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. On the other hand, CNNs and especially the deep CNNs, have acquired several solid theoretical basis; one of those significant basis is the neural tangent kernel (NTK) theory, which indeed can successfully explain the mechanism of various desirable properties of CNN, e.g., global convergence and good generalization properties. In this paper, we study a class of QCNN where NTK theory can be directly applied. The output of the proposed QCNN is a function of the projected quantum kernel, in the limit of large number of nodes of the CNN part; hence this scheme may have a potential quantum advantage. Also, because the parameters can be tuned only around the initial random variables chosen from unitary 2-design and Gaussian distributions, the proposed QCNN casts as a scheme that realizes the quantum kernel method with less computational complexity. Moreover, NTK is identical to the covariance matrix of a Gaussian process, which allows us to analytically study the learning process and as a consequence to have a condition of the dataset such that QCNN may perform better than the classical correspondence. These properties are all observed in a thorough numerical experiment.
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