In this work, we compare the performance of the Quantum Approximate Optimization Algorithm (QAOA) with state-of-the-art classical solvers such as Gurobi and MQLib to solve the combinatorial optimization problem MaxCut on 3-regular graphs. The goal is to identify under which conditions QAOA can achieve "quantum advantage" over classical algorithms, in terms of both solution quality and time to solution. One might be able to achieve quantum advantage on hundreds of qubits and moderate depth $p$ by sampling the QAOA state at a frequency of order 10 kHz. We observe, however, that classical heuristic solvers are capable of producing high-quality approximate solutions in linear time complexity. In order to match this quality for $\textit{large}$ graph sizes $N$, a quantum device must support depth $p>11$. Otherwise, we demonstrate that the number of required samples grows exponentially with $N$, hindering the scalability of QAOA with $p\leq11$. These results put challenging bounds on achieving quantum advantage for QAOA MaxCut on 3-regular graphs. Other problems, such as different graphs, weighted MaxCut, maximum independent set, and 3-SAT, may be better suited for achieving quantum advantage on near-term quantum devices.

翻译：本文比较了量子近似优化算法（QAOA）与Gurobi、MQLib等最先进经典求解器在解决3-正则图上的组合优化问题MaxCut时的性能表现。研究目标是从解质量与求解时间两个维度，明确QAOA在何种条件下能够对经典算法实现"量子优势"。通过在约10 kHz频率下对QAOA态进行采样，我们可能能够在数百量子比特和中等深度$p$条件下获得量子优势。然而，我们观察到经典启发式求解器能以线性时间复杂度生成高质量近似解。为在大规模图$N$上匹配此质量，量子设备必须支持深度$p>11$。否则，我们证明所需样本数将随$N$呈指数增长，阻碍深度$p\leq11$的QAOA可扩展性。这些结果为在3-正则图上实现QAOA MaxCut的量子优势设定了严峻的边界。其他问题（如不同图结构、加权MaxCut、最大独立集和3-SAT）或许更适合在近期量子设备上实现量子优势。