibbs sampling and matrix exponent updates. A de-quantization of the algorithm also leads to a faster classical solver. For generic instances, our quantum solver gives a nearly quadratic speedup over state-of-the-art algorithms. Such instances include approximating the ground state of spin glasses and MaxCut on Erd\"{o}s-R\'enyi graphs. We also provide an efficient randomized rounding procedure that converts approximately optimal SDP solutions into approximations of the original quadratic optimization problem.