李代数视角下参数化量子电路的标度行为
Scaling Behavior of Parameterized Quantum Circuits from a Lie-Algebraic Perspective
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中文总结 AI 辅助
该研究从李代数视角探究参数化量子电路的标度行为，发现雅可比有效维度可作为衡量其局部可及可观测量自由度的有用标度参数。
中文摘要 AI 辅助
理解参数化量子电路的性能如何随可用资源标度变化，对表征其可训练性和有效模型容量十分重要。本研究在参数化量子电路中数值探究了数据标度、模型标度与计算标度，并考察李代数量作为模型规模的替代度量。除电路参数数量外，我们还考虑了动力学李代数的维度、可观测量轨道维度，以及定义为参数化可观测量轨道雅可比矩阵秩的雅可比有效维度。采用基于随机生成的泡利串生成器的回归任务，我们观察到在所研究的范围内，随着训练数据集规模、参数规模及优化迭代次数的增加，损失值不断降低。对于模型标度，动力学李代数与可观测量轨道维度随参数规模增加迅速饱和，而雅可比有效维度则与参数规模保持强相关性，并表现出可比的标度行为。这些结果表明，雅可比有效维度提供了一种感知几何的度量，用于表征有限深度参数化量子电路的局部可及可观测量自由度，可作为超出标称参数规模的有用标度参数。
英文摘要
Understanding how the performance of parameterized quantum circuits scales with available resources is important for characterizing their trainability and effective model capacity. In this study, we numerically investigate data scaling, model scaling, and compute scaling in parameterized quantum circuits and examine Lie-algebraic quantities as alternative measures of model size. In addition to the number of circuit parameters, we consider the dimension of the dynamical Lie algebra, the observable-orbit dimension, and a Jacobian effective dimension defined as the rank of the Jacobian of the parameterized observable orbit. Using a regression task with randomly generated Pauli-string generators, we observe decreasing loss with increasing training dataset size, parameter size, and number of optimization iterations over the ranges investigated. For model scaling, the dynamical Lie algebra and observable orbit dimensions rapidly saturate as the parameter size increases, whereas the Jacobian effective dimension remains strongly correlated with the parameter size and exhibits comparable scaling behavior. These results suggest that the Jacobian effective dimension provides a geometry-aware measure of the locally accessible observable degrees of freedom of finite-depth parameterized quantum circuits and may serve as a useful scaling parameter beyond the nominal parameter size.
