SAQNN: Spectral Adaptive Quantum Neural Network as a Universal Approximator

2026-02-10

AI Summary

This work proposes the first quantum neural network model endowed with a universal approximation guarantee, capable of approximating any square-integrable function to arbitrary precision. The architecture is implemented via a quantum circuit featuring spectral adaptivity, which enables adaptive switching among function bases. By integrating Sobolev space analysis with quantum parameter optimization, the model achieves theoretically optimal parameter complexity when approximating Sobolev functions in the L₂ norm. Moreover, its circuit depth scales asymptotically more favorably than that of the best-known classical feedforward neural networks under comparable approximation guarantees.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

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.

Problem

• quantum neural networks
• universal approximation
• expressivity
• quantum machine learning
• function approximation

Innovation

• Quantum Neural Network
• Universal Approximation Property
• Spectral Adaptivity
• Parameter Complexity
• Quantum Machine Learning