Quantum machine learning (QML), as an interdisciplinary field bridging quantum computing and machine learning, has garnered significant attention in recent years. Currently, the field as a whole faces challenges due to incomplete theoretical foundations for the expressivity of quantum neural networks (QNNs). In this paper we propose a constructive QNN model and demonstrate that it possesses the universal approximation property (UAP), which means it can approximate any square-integrable function up to arbitrary accuracy. Furthermore, it supports switching function bases, thus adaptable to various scenarios in numerical approximation and machine learning. Our model has asymptotic advantages over the best classical feed-forward neural networks in terms of circuit size and achieves optimal parameter complexity when approximating Sobolev functions under $L_2$ norm.

翻译：量子机器学习（QML）作为连接量子计算与机器学习的交叉领域，近年来受到广泛关注。当前，由于量子神经网络（QNN）表达能力理论基础的不足，该领域整体面临挑战。本文提出了一种构造性QNN模型，并证明其具备通用逼近性质（UAP），即能够以任意精度逼近任何平方可积函数。此外，该模型支持切换函数基，从而可适应数值逼近与机器学习中的多种场景。在电路规模方面，我们的模型相对于最优经典前馈神经网络具有渐近优势，并在$L_2$范数下逼近Sobolev函数时实现了最优参数复杂度。