Hybrid quantum-classical algorithms for approximate graph coloring

TL;DR

This paper investigates the application of recursive quantum approximate optimization algorithms to graph coloring, compares their performance with classical algorithms, and finds that level-1 RQAOA shows promising results on certain graph ensembles.

Contribution

The paper demonstrates that level-1 RQAOA can outperform some classical algorithms on specific graph coloring problems and provides a classical simulation framework for testing hybrid quantum algorithms.

Findings

Level-1 RQAOA often surpasses classical SDP rounding algorithms in approximation ratio.

Standard QAOA fails to improve beyond random assignment on regular bipartite graphs.

Classical simulation of level-1 QAOA and RQAOA shows potential for near-term quantum devices.

Abstract

We show how to apply the recursive quantum approximate optimization algorithm (RQAOA) to MAX-$k$-CUT, the problem of finding an approximate $k$-vertex coloring of a graph. We compare this proposal to the best known classical and hybrid classical-quantum algorithms. First, we show that the standard (non-recursive) QAOA fails to solve this optimization problem for most regular bipartite graphs at any constant level $p$: the approximation ratio achieved by QAOA is hardly better than assigning colors to vertices at random. Second, we construct an efficient classical simulation algorithm which simulates level-$1$ QAOA and level-$1$ RQAOA for arbitrary graphs. In particular, these hybrid algorithms give rise to efficient classical algorithms, and no benefit arising from the use of quantum mechanics is to be expected. Nevertheless, they provide a suitable testbed for assessing the potential benefit of hybrid algorithm: We use the simulation algorithm to perform large-scale simulation of level-$1$ QAOA and RQAOA with up to $300$ qutrits applied to ensembles of randomly generated $3$-colorable constant-degree graphs. We find that level-$1$ RQAOA is surprisingly competitive: for the ensembles considered, its approximation ratios are often higher than those achieved by the best known generic classical algorithm based on rounding an SDP relaxation. This suggests the intriguing possibility that higher-level RQAOA may be a potentially useful algorithm for NISQ devices.