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 $1/2 + C/\sqrt{D}$ for $C=1/\sqrt{2} \approx 0.7071$. 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 $1/2 + P_*/\sqrt{D} + o(1/\sqrt{D})$ for $P_* \approx 0.7632$. (2) We show that there is a classical $k$-local algorithm that achieves a value of $1/2 + C/\sqrt{D} - O(1/\sqrt{k})$ for $D$-regular graphs of girth $> 2k+1$, where $C = 2/\pi \approx 0.6366$. 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.

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高周长图最大割的经典算法和量子限制

我们研究了局部量子算法的性能，例如用于最大割问题的量子近似优化算法 (QAOA)，以及它们与经典算法的关系。(1) 我们证明了每个（量子或经典）单局部算法在 $D$-周长 $>5$ 的正则图上实现了至多 $1/2 + C/\sqrt{D}$ 的最大切割C=1/\sqrt{2} \约 0.7071$。这是第一个这样的结果，表明单局部算法实现的值远离随机图的真实最优值，即 $1/2 + P_*/\sqrt{D} + o(1/\sqrt{D}) $ 为 $P_* \约 0.7632$。(2) 我们展示了经典的 $k$-local 算法，对于 $D$-regular 实现了 $1/2 + C/\sqrt{D} - O(1/\sqrt{k})$ 的值周长图 $> 2k+1$，其中 $C = 2/\pi \approx 0.6366$。这是里昂存在界的算法版本，与谢林顿-柯克帕特里克模型的 Aizenman、Lebowitz 和 Ruelle (ALR) 算法有关。这个界限比在高周长图上的 QAOA 的一个局部和两个局部版本实现的更好。(3) 通过计算实验，我们证明 ALR 算法在随机 $D$-正则图以及其他自然实例（包括确实具有短周期的图）上实现了比恒定局部性 QAOA 更好的性能。我们的实验工作表明，有可能超越我们的理论限制。这指出了$O(1)$-局部量子最大割算法可能被多项式时间经典算法*逐点支配*的诱人可能性，从某种意义上说，有一个经典算法在每个可能的实例上输出相同或更好质量的切割*。这与多项式时间算法无法模拟由局部量子算法引起的概率分布的证据形成对比。