Recently, quantum neural networks or quantum–classical neural networks (qcNN) have been actively studied, as a possible alternative to the conventional classical neural network (cNN), but their practical and theoretically-guaranteed performance is still to be investigated. In contrast, cNNs and especially deep cNNs, have acquired several solid theoretical basis; one of those basis is the neural tangent kernel (NTK) theory, which can successfully explain the mechanism of various desirable properties of cNNs, particularly the global convergence in the training process. In this paper, we study a class of qcNN composed of a quantum data-encoder followed by a cNN. The quantum part is randomly initialized according to unitary 2-designs, which is an effective feature extraction process for quantum states, and the classical part is also randomly initialized according to Gaussian distributions; then, in the NTK regime where the number of nodes of the cNN becomes infinitely large, the output of the entire qcNN becomes a nonlinear function of the so-called projected quantum kernel. That is, the NTK theory is used to construct an effective quantum kernel, which is in general nontrivial to design. Moreover, NTK defined for the qcNN is identical to the covariance matrix of a Gaussian process, which allows us to analytically study the learning process. These properties are investigated in thorough numerical experiments; particularly, we demonstrate that the qcNN shows a clear advantage over fully classical NNs and qNNs for the problem of learning the quantum data-generating process.

中文翻译：

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神经正切核机制中的量子经典混合神经网络

最近，量子神经网络或量子经典神经网络（qcNN）作为传统经典神经网络（cNN）的可能替代方案得到了积极的研究，但它们的实际和理论上保证的性能仍有待研究。相比之下，CNN，尤其是深度CNN，已经获得了一些坚实的理论基础；其中基础之一是神经正切核（NTK）理论，它可以成功解释CNN的各种理想特性的机制，特别是训练过程中的全局收敛。在本文中，我们研究了一类由量子数据编码器和 cNN 组成的 qcNN。量子部分根据酉2-设计随机初始化，这是一种有效的量子态特征提取过程，经典部分也是根据高斯分布随机初始化；然后，在 cNN 节点数量变得无限大的 NTK 体系中，整个 qcNN 的输出变成所谓的投影量子核的非线性函数。也就是说，NTK 理论用于构建有效的量子内核，这通常对于设计来说并不简单。此外，为 qcNN 定义的 NTK 与高斯过程的协方差矩阵相同，这使我们能够分析研究学习过程。这些特性是通过彻底的数值实验进行研究的；特别是，我们证明了 qcNN 在学习量子数据生成过程的问题上比完全经典的神经网络和 qNN 表现出明显的优势。