A Unified Frequency Principle for Quantum and Classical Machine Learning
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866912805209243648 |
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| author | Lu, Rundi Zhang, Ruiqi Li, Weikang Wei, Zhaohui Deng, Dong-Ling Liu, Zhengwei |
| author_facet | Lu, Rundi Zhang, Ruiqi Li, Weikang Wei, Zhaohui Deng, Dong-Ling Liu, Zhengwei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03169 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Unified Frequency Principle for Quantum and Classical Machine Learning Lu, Rundi Zhang, Ruiqi Li, Weikang Wei, Zhaohui Deng, Dong-Ling Liu, Zhengwei Quantum Physics 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. |
| title | A Unified Frequency Principle for Quantum and Classical Machine Learning |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2601.03169 |