Mitigating Frequency Learning Bias in Quantum Models via Multi‑Stage Residual Learning

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.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.

quant-pharXiv:2603.10083Mar 11, 20260 citations

A Unified Frequency Principle for Quantum and Classical Machine Learning

Rundi Lu, Ruiqi Zhang, Weikang Li +3 more

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.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.

quant-pharXiv:2601.03169Jan 7, 20260 citations

Spectral Bias in Variational Quantum Machine Learning

Callum Duffy, Marcin Jastrzebski

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.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.

arXiv:2506.22555Jun 27, 20250 citations

Fourier Analysis Perspective on Quantum Neural Networks

Seungcheol Oh, Emily Jimin Roh, Athanasios V. Vasilakos +2 more

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), 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.

Communications PhysicsMay 18, 20261 citation

Constrained and Vanishing Expressivity of Quantum Fourier Models

Hela Mhiri, Léo Monbroussou, Mario Herrero-Gonzalez +3 more

In this work, we highlight an unforeseen behavior of the expressivity of Parameterized Quantum Circuits (PQCs) for machine learning. A large class of these models, seen as Fourier series whose frequencies are derived from the encoding gates, were thought to have their Fourier coefficients mostly determined by the trainable gates. Here, we demonstrate a new correlation between the Fourier coefficients of the quantum model and its encoding gates. In addition, we display a phenomenon of vanishing expressivity in certain settings, where some Fourier coefficients vanish exponentially as the number of qubits grows. These two behaviors imply novel forms of constraints which limit the expressivity of PQCs, and therefore imply a new inductive bias for quantum models. The key concept in this work is the notion of a frequency redundancy in the Fourier series spectrum, which determines its importance. Those theoretical behaviors are observed in numerical simulations.In this work, we highlight an unforeseen behavior of the expressivity of Parameterized Quantum Circuits (PQCs) for machine learning. A large class of these models, seen as Fourier series whose frequencies are derived from the encoding gates, were thought to have their Fourier coefficients mostly determined by the trainable gates. Here, we demonstrate a new correlation between the Fourier coefficients of the quantum model and its encoding gates. In addition, we display a phenomenon of vanishing expressivity in certain settings, where some Fourier coefficients vanish exponentially as the number of qubits grows. These two behaviors imply novel forms of constraints which limit the expressivity of PQCs, and therefore imply a new inductive bias for quantum models. The key concept in this work is the notion of a frequency redundancy in the Fourier series spectrum, which determines its importance. Those theoretical behaviors are observed in numerical simulations.

arXiv:2403.09417QuantumSep 3, 202514 citations

Enhancing the expressivity of quantum neural networks with residual connections

Jingwei Wen, Zhiguo Huang, Dunbo Cai +1 more

Abstract In noisy intermediate-scale quantum era, the research on the combination of artificial intelligence and quantum computing has been greatly developed. Here we propose a quantum circuit-based algorithm to implement quantum residual neural networks, where the residual connection channels are constructed by introducing auxiliary qubits to data-encoding and trainable blocks in quantum neural networks. We prove that when this particular network architecture is applied to a l -layer data-encoding, the number of frequency generation forms extends from one, namely the difference of the sum of generator eigenvalues, to ${{{{{{{\mathcal{O}}}}}}}}({l}^{2})$ O ( l 2 ) , and the flexibility in adjusting Fourier coefficients can also be improved. It indicates that residual encoding can achieve better spectral richness and enhance the expressivity of various parameterized quantum circuits. Extensive numerical demonstrations in regression tasks and image classification are offered. Our work lays foundation for the complete quantum implementation of classical residual neural networks and offers a quantum feature map strategy for quantum machine learning.Abstract In noisy intermediate-scale quantum era, the research on the combination of artificial intelligence and quantum computing has been greatly developed. Here we propose a quantum circuit-based algorithm to implement quantum residual neural networks, where the residual connection channels are constructed by introducing auxiliary qubits to data-encoding and trainable blocks in quantum neural networks. We prove that when this particular network architecture is applied to a l -layer data-encoding, the number of frequency generation forms extends from one, namely the difference of the sum of generator eigenvalues, to $${{{{{{{\mathcal{O}}}}}}}}({l}^{2})$$ O ( l 2 ) , and the flexibility in adjusting Fourier coefficients can also be improved. It indicates that residual encoding can achieve better spectral richness and enhance the expressivity of various parameterized quantum circuits. Extensive numerical demonstrations in regression tasks and image classification are offered. Our work lays foundation for the complete quantum implementation of classical residual neural networks and offers a quantum feature map strategy for quantum machine learning.

arXiv:2401.15871Communications PhysicsJul 6, 202416 citations

\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.

{{{{{{{\mathcal{O}}}}}}}}({l}^{2})

O ( l 2 ) , and the flexibility in adjusting Fourier coefficients can also be improved. It indicates that residual encoding can achieve better spectral richness and enhance the expressivity of various parameterized quantum circuits. Extensive numerical demonstrations in regression tasks and image classification are offered. Our work lays foundation for the complete quantum implementation of classical residual neural networks and offers a quantum feature map strategy for quantum machine learning.