A Unified Frequency Principle for Quantum and Classical Machine Learning

TL;DR

This work introduces a unified frequency-principle (F-principle) that governs the early training dynamics of both classical deep neural networks and quantum neural networks on smooth data manifolds. By formulating gradient-flow dynamics on a shared geometric framework and using a spectral projector $E_{\lambda}$, the authors prove that low-frequency components of the target function are learned faster than high-frequency components, formalized through a bound on the decay rates $\mathcal{L}_{\lambda}$ relative to $\mathcal{L}$. They extend the theory to axis-aligned Pauli noise in parameterized quantum circuits, showing exponential suppression of high-frequency Fourier modes via a Pauli-path integral representation, which implies robustness of low-frequency learning and enables efficient classical simulation through frequency truncation with provable error guarantees. Numerical experiments on a one-dimensional regression task corroborate the theory, illustrating the distinct impacts of dephasing and depolarizing noise on frequency components during training. Collectively, the results provide a unifying, frequency-domain lens for understanding trainability, noise resilience, and classical simulability in near-term quantum and classical learning systems, with implications for designing noise-resilient QNNs and identifying tasks amenable to quantum or classical advantage.

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.

Figures (6)

• Figure 1: Illustration of the F-principle in classical and quantum neural networks. (a) Upper panel: The time-domain waveform of a target function $f(\bm x)$, exhibiting oscillatory behavior. Lower panel: Its corresponding frequency-domain representation obtained via Fourier transformation, showing distinct frequency components (peaks at different frequencies). (b) Architectures of classical and quantum neural networks that output $f_{\bm \theta}(\bm x)$, where $\bm x$ denotes the input and $\bm \theta$ denotes the set of tunable parameters. The classical model (left) consists of layered neurons, while the quantum model (right) is implemented as a parameterized quantum circuit followed by projective measurements. (c) Top: Frequency-domain representations of the QNN output $f_{\bm \theta}(\bm x)$ at different stages of training. From left to right: initial, early, and final stages. The orange lines represent the spectral components of $f_{\bm \theta}(\bm x)$. The dark blue boxes highlight low-frequency components, which are learned rapidly and converge during the early stage. The light blue boxes indicate high-frequency components, whose convergence is delayed until the late stage. This stage-wise convergence behavior illustrates the F-principle. Bottom: Schematic loss landscapes $\mathcal{L}(\bm \theta)$ for classical and quantum neural network models. The QNN loss is given by $\mathcal{L}(\bm \theta) = \sum_i \left(\langle O_{\bm \theta}(\bm x_i)\rangle - y_i\right)^2$, while the DNN loss takes the form $\mathcal{L}(\bm \theta) = \sum_i \left(f_{\bm \theta}(\bm x_i) - y_i\right)^2$. (d) Schematic of a parameterized quantum circuit under two noise models. Single-qubit operations ($H$, $S$, $R_Z(\theta_i)$) and CNOT gates are interleaved throughout the circuit. Solid dots mark points at which a local noise channel is applied. We consider two noise scenarios: (1) dephasing only at the blue-dot positions (black dots remain noise-free), and (2) depolarizing noise at both black and blue dot positions. The ellipses on the right indicate that the pattern repeats in subsequent layers.
• Figure 2: The QNN ansatz consists of two types of rotation gates: $R_Y(x)$ gates encode the input data, while $R_Y(\theta_i)$ gates contain trainable weights. To enhance the nonlinear expressive power of the QNN, we apply repeated encoding of the input $x$: the circuit module within the dark blue box is repeated 10 times. Additionally, to increase the number of trainable parameters and thereby improve the model's expressive capacity, the module within the light blue box is repeated 4 times. It is important to note that all rotation gates involving trainable parameters are independently parameterized, whereas all gates involving the input $x$ share the same parameter value across repetitions.
• Figure 3: Fourier spectrum of the QNN output $f_{\bm{\theta}}(x)$ during training under different noise models. Each panel plots the amplitude $\lvert\hat{f}_{\bm{\theta}}(\omega)\rvert$ versus frequency $\omega$; curves are colour-coded by iteration (dark $\rightarrow$ early stage, yellow = 500, i.e. near convergence). Low-frequency components converge after only a few updates, whereas high-frequency components require many more iterations, evidencing faster learning of low frequencies and noise-induced suppression of high frequencies.
• Figure 4: Evolution of Fourier amplitudes at selected frequencies (1, 5, 6, 13, 19, and 21 Hz, respectively) during QNN training under various noise models. The curves reflect how noise strength and type affect the learnability of frequency components across training iterations.
• Figure 5: Evolution of Fourier amplitudes at each frequency (from 1Hz to 21 Hz) during QNN training under various noise models, revealing how noise strength and channel type jointly govern the convergence rate and steady-state amplitude across training iterations.

Theorems & Definitions (9)

• Theorem 1: Informal
• Theorem 2
• Corollary 1
• Theorem 3
• Definition 1: Sobolev space $H^{m}(\mathcal{M})$
• Theorem 4: Rigorous unified F-principle
• proof
• lemma 1: fontana2025classical
• Theorem 5