发表机构
VSB – Technical University of Ostrava; IT4Innovations National Supercomputing Center, VSB – Technical University of Ostrava; Klaipeda University; Gran Sasso Science Institute; Indian Statistical Institute; Università dell’Aquila（俄斯特拉发科技大学; 俄斯特拉发科技大学 IT4创新国家超算中心; 克莱佩达大学; 格兰萨索科学研究所; 印度统计研究所; 阿奎拉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过分析VQE和QAOA中目标函数的几何特性，比较BFGS与自适应差分进化，发现参数绑定而非局部极小值数量决定全局搜索优势，为优化器选择提供实用流程。
AI 中文摘要
变分量子算法将哈密顿量、拟设和参数化的选择转化为一个经典的非凸优化问题。我们研究如何以有助于解释优化器行为的方式可视化和表征这一目标函数。我们区分了目标函数的两个性质：沿采样方向遇到的局部极小值数量，以及局部搜索端点之间质量的差异。然后，我们询问由BFGS代表的局部优化器与由jSO代表的自适应差分进化相比如何。我们不是将jSO与单次局部运行进行比较，而是允许BFGS在相同的函数评估预算内进行多次启动。这使局部搜索有多次机会探索不同的盆地，并为询问全局进化求解器何时有用提供了更强的基线。我们使用VQE和QAOA研究这些问题，重点关注阻挫、电路深度、参数绑定、混合局域性和非线性重参数化如何改变经典优化器所见的哈密顿量期望值目标。增加独立电路深度会提高采样到的局部极小值数量，但不会使全局搜索更有效。相比之下，参数绑定既产生更多重复的局部结构，也产生局部搜索端点之间质量上更大的差异，并且在这种机制下，自适应差分进化优于函数评估匹配的多启动BFGS。比较表明，局部极小值的数量本身并不能决定全局搜索是否有利：重要的区别在于不同盆地是导致同样好的解还是导致显著不同的目标值。这些结果为连接模型构建、目标函数几何、经验诊断和优化器选择提供了实用工作流程。
英文摘要
Variational quantum algorithms turn choices of Hamiltonian, ansatz, and parameterization into a classical nonconvex optimization problem. We study how this objective function can be visualized and characterized in ways that help explain optimizer behavior. We distinguish two properties of the objective: the number of local minima encountered along sampled directions and the differences in quality among local-search endpoints. We then ask how a local optimizer, represented by BFGS, compares with adaptive differential evolution, represented by jSO. Rather than comparing jSO with a single local run, we allow BFGS multiple starts within the same function-evaluation budget. This gives local search repeated opportunities to explore different basins and provides a stronger baseline for asking when a global evolutionary solver is useful. We study these questions using VQE and QAOA, focusing on how frustra- tion, circuit depth, parameter tying, mixed locality, and nonlinear repa- rameterization change the Hamiltonian expectation-value objective seen by the classical optimizer. Increasing independent circuit depth raises the sampled local-minimum count without making global search more effective. Parameter tying, by contrast, produces both more repeated local structure and much larger differences in quality among local-search endpoints, and in this regime adaptive differential evolution outperforms function-evaluation-matched multistart BFGS. The comparison shows that the number of local minima alone does not determine whether global search is advantageous: the important distinction is whether different basins lead to similarly good solutions or to substantially different objective values. These results provide a practical workflow for connecting model construction, objective- function geometry, empirical diagnostics, and optimizer choice.
