发表机构
Brno University of Technology; Institute of Physics of Materials, Czech Academy of Sciences（布尔诺理工大学; 捷克科学院材料物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过Wishart超环面随机场和Kac-Rice公式分析VQA损失景观，发现局部最小值分布的相变，并提出对称性约简以降低自由度，从而提升可训练性并解决更大问题。
AI 中文摘要
变分量子算法（VQAs）的训练效率由其损失景观的几何形状决定，这些损失景观可以通过将这些函数映射到流形上的随机场来进行形式化分析。我们可以不直接研究VQAs，而是研究相应的随机场，在我们的案例中，即Wishart超环面随机场（WHRFs）。我们主要关注WHRFs的临界点（尤其是局部最小值）的分布。为此，我们提出、重新表述并模拟了Kac-Rice公式。研究结果识别出局部最小值分布中的相变。超过特定阈值后，局部最小值在函数值上集中在全局最小值附近，这意味着即使是局部最小值也是全局最小值的良好近似。该阈值由问题哈密顿量的自由度与VQA中独立参数数量之间的比率决定。由于自由度参数呈指数增长，我们提出对称性约简操作以降低自由度。这在数学上降低了维度，缩小了所需参数阈值，从而能够解决更大的问题。
英文摘要
The training efficiency of Variational Quantum Algorithms (VQAs) is dictated by the geometry of their loss landscapes, which can be formally analysed by mapping these functions to random fields on manifolds. Instead of VQAs, we can then directly study the corresponding random fields, in our case, the Wishart Hypertoroidal Random Fields (WHRFs). We are mostly interested in the distribution of critical points (especially local minima) of WHRFs. For this purpose, the Kac-Rice formula is presented, reformulated, and simulated. The findings identify a phase transition in the distribution of local minima. Beyond a specific threshold, local minima concentrate near the global minimum in function value, meaning that even local minima are good approximators of the global one. The threshold is governed by the ratio between the problem Hamiltonian degrees of freedom and by the number of independent parameters in the VQA. Since the degrees of freedom parameter scales exponentially, we propose symmetry reduction operations to lower the degrees of freedom. This mathematically reduces the dimension, scaling down the required parameter threshold and enabling to solve bigger problems.
