The p -stage Quantum Approximate Optimization Algorithm (QAOA p ) is a promising approach for combinatorial optimization on noisy intermediate-scale quantum (NISQ) devices, but its theoretical behavior is not well understood beyond p=1 . We analyze QAOA 2 for the maximum cut problem (MAX-CUT), deriving a graph-size-independent expression for the expected cut fraction on any D -regular graph of girth >5 (i.e. without triangles, squares, or pentagons).We show that for all degrees D≥2 and every D -regular graph G of girth >5 , QAOA 2 has a larger expected cut fraction than QAOA 1 on G . However, we also show that there exists a 2 -local randomized classical algorithm A such that A has a larger expected cut fraction than QAOA 2 on all G . This supports our conjecture that for every constant p , there exists a local classical MAX-CUT algorithm that performs as well as QAOA p on all graphs.
The p -stage Quantum Approximate Optimization Algorithm (QAOA p ) is a promising approach for combinatorial optimization on noisy intermediate-scale quantum (NISQ) devices, but its theoretical behavior is not well understood beyond p=1 . We analyze QAOA 2 for the maximum cut problem (MAX-CUT), deriving a graph-size-independent expression for the expected cut fraction on any D -regular graph of girth >5 (i.e. without triangles, squares, or pentagons).We show that for all degrees D≥2 and every D -regular graph G of girth >5 , QAOA 2 has a larger expected cut fraction than QAOA 1 on G . However, we also show that there exists a 2 -local randomized classical algorithm A such that A has a larger expected cut fraction than QAOA 2 on all G . This supports our conjecture that for every constant p , there exists a local classical MAX-CUT algorithm that performs as well as QAOA p on all graphs.
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 D-regular graphs of girth >5 a maximum cut of at most 1/2+C/D for C=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) for P∗≈0.7632. (2) We show that there is a classical k-local algorithm that achieves a value of 1/2+C/D−O(1/k) for D-regular graphs of girth >2k+1, where C=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 D-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)-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-k-Cut and Quantum Advantage at Moderate Girth
Broadening the study of quantum optimization algorithms from binary to k-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-k-Cut on d-regular graphs of girth g. 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≥3. To evaluate the performance of QAOA under identical assumptions of girth and regularity, we develop tensor network techniques for general d and k≥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≥9, corresponding to girth g≥20, for both finite- and infinite-degree regimes. Thus, we obtain an apparent quantum advantage from applying QAOA to the Max-k-Cut problem.
Abstract Maximum cut (MaxCut) on graphs is a classic NP-hard problem. In quantum computing, Farhi, Gutmann, and Goldstone proposed the Quantum Approximate Optimization Algorithm (QAOA) for solving the MaxCut problem. Its guarantee on cut fraction (the fraction of edges in the output cut over all edges) was mainly studied for high-girth graphs, i.e., graphs with only long cycles. On the other hand, low-girth graphs are ubiquitous in theoretical computer science, including expander graphs being outstanding examples with wide applications in theory and beyond. In this paper, we apply QAOA to MaxCut on a set of expander graphs proposed by Mohanty and O'Donnell known as additive product graphs. Additionally, we apply multi-angle QAOA (ma-QAOA) to better utilize the graph structure of additive product graphs in ansatz design. In theory, we derive an iterative formula to calculate the expected cut fraction of such graphs. This formula also extends to the quantum MaxCut problem. On the other hand, we conduct numerical experiments to compare between best-known classical local algorithms and QAOA with constant depth. Our results demonstrate that QAOA outperforms the best-known classical algorithms by 0.3% to 5.2% on several additive product graphs, while ma-QAOA further enhances this advantage by an additional 0.6% to 2.5%. In particular, we observe cases that ma-QAOA exhibits superiority over best-known classical algorithms but QAOA does not. Furthermore, we extend our experiments to planar graphs such as tiling grid graphs, where QAOA also demonstrates an advantage.
Every 3-regular graph of girth at least g has a cut containing at least a fraction Mg of its edges, and the best rigorous bounds at large girth are 0.912≤limg→∞Mg≤0.92386. Lower bounds on Mg follow from the performance of local algorithms: on a graph of girth g≥2p+2 a depth-p local algorithm sees the same tree around every edge, so its expected cut fraction on that tree bounds Mg from below. We apply this strategy to the Variational Iterative Rotation Algorithm (VIRAL), a classical local algorithm, and evaluate it on the depth-p tree for every even depth up to p=32. The resulting bounds exceed those recently obtained from the Quantum Approximate Optimization Algorithm (QAOA) for every p≥4, by 0.005 to 0.011, and they extend the range of girths covered from g=36 to g=66. Notably, at g=66 we obtain M66≥0.9127 at 99.99999% confidence level, which raises the asymptotic lower bound from 0.912 to 0.9127.
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-p QAOA expectation on high-girth d-regular graphs of arbitrary size. The cost of evaluating the formula is exponential in the QAOA depth p but does not depend on the graph size. Evaluating this formula for Max-k-Cut problem for p≤4, we identify parameter regimes (k=3 with degree d≤10 and k=4 with d≤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≤20. Our results show that moving beyond binary to integer optimization problems can open up new avenues for quantum advantage.
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(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=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.