过参数化量子电路会因随机矩阵特性导致输出趋同,影响训练效果。
Random-Matrix-Induced Simplicity Bias in Over-parameterized Variational Quantum Circuits
- 从函数类视角揭示过参数化电路的随机矩阵本质
- 系统规模增大时,输出与梯度集中于常数附近,引发泛化崩溃
- 结构化电路设计可避免该问题,适合量子算法架构研究者
过参数化常用于提升变分量子线路(VQCs)的表达能力,但更深更参数化的线路常出现训练困难和泛化能力差的问题。本文从函数类角度提供理论解释:充分表达、无结构的变分试探态在系统规模增大时进入类似哈亚尔(Haar)分布的普适类,可观测值期望和参数梯度均随系统尺寸指数集中。因此,此类电路诱导的假设类以高概率坍缩为近似常数函数族,我们称之为简化偏差,而贫瘠平原是结果而非根源。利用随机矩阵理论和测度集中工具,我们严格刻画该普适类,并建立有限数据集上假设类的统一坍缩。进一步表明此坍缩并非不可避免:张量结构化VQCs(如张量网络或张量超网络参数化)位于该普适类之外,通过限制张量秩或键维数,可防止测度集中,保持局部可观测量输出多样性与非退化梯度信号,即使在过参数化情形下仍有效。我们的结果将贫瘠平原、表达能力限制与泛化坍缩统一于随机矩阵普适性这一结构性机制之下,凸显了架构归纳偏置在变分量子算法中的核心作用。
原文摘要 · Abstract (English)
Over-parameterization is commonly used to increase the expressivity of variational quantum circuits (VQCs), yet deeper and more highly parameterized circuits often exhibit poor trainability and limited generalization. In this work, we provide a theoretical explanation for this phenomenon from a function-class perspective. We show that sufficiently expressive, unstructured variational ansatze enter a Haar-like universality class in which both observable expectation values and parameter gradients concentrate exponentially with system size. As a consequence, the hypothesis class induced by such circuits collapses with high probability to a narrow family of near-constant functions, a phenomenon we term simplicity bias, with barren plateaus arising as a consequence rather than the root cause. Using tools from random matrix theory and concentration of measure, we rigorously characterize this universality class and establish uniform hypothesis-class collapse over finite datasets. We further show that this collapse is not unavoidable: tensor-structured VQCs, including tensor-network-based and tensor-hypernetwork parameterizations, lie outside the Haar-like universality class. By restricting the accessible unitary ensemble through bounded tensor rank or bond dimension, these architectures prevent concentration of measure, preserve output variability for local observables, and retain non-degenerate gradient signals even in over-parameterized regimes. Together, our results unify barren plateaus, expressivity limits, and generalization collapse under a single structural mechanism rooted in random-matrix universality, highlighting the central role of architectural inductive bias in variational quantum algorithms.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品;中文卡片由大模型生成,请以原文为准。