  In this study, we apply 1D quantum convolution to address the task of time
series forecasting. By encoding multiple points into the quantum circuit to
predict subsequent data, each point becomes a feature, transforming the problem
into a multidimensional one. Building on theoretical foundations from prior
research, which demonstrated that Variational Quantum Circuits (VQCs) can be
expressed as multidimensional Fourier series, we explore the capabilities of
different architectures and ansatz. This analysis considers the concepts of
circuit expressibility and the presence of barren plateaus. Analyzing the
problem within the framework of the Fourier series enabled the design of an
architecture that incorporates data reuploading, 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 expressibility and number of non-zero Fourier coefficients
consistently delivers favorable results across different scenarios, with
performance metrics improving as the number of qubits increases.
