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Communications in computer and information science

Geometric Origin of Fourier Coefficients in Data-Reuploading Single-Qubit Circuits

Seungcheol Oh, Chaemoon Im, Soohyun Park, Joongheon Kim

Published Jan 1, 2026• 0 citations
Publisher
DOI
10.1007/978-981-95-7829-0_4

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Abstract In this work, we apply 1D quantum convolution in the task of time series forecasting. By encoding multiple points into the quantum circuit to predict subsequent data, each point becomes a feature and the problem becomes a multidimensional one. Taking as basis previous theoretical works which demonstrated that Variational Quantum Circuits (VQCs) can be expressed as multidimensional Fourier series, the capabilities of different architectures and ansatz are explored. This analysis considers the concepts of circuit expressivity and the presence of barren plateaus. Analyzing the problem within the framework of the Fourier series enabled the incorporation of data reuploading in the architecture, resulting in enhanced performance. Rather than a strict requirement for the number of free parameters to exceed the degrees of freedom of the Fourier series, our findings suggest that even a limited number of parameters can produce Fourier functions of higher degrees. This highlights the remarkable expressive power of quantum circuits. This observation is also significant in reducing training times. The ansatz with greater expressivity and number of non-zero Fourier coefficients consistently delivers favorable results across different scenarios, with performance metrics improving as the number of qubits increases.
For quantum machine learning, the exact boundary between classical and quantum advantage is still poorly understood. Direct comparison between quantum neural networks (QNNs) and existing classical models, which encompass fundamentally different function classes, often fails to provide broader insight into the difference between the two. Inspired by the techniques of Neural Quantum States and Random Fourier Features, this work introduces Neural Fourier Surrogates (NFS), a stochastic classical neural network architecture for efficiently learning coefficients over the same finite Fourier series support as quantum neural networks. Testing on a selection of tabular benchmark datasets, we find that NFS is an effective classifier architecture broadly competitive with established classical baselines, including a comparable Random Fourier Features model, and possessing comparable performance to data-reuploading QNNs; combined with additional analysis comparing the learned Fourier spectra of QNNs and NFS on synthetic data, these results establish NFS as a natural classical baseline for evaluating QNN performance.