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.
翻译:量子神经网络(QNNs)是建立机器学习、化学和优化应用的主导战略。虽然QNN的应用受到广泛调查,但其理论基础仍然不甚为人理解。在本文件中,我们为数据再加载量子神经网络的显性能力制定了理论框架,这种网络由不同编码的电路块和可训练的电路块组成。首先,我们证明单位量子神经网络可以通过将该模型绘制成部分Fourier系列来接近任何单方位功能。我们特别建立了可训练门参数和四列系数参数之间的确切关联,解决了QNNN的普遍近似特性的开放问题。第二,我们通过分析频率频谱和四列系数的灵活性来讨论单位本地QNNs对相近功能的局限性。我们进一步通过数字实验来展示单位本地QNNs的显性和局限性。我们认为这些结果将改进我们对QNPs的理解,并为设计强大的QNNS机器学习任务提供有用的指导。