Fourier series guided design of quantum convolutional neural networks for enhanced time series forecasting

In this work, we explore the application of 1D quantum convolutional layers for time series forecasting, framing the problem within the context of machine learning and quantum circuit expressivity. By encoding multiple time steps as features into a quantum circuit, the forecasting task becomes intrinsically multidimensional. We leverage the recent theoretical results that describe Variational Quantum Circuits (VQCs) as multidimensional Fourier series, to design a more expressive 1D quantum convolutional neural network in which an enhanced performance is obtained by reuploading the data. We aim to relate the number of accessible coefficients in each architecture design with the obtained metrics. Our analysis incorporates key machine learning concepts such as expressivity and the presence of barren plateaus, evaluating their impact on generalization and training efficiency. Notably, we show that even with a limited number of trainable parameters, quantum circuits can approximate high-degree Fourier functions, underscoring their expressive capacity and computational efficiency. Our results demonstrate that more expressive ansatz, characterized by richer Fourier spectra, consistently yield improved predictive performance as the number of qubits increases. This study bridges theoretical insights from quantum physics with practical advances in machine learning model design for time series forecasting.