SAQNN：谱适应量子神经网络作为通用逼近器
SAQNN: Spectral Adaptive Quantum Neural Network as a Universal Approximator
AI总结：
本文提出了一种谱适应量子神经网络模型，证明其具备通用逼近能力，并在电路规模和参数复杂度方面优于经典神经网络。
摘要：
量子机器学习（QML）作为连接量子计算和机器学习的交叉学科，近年来引起了广泛关注。目前，该领域整体面临挑战，因为量子神经网络（QNNs）的表达性理论基础尚不完整。在本文中，我们提出了一种构造性的QNN模型，并证明其具有通用逼近性质（UAP），这意味着它可以以任意精度逼近任何平方可积函数。此外，它支持切换函数基，从而适应数值逼近和机器学习中的各种场景。我们的模型在电路规模方面相对于最佳经典前馈神经网络具有渐近优势，并且在以L2范数逼近Sobolev函数时达到最优参数复杂度。
英文摘要：
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.
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