Fourier Series Guided Design of Quantum Convolutional Neural Networks for Enhanced Time Series Forecasting]{Fourier Series Guided Design of Quantum Convolutional Neural Networks for Enhanced Time Series Forecasting

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.