Quantum Lazy Training

TL;DR

This paper demonstrates that large-scale geometrically local parameterized quantum circuits exhibit lazy training behavior, where parameters change minimally and linear approximations remain accurate, akin to classical neural networks.

Contribution

It provides the first theoretical analysis showing lazy training occurs in quantum circuits with many qubits, including bounds on parameter changes and approximation accuracy.

Findings

Parameters change minimally in large qubit regimes.

Linear approximation remains accurate as qubit number increases.

Numerical simulations support theoretical bounds.

Abstract

In the training of over-parameterized model functions via gradient descent, sometimes the parameters do not change significantly and remain close to their initial values. This phenomenon is called lazy training, and motivates consideration of the linear approximation of the model function around the initial parameters. In the lazy regime, this linear approximation imitates the behavior of the parameterized function whose associated kernel, called the tangent kernel, specifies the training performance of the model. Lazy training is known to occur in the case of (classical) neural networks with large widths. In this paper, we show that the training of geometrically local parameterized quantum circuits enters the lazy regime for large numbers of qubits. More precisely, we prove bounds on the rate of changes of the parameters of such a geometrically local parameterized quantum circuit in the training process, and on the precision of the linear approximation of the associated quantum model function; both of these bounds tend to zero as the number of qubits grows. We support our analytic results with numerical simulations.