  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.
