Maximum cut (MaxCut) on graphs is a classic NP-hard problem. In quantum computing, Farhi, Gutmann, and Goldstone proposed the Quantum Approximate Optimization Algorithm (QAOA) for solving the MaxCut problem. Its guarantee on cut fraction (the fraction of edges in the output cut over all edges) was mainly studied for high-girth graphs, i.e., graphs with only long cycles. On the other hand, low-girth graphs are ubiquitous in theoretical computer science, including expander graphs being outstanding examples with wide applications in theory and beyond. In this paper, we apply QAOA to MaxCut on a set of expander graphs proposed by Mohanty and O'Donnell known as additive product graphs. Additionally, we apply multi-angle QAOA (ma-QAOA) to better utilize the graph structure of additive product graphs in ansatz design. In theory, we derive an iterative formula to calculate the expected cut fraction of such graphs. This formula also extends to the quantum MaxCut problem. On the other hand, we conduct numerical experiments to compare between best-known classical local algorithms and QAOA with constant depth. Our results demonstrate that QAOA outperforms the best-known classical algorithms by 0.3% to 5.2% on several additive product graphs, while ma-QAOA further enhances this advantage by an additional 0.6% to 2.5%. In particular, we observe cases that ma-QAOA exhibits superiority over best-known classical algorithms but QAOA does not. Furthermore, we extend our experiments to planar graphs such as tiling grid graphs, where QAOA also demonstrates an advantage.

翻译：图上的最大割（MaxCut）是一个经典的NP难问题。在量子计算领域，Farhi、Gutmann和Goldstone提出了用于求解MaxCut问题的量子近似优化算法（QAOA）。该算法在割分数（输出割中边数占总边数的比例）上的性能保证主要针对高围长图（即仅包含长环的图）进行了研究。另一方面，低围长图在理论计算机科学中普遍存在，其中扩展图是理论和实际应用中具有广泛用途的突出例子。本文中，我们将QAOA应用于由Mohanty和O'Donnell提出的加法积图（一类扩展图）上的MaxCut问题。此外，我们应用多角度QAOA（ma-QAOA）以在ansatz设计中更好地利用加法积图的结构。在理论上，我们推导了一个迭代公式来计算此类图的期望割分数。该公式也可推广至量子MaxCut问题。另一方面，我们进行了数值实验，以比较最著名的经典局部算法与恒定深度的QAOA。我们的结果表明，在多个加法积图上，QAOA的性能优于最著名的经典算法0.3%至5.2%，而ma-QAOA进一步将这一优势提升了0.6%至2.5%。特别地，我们观察到在某些情况下，ma-QAOA展现出优于最著名经典算法的性能，而QAOA则未能实现。此外，我们将实验扩展至平面图（如平铺网格图），QAOA在这些图上同样表现出优势。