A Unified Frequency Principle for Quantum and Classical Machine Learning

• URL: http://arxiv.org/abs/2601.03169v1
• Date: Tue, 06 Jan 2026 16:44:22 GMT
• Title: A Unified Frequency Principle for Quantum and Classical Machine Learning
• Authors: Rundi Lu, Ruiqi Zhang, Weikang Li, Zhaohui Wei, Dong-Ling Deng, Zhengwei Liu,
• Abstract summary: We present a unified theoretical framework for the frequency principle (F-principle) that characterizes the training dynamics of quantum neural networks.<n>Within this framework, we prove that quantum neural networks exhibit a spectral bias toward learning low-frequency components of target functions.<n>Our results provide a frequency-domain lens that unifies classical and quantum learning dynamics, clarifies the role of noise in shaping trainability, and guides the design of noise-resilient quantum neural networks.
• Score: 9.529771617722703
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Quantum neural networks constitute a key class of near-term quantum learning models, yet their training dynamics remain not fully understood. Here, we present a unified theoretical framework for the frequency principle (F-principle) that characterizes the training dynamics of both classical and quantum neural networks. Within this framework, we prove that quantum neural networks exhibit a spectral bias toward learning low-frequency components of target functions, mirroring the behavior observed in classical deep networks. We further analyze the impact of noise and show that, when single-qubit noise is applied after encoding-layer rotations and modeled as a Pauli channel aligned with the rotation axis, the Fourier component labeled by $\boldsymbolω$ is suppressed by a factor $(1-2γ)^{\|\boldsymbolω\|_1}$. This leads to exponential attenuation of high-frequency terms while preserving the learnability of low-frequency structure. In the same setting, we establish that the resulting noisy circuits admit efficient classical simulation up to average-case error. Numerical experiments corroborate our theoretical predictions: Quantum neural networks primarily learn low-frequency features during early optimization and maintain robustness against dephasing and depolarizing noise acting on the encoding layer. Our results provide a frequency-domain lens that unifies classical and quantum learning dynamics, clarifies the role of noise in shaping trainability, and guides the design of noise-resilient quantum neural networks.

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