Quantum-enhanced neural networks in the neural tangent kernel framework

TL;DR

This paper integrates quantum neural networks with the neural tangent kernel framework, providing a theoretical basis for their performance and demonstrating advantages over classical models in learning quantum data.

Contribution

It introduces a quantum-classical neural network model analyzed via NTK theory, linking quantum kernels to classical Gaussian processes for the first time.

Findings

Quantum-classical neural networks can be analyzed using NTK theory.

The quantum kernel in the model is effectively a nonlinear function of a projected quantum kernel.

The proposed qcNN outperforms classical neural networks in learning quantum data-generating processes.

Abstract

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.