On constant-time quantum annealing and guaranteed approximations for graph optimization problems

TL;DR

This paper introduces a theoretical framework using Lieb-Robinson bounds to guarantee constant-factor approximations for certain graph optimization problems with constant-time quantum annealing, advancing the understanding of QA's solution quality.

Contribution

It develops new tools based on Lieb-Robinson bounds to provide approximation guarantees for quantum annealing on bounded degree graphs, specifically for MaxCut and Maximum Independent Set.

Findings

Constant-time QA guarantees constant-factor approximations on bounded degree graphs.

Explicit approximation ratios and runtimes are derived for MaxCut and Maximum Independent Set.

Results are comparable to those in the QAOA framework.

Abstract

Quantum Annealing (QA) is a computational framework where a quantum system's continuous evolution is used to find the global minimum of an objective function over an unstructured search space. It can be seen as a general metaheuristic for optimization problems, including NP-hard ones if we allow an exponentially large running time. While QA is widely studied from a heuristic point of view, little is known about theoretical guarantees on the quality of the solutions obtained in polynomial time. In this paper we use a technique borrowed from theoretical physics, the Lieb-Robinson (LR) bound, and develop new tools proving that short, constant time quantum annealing guarantees constant factor approximations ratios for some optimization problems when restricted to bounded degree graphs. Informally, on bounded degree graphs the LR bound allows us to retrieve a (relaxed) locality argument, through which the approximation ratio can be deduced by studying subgraphs of bounded radius. We illustrate our tools on problems MaxCut and Maximum Independent Set for cubic graphs, providing explicit approximation ratios and the runtimes needed to obtain them. Our results are of similar flavor to the well-known ones obtained in the different but related QAOA (quantum optimization algorithms) framework. Eventually, we discuss theoretical and experimental arguments for further improvements.