Finding Maximum Cliques on the D-Wave Quantum Annealer

TL;DR

This study evaluates the D-Wave 2X quantum annealer's ability to find maximum cliques, comparing its performance with classical algorithms and analyzing the potential for quantum speedup on different graph types.

Contribution

It provides formulations of the maximum clique problem for quantum annealing, compares quantum and classical methods, and assesses the quantum speedup potential on specific graph instances.

Findings

No quantum speedup on random graphs that fit DW.

Significant speed-ups observed on specially structured graphs.

Quantum annealer's performance depends on graph structure.

Abstract

This paper assesses the performance of the D-Wave 2X (DW) quantum annealer for finding a maximum clique in a graph, one of the most fundamental and important NP-hard problems. Because the size of the largest graphs DW can directly solve is quite small (usually around 45 vertices), we also consider decomposition algorithms intended for larger graphs and analyze their performance. For smaller graphs that fit DW, we provide formulations of the maximum clique problem as a quadratic unconstrained binary optimization (QUBO) problem, which is one of the two input types (together with the Ising model) acceptable by the machine, and compare several quantum implementations to current classical algorithms such as simulated annealing, Gurobi, and third-party clique finding heuristics. We further estimate the contributions of the quantum phase of the quantum annealer and the classical post-processing phase typically used to enhance each solution returned by DW. We demonstrate that on random graphs that fit DW, no quantum speedup can be observed compared with the classical algorithms. On the other hand, for instances specifically designed to fit well the DW qubit interconnection network, we observe substantial speed-ups in computing time over classical approaches.