The Spectral Amplitude Principle for Dynamics of Quantum Neural Networks

TL;DR

This paper reveals that Quantum Neural Networks prioritize spectral amplitude over frequency during training, enabling them to efficiently learn high-frequency functions and outperform classical neural networks in complex spectral tasks.

Contribution

It introduces the Spectral Amplitude Priority principle, a novel mechanism governing QNN training dynamics, supported by theoretical analysis and empirical validation.

Findings

QNNs focus on spectral amplitude rather than frequency.

QNNs outperform DNNs in high-frequency function learning.

QNNs can efficiently capture complex spectral features.

Abstract

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.