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arXiv:2110.14206 · Leibniz international proceedings in informatics

The Quantum Approximate Optimization Algorithm at High Depth for MaxCut on Large-Girth Regular Graphs and the Sherrington-Kirkpatrick Model

Joao Basso, Edward Farhi, Kunal Marwaha, Benjamin Villalonga, Leo Zhou

Published Oct 27, 2021• 43 citations
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Abstract

The Quantum Approximate Optimization Algorithm (QAOA) finds approximate solutions to combinatorial optimization problems. Its performance monotonically improves with its depth $p$. We apply the QAOA to MaxCut on large-girth $D$-regular graphs. We give an iterative formula to evaluate performance for any $D$ at any depth $p$. Looking at random $D$-regular graphs, at optimal parameters and as $D$ goes to infinity, we find that the $p=11$ QAOA beats all classical algorithms (known to the authors) that are free of unproven conjectures. While the iterative formula for these $D$-regular graphs is derived by looking at a single tree subgraph, we prove that it also gives the ensemble-averaged performance of the QAOA on the Sherrington-Kirkpatrick (SK) model defined on the complete graph. We also generalize our formula to Max-$q$-XORSAT on large-girth regular hypergraphs. Our iteration is a compact procedure, but its computational complexity grows as $O(p^2 4^p)$. This iteration is more efficient than the previous procedure for analyzing QAOA performance on the SK model, and we are able to numerically go to $p=20$. Encouraged by our findings, we make the optimistic conjecture that the QAOA, as $p$ goes to infinity, will achieve the Parisi value. We analyze the performance of the quantum algorithm, but one needs to run it on a quantum computer to produce a string with the guaranteed performance.

DOI
10.4230/lipics.tqc.2022.7

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The Quantum Approximate Optimization Algorithm (QAOA) finds approximate solutions to combinatorial optimization problems. Its performance monotonically improves with its depth ppp. We apply the QAOA to MaxCut on large-girth DDD-regular graphs. We give an iterative formula to evaluate performance for any DDD at any depth ppp. Looking at random DDD-regular graphs, at optimal parameters and as DDD goes to infinity, we find that the p=11p=11p=11 QAOA beats all classical algorithms (known to the authors) that are free of unproven conjectures. While the iterative formula for these DDD-regular graphs is derived by looking at a single tree subgraph, we prove that it also gives the ensemble-averaged performance of the QAOA on the Sherrington-Kirkpatrick (SK) model defined on the complete graph. We also generalize our formula to Max-qqq-XORSAT on large-girth regular hypergraphs. Our iteration is a compact procedure, but its computational complexity grows as O(p24p)O(p^2 4^p)O(p24p). This iteration is more efficient than the previous procedure for analyzing QAOA performance on the SK model, and we are able to numerically go to p=20p=20p=20. Encouraged by our findings, we make the optimistic conjecture that the QAOA, as ppp goes to infinity, will achieve the Parisi value. We analyze the performance of the quantum algorithm, but one needs to run it on a quantum computer to produce a string with the guaranteed performance.
The Quantum Approximate Optimization Algorithm (QAOA) is a general-purpose algorithm for combinatorial optimization problems whose performance can only improve with the number of layers p . While QAOA holds promise as an algorithm that can be run on near-term quantum computers, its computational power has not been fully explored. In this work, we study the QAOA applied to the Sherrington-Kirkpatrick (SK) model, which can be understood as energy minimization of n spins with all-to-all random signed couplings. There is a recent classical algorithm by Montanari that, assuming a widely believed conjecture, can efficiently find an approximate solution for a typical instance of the SK model to within ( 1 − ϵ ) times the ground state energy. We hope to match its performance with the QAOA.Our main result is a novel technique that allows us to evaluate the typical-instance energy of the QAOA applied to the SK model. We produce a formula for the expected value of the energy, as a function of the 2 p QAOA parameters, in the infinite size limit that can be evaluated on a computer with O ( 16 p ) complexity. We evaluate the formula up to p = 12 , and find that the QAOA at p = 11 outperforms the standard semidefinite programming algorithm. Moreover, we show concentration: With probability tending to one as n → ∞ , measurements of the QAOA will produce strings whose energies concentrate at our calculated value. As an algorithm running on a quantum computer, there is no need to search for optimal parameters on an instance-by-instance basis since we can determine them in advance. What we have here is a new framework for analyzing the QAOA, and our techniques can be of broad interest for evaluating its performance on more general problems where classical algorithms may fail.
Combinatorial optimization is regarded as a potentially promising application of near and long-term quantum computers. The best-known heuristic quantum algorithm for combinatorial optimization on gate-based devices, the Quantum Approximate Optimization Algorithm (QAOA), has been the subject of many theoretical and empirical studies. Unfortunately, its application to specific combinatorial optimization problems poses several difficulties: among these, few performance guarantees are known, and the variational nature of the algorithm makes it necessary to classically optimize a number of parameters. In this work, we partially address these issues for a specific combinatorial optimization problem: diluted spin models, with MAX-CUT as a notable special case. Specifically, generalizing the analysis of the Sherrington-Kirkpatrick model by Farhi et al., we establish an explicit algorithm to evaluate the performance of QAOA on MAX-CUT applied to random Erdos-Renyi graphs of expected degree ddd for an arbitrary constant number of layers ppp and as the problem size tends to infinity. This analysis yields an explicit mapping between QAOA parameters for MAX-CUT on Erdos-Renyi graphs of expected degree ddd, in the limit d→∞d \to \inftyd→∞, and the Sherrington-Kirkpatrick model, and gives good QAOA variational parameters for MAX-CUT applied to Erdos-Renyi graphs. We then partially generalize the latter analysis to graphs with a degree distribution rather than a single degree ddd, and finally to diluted spin-models with DDD-body interactions (D≥3D \geq 3D≥3). We validate our results with numerical experiments suggesting they may have a larger reach than rigorously established; among other things, our algorithms provided good initial, if not nearly optimal, variational parameters for very small problem instances where the infinite-size limit assumption is clearly violated.
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.
We study the performance of local quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) for the maximum cut problem, and their relationship to that of classical algorithms. (1) We prove that every (quantum or classical) one-local algorithm achieves on DDD-regular graphs of girth >5> 5>5 a maximum cut of at most 1/2+C/D1/2 + C/\sqrt{D}1/2+C/D​ for C=1/2≈0.7071C=1/\sqrt{2} \approx 0.7071C=1/2​≈0.7071. This is the first such result showing that one-local algorithms achieve a value bounded away from the true optimum for random graphs, which is 1/2+P∗/D+o(1/D)1/2 + P_*/\sqrt{D} + o(1/\sqrt{D})1/2+P∗​/D​+o(1/D​) for P∗≈0.7632P_* \approx 0.7632P∗​≈0.7632. (2) We show that there is a classical kkk-local algorithm that achieves a value of 1/2+C/D−O(1/k)1/2 + C/\sqrt{D} - O(1/\sqrt{k})1/2+C/D​−O(1/k​) for DDD-regular graphs of girth >2k+1> 2k+1>2k+1, where C=2/π≈0.6366C = 2/π\approx 0.6366C=2/π≈0.6366. This is an algorithmic version of the existential bound of Lyons and is related to the algorithm of Aizenman, Lebowitz, and Ruelle (ALR) for the Sherrington-Kirkpatrick model. This bound is better than that achieved by the one-local and two-local versions of QAOA on high-girth graphs. (3) Through computational experiments, we give evidence that the ALR algorithm achieves better performance than constant-locality QAOA for random DDD-regular graphs, as well as other natural instances, including graphs that do have short cycles. Our experimental work suggests that it could be possible to extend beyond our theoretical constraints. This points at the tantalizing possibility that O(1)O(1)O(1)-local quantum maximum-cut algorithms might be *pointwise dominated* by polynomial-time classical algorithms, in the sense that there is a classical algorithm outputting cuts of equal or better quality *on every possible instance*. This is in contrast to the evidence that polynomial-time algorithms cannot simulate the probability distributions induced by local quantum algorithms.
Provable Classical and Quantum Local Algorithms for Max-kkk-Cut and Quantum Advantage at Moderate Girth
Broadening the study of quantum optimization algorithms from binary to kkk-element alphabets has been shown to open new avenues for potential quantum advantage. A quantum advantage claim for approximate optimization requires showing that, under the same assumptions, an efficient quantum algorithm provably achieves a better performance than can be proven for the best known efficient classical algorithms. We study local classical and quantum algorithms for Max-kkk-Cut on ddd-regular graphs of girth ggg. We advance classical algorithms for this problem by developing a local vector algorithm based on the explicit vector construction of Thompson, Parekh, and Marwaha (TPM). Our algorithm gives the best provable cut fraction guarantee among known efficient classical algorithms on regular graphs for k≥3k\geq3k≥3. To evaluate the performance of QAOA under identical assumptions of girth and regularity, we develop tensor network techniques for general ddd and k≥2k \geq 2k≥2. In addition, using an equivalence to a coupled qudit--boson system, we compute the QAOA performance in the infinite-degree limit. Together, these techniques give provable guarantees on QAOA performance on large graphs. Despite the improvements we introduce to the classical algorithm, QAOA achieves a better cut fraction guarantee for depths p≥9p\geq 9p≥9, corresponding to girth g≥20g\geq 20g≥20, for both finite- and infinite-degree regimes. Thus, we obtain an apparent quantum advantage from applying QAOA to the Max-kkk-Cut problem.
We explore strategies aimed at reducing the amount of computation, both quantum and classical, required to run the Quantum Approximate Optimization Algorithm (QAOA). First, following Wurtz et al. [Phys.Rev A 104:052419], we consider the standard QAOA with instance-independent "tree" parameters chosen in advance. These tree parameters are chosen to optimize the MaxCut expectation for large girth graphs. We provide extensive numerical evidence supporting the performance guarantee for tree parameters conjectured in [Phys.Rev A 103:042612] and see that the approximation ratios obtained with tree parameters are typically well beyond the conjectured lower bounds, often comparable to performing a full optimization. This suggests that in practice, the QAOA can achieve near-optimal performance without the need for parameter optimization. Next, we modify the warm-start QAOA of Tate et al. [Quantum 7:1121]. The starting state for the QAOA is now an optimized product state associated with a solution of the Goemans-Williamson (GW) algorithm. Surprisingly, the tree parameters continue to perform well for the warm-start QAOA. We find that for random 3-regular graphs with hundreds of vertices, the expected cut obtained by the warm-start QAOA at depth p≳3p \gtrsim 3p≳3 is comparable to that of the standard GW algorithm. Our numerics on random instances do not provide general performance guarantees but do provide substantial evidence that there exists a regime of instance sizes in which the QAOA finds good solutions at low depth without the need for parameter optimization. For each instance studied, we classically compute the expected size of the QAOA distribution of cuts; producing the actual cuts requires running on a quantum computer.