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Efficiently generating ground states is hard for postselected quantum computation
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Evaluation of Quantum Approximate Optimization Algorithm based on the approximation ratio of single samples
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Spin Summations: A HighPerformance Perspective
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Spectral Planting and the Hardness of Refuting Cuts, Colorability, and Communities in Random Graphs
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An explicit vector algorithm for highgirth MaxCut
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Classical algorithms and quantum limitations for maximum cut on highgirth 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) onelocal algorithm achieves on Dregular 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 onelocal 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 klocal algorithm that achieves a value of 1/2 + C/√(D)  O(1/√(k)) for Dregular 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 SherringtonKirkpatrick model. This bound is better than that achieved by the onelocal and twolocal versions of QAOA on highgirth graphs. (3) Through computational experiments, we give evidence that the ALR algorithm achieves better performance than constantlocality QAOA for random Dregular 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 maximumcut algorithms might be *pointwise dominated* by polynomialtime 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 polynomialtime algorithms cannot simulate the probability distributions induced by local quantum algorithms.
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