The Quantum Approximate Optimization Algorithm (QAOA) is a general purpose quantum algorithm designed for combinatorial optimization. We analyze its expected performance and prove concentration properties at any constant level (number of layers) on ensembles of random combinatorial optimization problems in the infinite size limit. These ensembles include mixed spin models and Max-$q$-XORSAT on sparse random hypergraphs. Our analysis can be understood via a saddle-point approximation of a sum-over-paths integral. This is made rigorous by proving a generalization of the multinomial theorem, which is a technical result of independent interest. We then show that the performance of the QAOA at constant levels for the pure $q$-spin model matches asymptotically the ones for Max-$q$-XORSAT on random sparse Erd\H{o}s-R\'{e}nyi hypergraphs and every large-girth regular hypergraph. Through this correspondence, we establish that the average-case value produced by the QAOA at constant levels is bounded away from optimality for pure $q$-spin models when $q\ge 4$ and is even. This limitation gives a hardness of approximation result for quantum algorithms in a new regime where the whole graph is seen.
翻译:QAOA 是一个通用的量子算法,用于组合优化。 我们分析其预期性能, 并证明在无限大小限制范围内随机组合优化问题的组合中, 任意组合优化问题在任何恒定水平( 层数) 上的浓度属性。 这些组合包括混合旋转模型和随机随机高光仪上的最大- Q$- XORSAT。 我们的分析可以通过一个总和超正方形组合的支撑点近似值来理解。 通过这个对函, 我们通过证明多名论的概括化而使分析变得严格。 这是独立兴趣的技术结果。 然后我们显示, 纯$- 平面模型在恒定水平上的QAAAA, 与随机稀疏的 Max- q$- XORSAT 的常态匹配。 随机稀薄的 Erd\ H{s- R\\ { { { nyi}i 高光度和每个大色常规高光谱可以理解。 我们通过这个对应, 我们确定, 当最优的美元正值在最大正值中, QAA a a a oral awayd res respalal ex Exalal dealal 。