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arXiv:2601.03169

A Unified Frequency Principle for Quantum and Classical Machine Learning

Rundi Lu, Ruiqi Zhang, Weikang Li, Zhaohui Wei, Dong-Ling Deng, Zhengwei Liu

quant-phSubmitted Jan 6, 2026• Announced Jan 7, 2026• 0 citations
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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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26 pages, 6 figures. Comments are welcome

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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 ω\boldsymbolωω is suppressed by a factor (1−2γ)∥ω∥1(1-2γ)^{\|\boldsymbolω\|_1}(1−2γ)∥ω∥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.
The mechanism governing the training dynamics of Quantum Neural Networks (QNNs) remains under-explored. In classical Deep Neural Networks (DNNs), training is dominated by "Spectral Bias," i.e. prioritizing learning low-frequency components and struggle for high-frequency details. In this work, we theoretically and empirically identify a distinct mechanism in QNNs, which we term Spectral Amplitude Priority. By analyzing the frequency-domain gradients and residual dynamics via the Quantum Neural Tangent Kernel (QNTK), we prove that QNN training is governed primarily by the magnitude of spectral components rather than their frequency indices. Consequently, QNNs can efficiently capture high-frequency functions-provided they have significant amplitude-thereby overcoming the inherent limitations of their classical counterparts. We validate this principle on both synthetic high-frequency functions and quantum-advantage tasks. The results show that QNNs significantly outperform DNNs in high-frequency tasks, offering an explanation for QNNs' superior expressivity in complex spectral landscapes.
Quantum neural networks (QNNs), typically implemented via parameterized quantum circuits (PQCs), offer a potential route to quantum advantage in learning. We review recent progress through the Fourier structure of PQCs: the encoding Hamiltonian fixes the accessible frequency set, while training estimates the corresponding Fourier coefficients. This perspective explains how QNNs can mitigate the spectral bias that limits classical deep neural networks and motivates task-specialized designs. In particular, quantum ordinary differential equation (QODE) solver, quantum convolutional neural networks (QCNNs) and quantum reinforcement learning (QRL) exploit frequency pinching to better capture high-frequency image features and structured value landscapes. We further discuss trainability on noisy intermediate-scale quantum (NISQ) hardware, emphasizing Lie-algebraic and Fourier views of barren plateaus and practical mitigation strategies. Together, the Fourier and Lie perspectives yield concrete design principles and motivate quantum-native architectures built around spectral bias avoidance rather than classical imitation. Quantum neural networks (QNNs), implemented via parameterized quantum circuits (PQCs), promise a path to quantum advantage in learning by addressing the limitations of classical deep neural networks. Here, the authors explore the Fourier structure of PQCs to mitigate spectral bias, enhancing applications like quantum ODE solvers and quantum reinforcement learning, with implications for designing quantum-native architectures.
Quantum neural networks (QNNs) have emerged as a leading strategy to establish applications in machine learning, chemistry, and optimization. While the applications of QNN have been widely investigated, its theoretical foundation remains less understood. In this paper, we formulate a theoretical framework for the expressive ability of data re-uploading quantum neural networks that consist of interleaved encoding circuit blocks and trainable circuit blocks. First, we prove that single-qubit quantum neural networks can approximate any univariate function by mapping the model to a partial Fourier series. We in particular establish the exact correlations between the parameters of the trainable gates and the Fourier coefficients, resolving an open problem on the universal approximation property of QNN. Second, we discuss the limitations of single-qubit native QNNs on approximating multivariate functions by analyzing the frequency spectrum and the flexibility of Fourier coefficients. We further demonstrate the expressivity and limitations of single-qubit native QNNs via numerical experiments. We believe these results would improve our understanding of QNNs and provide a helpful guideline for designing powerful QNNs for machine learning tasks.
Quantum machine learning models based on parameterized circuits can be viewed as Fourier series approximators. However, they often struggle to learn functions with multiple frequency components, particularly high-frequency or non-dominant ones; a phenomenon we term the quantum Fourier parameterization bias. Inspired by recent advances in classical Fourier neural operators (FNOs), we adapt the multi-stage residual learning idea to the quantum domain, iteratively training additional quantum modules on the residuals of previous stages. We evaluate our method on a synthetic benchmark composed of spatially localized frequency components with diverse envelope shapes (Gaussian, Lorentzian, triangular). Systematic experiments show that the number of qubits, the encoding scheme, and residual learning are all crucial for resolving multiple frequencies; residual learning alone can improve test MSE significantly over a single-stage baseline trained for the same total number of epochs. Our work provides a practical framework for enhancing the spectral expressivity of quantum models and offers new insights into their frequency-learning behavior.
In this work, we investigate the phenomenon of spectral bias in quantum machine learning, where, in classical settings, models tend to fit low-frequency components of a target function earlier during training than high-frequency ones, demonstrating a frequency-dependent rate of convergence. We study this effect specifically in parameterised quantum circuits (PQCs). Leveraging the established formulation of PQCs as Fourier series, we prove that spectral bias in this setting arises from the ``redundancy'' of the Fourier coefficients, which denotes the number of terms in the analytical form of the model contributing to the same frequency component. The choice of data encoding scheme dictates the degree of redundancy for a Fourier coefficient. We find that the magnitude of the Fourier coefficients' gradients during training strongly correlates with the coefficients' redundancy. We then further demonstrate this empirically with three different encoding schemes. Additionally, we demonstrate that PQCs with greater redundancy exhibit increased robustness to random perturbations in their parameters at the corresponding frequencies. We investigate how design choices affect the ability of PQCs to learn Fourier sums, focusing on parameter initialization scale and entanglement structure, finding large initializations and low-entanglement schemes tend to slow convergence.
Quantum machine learning models based on parameterized circuits can be viewed as Fourier series approximators. However, they often struggle to learn functions with multiple frequency components, particularly high‑frequency or non-dominant ones - a phenomenon we term the \textit{quantum Fourier parameterization bias}. Inspired by recent advances in classical Fourier neural operators (FNOs), we adapt the multi-stage residual learning idea to the quantum domain, iteratively training additional quantum modules on the residuals of previous stages. We evaluate our method on a synthetic benchmark composed of spatially localized frequency components with diverse envelope shapes (Gaussian, Lorentzian, triangular). Systematic experiments show that the number of qubits, the encoding scheme, and residual learning are all crucial for resolving multiple frequencies; residual learning alone can improve test MSE significantly over a single‑stage baseline trained for the same total number of epochs. Our work provides a practical framework for enhancing the spectral expressivity of quantum models and offers new insights into their frequency-learning behavior.