We show through numerical simulation that the Quantum Approximate Optimization Algorithm (QAOA) for higher-order, random-coefficient, heavy-hex compatible spin glass Ising models has strong parameter concentration across problem sizes from $16$ up to $127$ qubits for $p=1$ up to $p=5$, which allows for straight-forward transfer learning of QAOA angles on instance sizes where exhaustive grid-search is prohibitive even for $p>1$. We use Matrix Product State (MPS) simulation at different bond dimensions to obtain confidence in these results, and we obtain the optimal solutions to these combinatorial optimization problems using CPLEX. In order to assess the ability of current noisy quantum hardware to exploit such parameter concentration, we execute short-depth QAOA circuits (with a CNOT depth of 6 per $p$, resulting in circuits which contain $1420$ two qubit gates for $127$ qubit $p=5$ QAOA) on $100$ higher-order (cubic term) Ising models on IBM quantum superconducting processors with $16, 27, 127$ qubits using QAOA angles learned from a single $16$-qubit instance. We show that (i) the best quantum processors generally find lower energy solutions up to $p=3$ for 27 qubit systems and up to $p=2$ for 127 qubit systems and are overcome by noise at higher values of $p$, (ii) the best quantum processors find mean energies that are about a factor of two off from the noise-free numerical simulation results. Additional insights from our experiments are that large performance differences exist among different quantum processors even of the same generation and that dynamical decoupling significantly improve performance for some, but decrease performance for other quantum processors. Lastly we show $p=1$ QAOA angle mean energy landscapes computed using up to a $414$ qubit quantum computer, showing that the mean QAOA energy landscapes remain very similar as the problem size changes.

翻译：我们通过数值模拟证明，针对高阶、随机系数、兼容重六边形结构的自旋玻璃伊辛模型，量子近似优化算法（QAOA）在问题规模从16量子比特到127量子比特、参数p=1到p=5的范围内表现出强烈的参数集中特性。这使得在实例规模上（即使对于p>1的情况，穷举网格搜索也难以实现）能够直接进行QAOA角度的迁移学习。我们采用不同键维度的矩阵乘积态（MPS）模拟来验证这些结果的可靠性，并使用CPLEX求解器获得这些组合优化问题的最优解。为评估当前含噪声量子硬件利用此类参数集中特性的能力，我们在IBM超导量子处理器（16、27、127量子比特规模）上执行浅层QAOA电路（每个p的CNOT深度为6，对应127量子比特p=5的电路包含1420个双量子比特门），针对100个高阶（三次项）伊辛模型，使用从单个16量子比特实例学习得到的QAOA角度进行实验。结果表明：（i）最优量子处理器在27量子比特系统中最高达到p=3、127量子比特系统中最高达到p=2时能找到更低能量解，但在更高p值时被噪声效应压制；（ii）最优量子处理器获得的平均能量与无噪声数值模拟结果相差约两倍。实验还发现：同代量子处理器之间存在显著性能差异，且动态解耦技术对部分处理器性能有显著提升，但对其他处理器反而降低性能。最后我们展示了使用最高414量子比特量子计算机计算的p=1 QAOA角度平均能量分布，证明QAOA平均能量分布随问题规模变化保持高度相似性。