AI 中文总结
针对变分量子本征求解器的贫瘠高原问题，本文构建理论框架表征其对SPSA算法优化动力学的影响，量化迭代复杂度与测量预算，揭示梯度衰减会使迭代次数和测量预算指数级增长。
AI 中文摘要
贫瘠高原（BP）现象对变分量子本征求解器（VQE）的可训练性构成了根本性挑战，随着系统规模增大，该现象会导致梯度呈指数级消失。尽管已有大量研究探讨了BP的几何起源，但人们对其在有限 shot 测量下对实际算法的优化动力学及复杂度的影响仍知之甚少。本文构建了一个理论框架，用于表征BP如何影响同时扰动随机近似（SPSA）算法的优化动力学，并量化由此产生的迭代复杂度和测量预算。我们推导了SPSA梯度估计量的非渐近偏差与方差表征，引入信噪比分析以量化梯度可靠性，并建立了有限 shot 测量下SPSA的收敛保证。研究结果表明，与BP相关的梯度能量指数级衰减，会导致达到固定相对优化精度所需的迭代次数呈指数级增加，进而使总测量预算也呈指数级增长。
英文摘要
The barren plateau (BP) phenomenon poses a fundamental challenge to the trainability of variational quantum eigensolvers (VQEs) by causing exponentially vanishing gradients as the system size increases. While extensive studies have investigated the geometric origins of BP, its impact on the optimization dynamics and complexity of practical algorithms under finite-shot measurements remains poorly understood. In this paper, we develop a theoretical framework that characterizes how the BP affects the optimization dynamics of the Simultaneous Perturbation Stochastic Approximation (SPSA) algorithm and quantifies the resulting iteration complexity and measurement budget. We derive non-asymptotic bias and variance characterizations of the SPSA gradient estimator, introduce a signal-to-noise ratio analysis to quantify gradient reliability, and establish convergence guarantees for SPSA under finite-shot measurements. Our results show that the exponentially decaying gradient energy associated with BP leads to an exponential increase in the number of iterations required to achieve a fixed relative optimization accuracy, which in turn results in an exponential increase in the total measurement budget.
