Approximation rates of quantum neural networks for periodic functions via Jackson's inequality

TL;DR

This paper investigates the approximation capabilities of quantum neural networks for periodic functions, demonstrating that they can achieve better efficiency and accuracy, especially for smoother functions, by leveraging Jackson's inequality.

Contribution

The paper introduces a novel analysis of QNNs for periodic functions, showing quadratic parameter reduction and improved approximation results using Jackson's inequality.

Findings

Quadratic reduction in the number of parameters needed for approximation.

Better approximation results for smoother functions.

Effective use of Jackson's inequality in quantum neural network analysis.

Abstract

Quantum neural networks (QNNs) are an analog of classical neural networks in the world of quantum computing, which are represented by a unitary matrix with trainable parameters. Inspired by the universal approximation property of classical neural networks, ensuring that every continuous function can be arbitrarily well approximated uniformly on a compact set of a Euclidean space, some recent works have established analogous results for QNNs, ranging from single-qubit to multi-qubit QNNs, and even hybrid classical-quantum models. In this paper, we study the approximation capabilities of QNNs for periodic functions with respect to the supremum norm. We use the Jackson inequality to approximate a given function by implementing its approximating trigonometric polynomial via a suitable QNN. In particular, we see that by restricting to the class of periodic functions, one can achieve a quadratic reduction of the number of parameters, producing better approximation results than in the literature. Moreover, the smoother the function, the fewer parameters are needed to construct a QNN to approximate the function.