  Quantum neural networks (QNNs) have emerged as a leading strategy to
establish applications in machine learning, chemistry, and optimization. While
the applications of QNN have been widely investigated, its theoretical
foundation remains less understood. In this paper, we formulate a theoretical
framework for the expressive ability of data re-uploading quantum neural
networks that consist of interleaved encoding circuit blocks and trainable
circuit blocks. First, we prove that single-qubit quantum neural networks can
approximate any univariate function by mapping the model to a partial Fourier
series. We in particular establish the exact correlations between the
parameters of the trainable gates and the Fourier coefficients, resolving an
open problem on the universal approximation property of QNN. Second, we discuss
the limitations of single-qubit native QNNs on approximating multivariate
functions by analyzing the frequency spectrum and the flexibility of Fourier
coefficients. We further demonstrate the expressivity and limitations of
single-qubit native QNNs via numerical experiments. We believe these results
would improve our understanding of QNNs and provide a helpful guideline for
designing powerful QNNs for machine learning tasks.
