Power and limitations of single-qubit native quantum neural networks
This work addresses a foundational gap in quantum machine learning by providing theoretical insights into the expressive power of QNNs, which is incremental but crucial for guiding future designs.
The paper tackles the theoretical understanding of quantum neural networks (QNNs) by proving that single-qubit QNNs can approximate any univariate function via a Fourier series mapping and analyzing their limitations for multivariate functions, with numerical experiments to demonstrate these findings.
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.