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. Beyond previous works' understanding of existence, we in particular establish the exact correlations between the parameters of the trainable gates and the working Fourier coefficients, by exploring connections to quantum signal processing. 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. As applications, we introduce natural extensions to multi-qubit quantum neural networks, which exhibit the capability of classifying real-world multi-dimensional data. 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的应用已经受到广泛调查,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,数据再加载量子神经网络由内分编码电路块和可训练电路块组成。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系列来接近任何单位功能。除了先前的工作对存在的理解外,我们还特别通过探索量子信号处理的连接,在可训练门的参数和工作四倍数系数之间建立了确切的关联。第二,我们通过分析频率频谱和四倍系数的灵活性,讨论单位量子本地QNNNNs在接近多变量功能上的局限性。我们通过数字实验进一步展示单位本地QNNNPs的明示性和局限性和局限性。作为应用,我们将多位量子的量子神经网络引入自然扩展,通过探索量子信号处理量子信号信号处理过程的能力。我们对机器的模型进行分级研究。