SAQNN: Spectral Adaptive Quantum Neural Network as a Universal Approximator

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.

💡 Research Summary

The paper introduces the Spectral Adaptive Quantum Neural Network (SAQNN), a constructive quantum neural network architecture inspired by the linear combination of unitaries (LCU) technique. SAQNN consists of three modular blocks: (i) state preparation using multiplexed R_y gates to generate arbitrary real‑amplitude quantum states, (ii) spectrum selection that repeatedly encodes the input vector x via data re‑uploading into rotation angles, thereby selecting Fourier frequency components, and (iii) phase injection where trainable Z‑rotations apply scaling coefficients before measurement. The observable O = |0⟩⟨0| is measured, and the expectation value ⟨0|U† O U|0⟩¹ᐟ² serves as the network output.

Two main theorems are proved. Theorem 1 establishes a universal approximation property (UAP) for any square‑integrable multivariate function f :