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 is suppressed by a factor
(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.
Abstract Quantum neural networks (QNNs) are emerging as a promising paradigm at the intersection of quantum computing and artificial intelligence. However, their conceptual abstraction and physical implementation remain challenging in educational contexts. In this work, we present a didactic framework to teach the fundamentals of QNNs by leveraging classical artificial neural networks (ANNs), Python simulation with Qiskit, and analysis of training dynamics using the Iris dataset. The paper begins with a theoretical comparison between classical and quantum neurons, followed by the design of a full QNN training algorithm implemented in Python. Through simulated results and pedagogical discussion, we show how learners can explore key aspects of quantum learning—including superposition, variational circuits, probabilistic measurement, and optimization—without access to quantum hardware. The use of graphical outputs and code-structure correspondence enhances student engagement and promotes algorithmic thinking. This approach supports quantum science education by offering a reproducible, interactive, and conceptually aligned method for introducing QNNs in undergraduate and graduate classrooms.