Combinatorial optimization is regarded as a potentially promising application of near and long-term quantum computers. The best-known heuristic quantum algorithm for combinatorial optimization on gate-based devices, the Quantum Approximate Optimization Algorithm (QAOA), has been the subject of many theoretical and empirical studies. Unfortunately, its application to specific combinatorial optimization problems poses several difficulties: among these, few performance guarantees are known, and the variational nature of the algorithm makes it necessary to classically optimize a number of parameters. In this work, we partially address these issues for a specific combinatorial optimization problem: diluted spin models, with MAX-CUT as a notable special case. Specifically, generalizing the analysis of the Sherrington-Kirkpatrick model by Farhi et al., we establish an explicit algorithm to evaluate the performance of QAOA on MAX-CUT applied to random Erdos-Renyi graphs of expected degree $d$ for an arbitrary constant number of layers $p$ and as the problem size tends to infinity. This analysis yields an explicit mapping between QAOA parameters for MAX-CUT on Erdos-Renyi graphs of expected degree $d$, in the limit $d \to \infty$, and the Sherrington-Kirkpatrick model, and gives good QAOA variational parameters for MAX-CUT applied to Erdos-Renyi graphs. We then partially generalize the latter analysis to graphs with a degree distribution rather than a single degree $d$, and finally to diluted spin-models with $D$-body interactions ($D \geq 3$). We validate our results with numerical experiments suggesting they may have a larger reach than rigorously established; among other things, our algorithms provided good initial, if not nearly optimal, variational parameters for very small problem instances where the infinite-size limit assumption is clearly violated.
翻译:组合优化被视为近长期量子计算机的潜在有希望的应用。 在这项工作中,我们部分解决了在基于门的装置上组合优化最著名的超光速量量算法(QAOA)是许多理论和经验研究的主题。 不幸的是,在特定的组合优化问题中应用它带来了一些困难:在这些困难中,很少有性能保障,而且算法的变异性性质使得有必要传统地优化一些参数。在这项工作中,我们部分地解决了这些问题,解决了一个特定的组合优化问题:稀释的旋转模型,以MAX-CUT为显著的特例。具体地说,将Sherrington-Kirkprialimation Aalth(QA)模型的分析概括起来,我们为随机的Erdos-Renydal-Reny图形应用了QD(美元) 模型的性能性能, 以任意的不变的数值降价值为美元(美元),而以问题大小为常数的数值来计算。本次分析得出一个清晰的A-Kirk-QA(美元)模型分析结果,以直径为直径直到数字的数值。