Representation Learning via Quantum Neural Tangent Kernels

TL;DR

This paper introduces quantum neural tangent kernels to analyze variational quantum circuits, providing analytical insights into their training dynamics and performance in quantum machine learning tasks.

Contribution

It defines quantum neural tangent kernels and derives their dynamical equations, extending analysis to hybrid quantum-classical architectures and the lazy training regime.

Findings

Analytical solutions for quantum neural tangent kernel dynamics.

Extension to hybrid quantum-classical neural networks.

Numerical validation of theoretical predictions.

Abstract

Variational quantum circuits are used in quantum machine learning and variational quantum simulation tasks. Designing good variational circuits or predicting how well they perform for given learning or optimization tasks is still unclear. Here we discuss these problems, analyzing variational quantum circuits using the theory of neural tangent kernels. We define quantum neural tangent kernels, and derive dynamical equations for their associated loss function in optimization and learning tasks. We analytically solve the dynamics in the frozen limit, or lazy training regime, where variational angles change slowly and a linear perturbation is good enough. We extend the analysis to a dynamical setting, including quadratic corrections in the variational angles. We then consider hybrid quantum-classical architecture and define a large-width limit for hybrid kernels, showing that a hybrid quantum-classical neural network can be approximately Gaussian. The results presented here show limits for which analytical understandings of the training dynamics for variational quantum circuits, used for quantum machine learning and optimization problems, are possible. These analytical results are supported by numerical simulations of quantum machine learning experiments.