We identify a sharp interaction-degree threshold for the classical simulation of QAOA with $2$-local cost functions. At degree $3$, classical sampling from depth-$1$ QAOA with small multiplicative error would collapse the polynomial hierarchy to its third level. At degree $2$, exact classical sampling from depth-$p$ QAOA on $n$ qubits runs in time $n^{O(1)}$ whenever $p = O(\log n)$. The hard degree-$3$ instances have trivially optimizable cost functions, so sampling hardness does not by itself imply a quantum optimization advantage.

翻译：我们识别出在$2$-局部代价函数的QAOA经典模拟中一个尖锐的交互度阈值。当交互度为$3$时，对深度为$1$的QAOA进行具有小乘法误差的经典采样会将多项式层级坍缩至其第三层。当交互度为$2$时，对$n$量子比特上深度为$p$的QAOA进行精确经典采样可在$n^{O(1)}$时间内完成，只要$p = O(\log n)$。这些难度为度$3$的实例具有可平凡优化的代价函数，因此采样硬度本身并不暗示量子优化优势。