SAQNN: Spectral Adaptive Quantum Neural Network as a Universal Approximator

TL;DR

The paper addresses the lack of a rigorous approximation theory for quantum neural networks by introducing SAQNN, a constructive quantum model that implements truncated Fourier and Chebyshev bases via a linear combination of unitaries. It proves a universal approximation property for multivariate square-integrable functions and derives explicit $L_2$ Sobolev error bounds, with detailed resource scaling: circuit width $O(\log n)$, depth $O(n\log n)$, and parameter complexity $O(n)$, where $n=O((d+1/\\epsilon)^{16(1/\\epsilon)^{2/s}})$. SAQNN’s spectral adaptivity and explicit basis switching offer a versatile framework for function approximation in high dimensions, and numerical experiments in $d=2$ demonstrate practical feasibility for Fourier and Chebyshev targets. By connecting quantum circuit design to canonical approximation bases, the work provides both theoretical foundations and actionable architectures that may yield polynomial-resource advantages over classical neural networks in high-dimensional settings. These contributions advance quantum learning theory and offer a blueprint for reliable, scalable QNNs in complex numerical tasks.

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.

Figures (2)

• Figure 1: The construction of SAQNN. The state preparation block, spectrum selection block and phase injection are marked blue, green and red respectively. When approximating functions of $d$ variables using $n$ terms of truncated Fourier series, the whole circuit consists of $m+d$ qubits, with state preparation block $P(\boldsymbol{\theta})$, $n$ layers of spectrum selection block accompanied by phase injection, a padding layer and an inversion of $P(\boldsymbol{\theta})$ applied before final measurement. Here $m=\lceil\log n\rceil$ and $c_1,c_2,\cdots,c_n$ denotes coefficients of Fourier series with $n$ terms.
• Figure 2: Numerical results of Fourier and Chebyshev series approximating target functions. The target functions are visualized in the left graphics and the right ones are learned from our model. Both of them are plotted with 50$\times$50 grid samples of pre-defined intervals.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Lemma 3: Decomposition of multiplexor rotation gates bullock2003Mottonen2004Shende2006
• Lemma 4: Fourier approximation stein1971rivlin1981weisz2012
• Theorem 4
• Definition 5: Sobolev function spaces adams2003canuto2006
• Theorem 5
• Lemma 6: sun2023
• Theorem 6
• proof