Fourier analysis of the multivariate variational quantum algorithms

Variational quantum algorithms (VQAs) have shown advantages over classical approaches in solving many problems, yet their quantum neural networks (QNNs) models still perform poorly in complex tasks like multi-class classification and high-frequency time-series prediction, raising questions about their feasibility. While existing studies have characterized their performance from several perspectives, but research on their learning performance and visualization remains scarce. This work develops the universal approximation theory of variational quantum algorithms for multivariate function fitting, supported by a verification in visual training analysis on two-input binary datasets. We first visualize how a single-qubit VQA with single-encoding process four types of binary data, revealing inherent limitations, and then formulate the measurement expectation of VQA as a partial Fourier series, establishing its universal approximation capacity and showing that the Fourier spectrum is determined by the data encoding strategy: multiple encoding layers broaden the spectrum and enhance learning performance. Guided by this theory, we adjust the circuit structure and achieve nearly 100% accuracy across all four binary classification tasks.