A Unified Frequency Principle for Quantum and Classical Machine Learning

TL;DR

This paper introduces a unified theoretical framework for the frequency principle in both classical and quantum neural networks, revealing how noise influences their learning dynamics and spectral bias.

Contribution

It establishes a spectral bias towards low-frequency learning in quantum neural networks and analyzes noise effects, unifying classical and quantum training dynamics.

Findings

Quantum neural networks favor low-frequency components during training.

Noise suppresses high-frequency Fourier components exponentially.

Quantum neural networks are robust against certain noise models and can be classically simulated efficiently.

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 $\boldsymbol{\omega}$ is suppressed by a factor $(1-2\gamma)^{\|\boldsymbol{\omega}\|_1}$. 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.