We study MaxCut on 3-regular graphs of minimum girth $g$ for various $g$'s. We obtain new lower bounds on the maximum cut achievable in such graphs by analyzing the Quantum Approximate Optimization Algorithm (QAOA). For $g \geq 16$, at depth $p \geq 7$, the QAOA improves on previously known lower bounds. Our bounds are established through classical numerical analysis of the QAOA's expected performance. This analysis does not produce the actual cuts but establishes their existence. When implemented on a quantum computer, the QAOA provides an efficient algorithm for finding such cuts, using a constant-depth quantum circuit. To our knowledge, this gives an exponential speedup over the best known classical algorithm guaranteed to achieve cuts of this size on graphs of this girth. We also apply the QAOA to the Maximum Independent Set problem on the same class of graphs.

翻译：我们研究了在不同围长$g$下三正则图的最小围长$g$上的最大割问题。通过分析量子近似优化算法（QAOA），我们获得了此类图中可实现的最大割的新下界。对于$g \geq 16$且深度$p \geq 7$的情况，QAOA改进了先前已知的下界。这些下界是通过对QAOA期望性能的经典数值分析建立的。该分析并不产生实际的割，但证明了其存在性。当在量子计算机上实现时，QAOA通过恒定深度的量子电路，为寻找此类割提供了高效算法。据我们所知，这相比已知在相同围长图上保证能达到该割尺寸的最佳经典算法，实现了指数级加速。我们还将QAOA应用于同一类图上的最大独立集问题。