这篇论文点破了一个反直觉事实：量子模型容量越大越难训练，并给出了用对称性绕过贫瘠高原的数学方法。
该研究指出参数化量子电路(PQCs)的巨大希尔伯特空间容量是导致贫瘠高原(BPs)（梯度指数平坦）的直接数学原因，并建立了动力学李代数(DLAs)与几何量子ML的框架。在非线性二分类任务中验证，无结构架构通过不可扩展参数化达到近完美训练精度（量子过拟合），而嵌入群论几何先验作为结构正则化器。通过将DLA增长限制为多项式级别，牺牲原始记忆能力来保证可扩展、梯度丰富的训练景观，提供了可扩展量子神经网络的“按设计可训练”路径。
Beyond the Expressivity-Trainability Paradox: A Dynamical Lie Algebra Perspective on Navigating Barren Plateaus in Quantum Machine Learning
As Quantum Machine Learning (QML) transitions toward practical implementation, the field faces a critical architectural bottleneck that challenges the fundamental assumptions of classical statistical learning theory. In classical deep learning, increasing model capacity typically risks overfitting. However, this study advances a counter-intuitive paradigm: unstructured contemporary QML architectures suffer from a profound state of quantum underfitting, driven by the "expressivity-trainability paradox." We demonstrate that the vast Hilbert space capacity of Parameterized Quantum Circuits (PQCs)-traditionally chased as the source of quantum advantage is the direct mathematical cause of Barren Plateaus (BPs), where gradient landscapes become exponentially flat. By synthesizing recent breakthroughs in Dynamical Lie Algebras (DLAs) and Geometric QML, we establish a comprehensive framework linking the algebraic dimension of circuit generators to their optimization dynamics. Furthermore, we empirically validate this framework on a non-linear binary classification task, illuminating a uniquely quantum manifestation of the bias-variance tradeoff: while unstructured architectures achieve near-perfect training accuracy via unscalable parameterization (quantum overfitting), embedding group-theoretic geometric priors acts as a structural regularizer. By restricting the DLA growth to a polynomial regime, our symmetry-preserving approach sacrifices raw memorization capacity to guarantee scalable, gradient-rich training landscapes, offering a robust roadmap for "Trainability-by-Design" in scalable quantum neural networks.