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.
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, 以最优参数, 并随着美元进入无限度, 我们发现, $p= 11$ QAAA 的性能比所有经典算法( 作者们知道) 的更深的 $p美元 。 我们用大毛色图将 QAAA 的迭接公式应用到 MaxCut MaxCut 大毛色 $D$ 常规图形。 我们用一个迭接式公式来评估任何美元( 以最佳参数计算, 以美元计算到无限值 QA) 的性能。 我们用一个通用公式到 Max- $XOSAT ( 作者们知道 ), 在大毛色色图中, 将我们方的性能 QA 的 迭代用 QA 的性能 。 将我们用 IMINA 的性能化到 。 以 QO QQA 进行常规的性 。