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arXiv:2409.06682

The Spectral Amplitude Principle for Dynamics of Quantum Neural Networks

Xu, Yi-hang, Dan-Bo Zhang, Yan, Junchi

Published Sep 10, 2024• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2409.06682

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Quantum machine learning models based on parameterized circuits can be viewed as Fourier series approximators. However, they often struggle to learn functions with multiple frequency components, particularly high‑frequency or non-dominant ones - a phenomenon we term the \textit{quantum Fourier parameterization bias}. Inspired by recent advances in classical Fourier neural operators (FNOs), we adapt the multi-stage residual learning idea to the quantum domain, iteratively training additional quantum modules on the residuals of previous stages. We evaluate our method on a synthetic benchmark composed of spatially localized frequency components with diverse envelope shapes (Gaussian, Lorentzian, triangular). Systematic experiments show that the number of qubits, the encoding scheme, and residual learning are all crucial for resolving multiple frequencies; residual learning alone can improve test MSE significantly over a single‑stage baseline trained for the same total number of epochs. Our work provides a practical framework for enhancing the spectral expressivity of quantum models and offers new insights into their frequency-learning behavior.
Parametrized quantum circuits can be used as quantum neural networks and have the potential to outperform their classical counterparts when trained for addressing learning problems. To date, much of the results on their performance on practical problems are heuristic in nature. In particular, the convergence rate for the training of quantum neural networks is not fully understood. Here, we analyze the dynamics of gradient descent for the training error of a class of variational quantum machine learning models. We define wide quantum neural networks as parametrized quantum circuits in the limit of a large number of qubits and variational parameters. Then, we find a simple analytic formula that captures the average behavior of their loss function and discuss the consequences of our findings. For example, for random quantum circuits, we predict and characterize an exponential decay of the residual training error as a function of the parameters of the system. Finally, we validate our analytic results with numerical experiments.
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