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arXiv:2011.13420 · Quantum

Hybrid quantum-classical algorithms for approximate graph coloring

Sergey B. Bravyi, Alexander Kliesch, Robert König, Eugene Tang

Published Mar 30, 2022• 100 citations
arXivHTMLPDF

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.

DOI
10.22331/q-2022-03-30-678

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Abstract Quantum approximate optimization algorithm (QAOA) is a quantum–classical hybrid algorithm proposed with the goal of approximately solving combinatorial optimization problems such as the MAX-CUT problem. It has been considered a potential candidate for achieving quantum advantage in the noisy intermediate-scale quantum era and has been extensively studied. However, the performance limitations of low-level QAOA have also been demonstrated across various instances. In this work, we first analytically prove the performance limitations of the standard level-1 QAOA in solving the MAX-CUT problem on bipartite graphs. To this end, we derive an upper bound for the approximation ratio based on the average degree of bipartite graphs. Second, we demonstrate that recursive QAOA (RQAOA), which recursively reduces graph size using QAOA as a subroutine, outperforms the standard level-1 QAOA. However, the performance of RQAOA exhibits limitations as the graph size increases. Finally, we show that RQAOA with a restricted parameter regime can fully address these limitations. Surprisingly, this modified RQAOA always finds the exact maximum cut for any bipartite graphs and even for a more general graph with parity-signed weights.
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In the era of Noisy Intermediate Scale Quantum (NISQ) computing, available quantum resources are limited. Many NP-hard problems can be efficiently addressed using hybrid classical and quantum computational methods. This paper proposes a hybrid variational quantum algorithm designed to solve the kkk-coloring problem of graph vertices. The hybrid classical and quantum algorithms primarily partition the graph into multiple subgraphs through hierarchical techniques. The Quantum Approximate Optimization Algorithm (QAOA) is employed to determine the coloring within the subgraphs, while a classical greedy algorithm is utilized to find the coloring of the interaction graph. Fixed coloring is applied to the interaction graph, and feedback is provided to correct any conflicting colorings within the subgraphs. The merging process into the original graph is iteratively optimized to resolve any arising conflicts. We employ a hierarchical framework that integrates feedback correction and conflict resolution to achieve kkk-coloring of arbitrary graph vertices. Through experimental analysis, we demonstrate the effectiveness of the algorithm, highlighting the rapid convergence of conflict evolution and the fact that iterative optimization allows the classical algorithm to approximate the number of colorings. Finally, we apply the proposed algorithm to optimize the scheduling of a subway transportation network, demonstrating a high degree of fairness.
Quantum algorithms for binary optimization problems have been the subject of extensive study. However, the application of quantum algorithms to integer optimization problems remains comparatively unexplored. In this paper, we study the Quantum Approximate Optimization Algorithm (QAOA) applied to integer problems on graphs, with each integer variable encoded in a qudit. We derive a general iterative formula for depth-ppp QAOA expectation on high-girth ddd-regular graphs of arbitrary size. The cost of evaluating the formula is exponential in the QAOA depth ppp but does not depend on the graph size. Evaluating this formula for Max-kkk-Cut problem for p≤4p\leq 4p≤4, we identify parameter regimes (k=3k=3k=3 with degree d≤10d \leq 10d≤10 and k=4k=4k=4 with d≤40d \leq 40d≤40) in which QAOA outperforms the Frieze-Jerrum semi-definite programming (SDP) algorithm, which provides the best worst-case guarantee on the approximation ratio. To strengthen the classical baseline, we introduce a new heuristic algorithm based on the degree-of-saturation that achieves strong results on the \texttt{GSet} benchmark with quasi-linear runtime in the number of edges. It empirically outperforms both the Frieze-Jerrum algorithm and shallow-depth QAOA on regular graphs. Nevertheless, we provide numerical evidence that QAOA may overtake this heuristic at depth p≤20p\leq 20p≤20. Our results show that moving beyond binary to integer optimization problems can open up new avenues for quantum advantage.
Abstract The quantum approximate optimization algorithm (QAOA) is a hybrid quantum–classical algorithm to solve binary-variable optimization problems. Due to the short circuit depth and its expected robustness to systematic errors it is a promising candidate likely to run on near-term quantum devices. We simulate the performance of QAOA applied to the Max-Cut problem and compare it with some of the best classical alternatives. When comparing solvers, their performance is characterized by the computational time taken to achieve a given quality of solution. Since QAOA is based on sampling, we utilize performance metrics based on the probability of observing a sample above a certain quality. In addition, we show that the QAOA performance varies significantly with the graph type. In particular for three-regular random graphs, QAOA performance shows improvement by up to two orders of magnitude compared to previous estimates, strongly reducing the performance gap with classical alternatives. This was possible by reducing the number of function evaluations per iteration and optimizing the variational parameters on small graph instances and transferring to large via training. Because QAOA’s performance guarantees are only known for limited applications and contexts, we utilize a framework for the search for quantum advantage which incorporates a large number of problem instances and all three classical solver modalities: exact, approximate, and heuristic.
The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical algorithm to solve binary-variable optimization problems. Due to its expected robustness to systematic errors and the short circuit depth, it is one of the promising candidates likely to run on near-term quantum devices. We project the performance of QAOA applied to the Max-Cut problem and compare it with some of the best classical alternatives, both for exact or approximate solution. When comparing approximate solvers, their performance is characterized by the computational time taken to achieve a given quality of solution. Since QAOA is based on sampling, we introduce performance metrics based on the probability of observing a sample above a certain quality. In addition, we show that the QAOA performance varies significantly with the graph type. By selecting a suitable optimizer for the variational parameters and reducing the number of function evaluations, QAOA performance improves by up to 2 orders of magnitude compared to previous estimates. Especially for 3-regular random graphs, this setting decreases the performance gap with classical alternatives.