The Spectral Amplitude Principle for Dynamics of Quantum Neural Networks

The mechanism governing the training dynamics of Quantum Neural Networks (QNNs) remains under-explored. In classical Deep Neural Networks (DNNs), training is known dominated by "Spectral Bias,'' i.e. prioritizing learning low-frequency while struggling with high-frequency. QNNs exhibit similar spectral limitations when the target function possesses flat spectral amplitudes. However, in this work, considering target problems with different spectral amplitudes, 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. These results show that QNNs notably outperform DNNs in high-frequency tasks, offering an explanation for QNNs' superior expressivity.