Power and limitations of single-qubit native quantum neural networksDownload PDF

Published: 31 Oct 2022, Last Modified: 12 Oct 2022NeurIPS 2022 AcceptReaders: Everyone
Keywords: quantum machine learning, quantum neural networks, expressivity, function approximation, universal approximation, Fourier series
Abstract: 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.
TL;DR: This work establishes a theoretical framework for the expressivity of basic quantum neural networks and arrives at the universal approximation property.
Supplementary Material: zip
13 Replies