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. 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.

翻译：QAOA 使用 QAOA 来评估任何深度的$D的性能。 以随机的 $D 常规图形, 以最优的参数和以美元表示的乐观值, 我们发现, $p= 11$ QAOA 的算法比所有的经典算法( 作者们知道的) 更加精细。 我们用大毛线将 QAAA 的计算法( 作者们知道的) 比所有经典算法更加简单化。 我们用大毛线图对 Max- $ DO A 应用 QAA 的迭接式公式, 以大毛线图来计算这些美元常规图形。 我们的计算法性能也比常规的ODA 高。 我们的计算法在SQOA 上也比常规的计算法更精准化。