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arXiv:2307.12904 · IEEE Transactions on Neural Networks and Learning Systems

Universal Approximation Theorem and Error Bounds for Quantum Neural Networks and Quantum Reservoirs

Lukas Gonon, Antoine Jacquier

Published Apr 9, 2025• 9 citations
arXivHTMLPDF

Abstract

Universal approximation theorems are the foundations of classical neural networks, providing theoretical guarantees that the latter are able to approximate maps of interest. Recent results have shown that this can also be achieved in a quantum setting, whereby classical functions can be approximated by parameterized quantum circuits. We provide here precise error bounds for specific classes of functions and extend these results to the interesting new setup of randomized quantum circuits, mimicking classical reservoir neural networks. Our results show in particular that a quantum neural network with ${\mathcal {O}}(\varepsilon ^{-2})$ weights and ${\mathcal {O}} (\lceil \log _{2}(\varepsilon ^{-1})\rceil)$ qubits suffices to achieve approximation error $\varepsilon \gt 0$ when approximating functions with integrable Fourier transforms.

DOI
10.1109/tnnls.2025.3552223

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Universal approximation theorems are the foundations of classical neural networks, providing theoretical guarantees that the latter are able to approximate maps of interest. Recent results have shown that this can also be achieved in a quantum setting, whereby classical functions can be approximated by parameterized quantum circuits. We provide here precise error bounds for specific classes of functions and extend these results to the interesting new setup of randomized quantum circuits, mimicking classical reservoir neural networks. Our results show in particular that a quantum neural network with O(ε−2){\mathcal {O}}(\varepsilon ^{-2})O(ε−2) weights and O(⌈log⁡2(ε−1)⌉){\mathcal {O}} (\lceil \log _{2}(\varepsilon ^{-1})\rceil)O(⌈log2​(ε−1)⌉) qubits suffices to achieve approximation error ε>0\varepsilon \gt 0ε>0 when approximating functions with integrable Fourier transforms.
Parameterized quantum circuits (PQCs) have emerged as a promising approach for quantum neural networks. However, understanding their expressive power in accomplishing machine learning tasks remains a crucial question. This paper investigates the expressivity of PQCs for approximating general multivariate function classes. Unlike previous Universal Approximation Theorems for PQCs, which are either nonconstructive or rely on parameterized classical data processing, we explicitly construct data re-uploading PQCs for approximating multivariate polynomials and smooth functions. We establish the first non-asymptotic approximation error bounds for these functions in terms of the number of qubits, quantum circuit depth, and number of trainable parameters. Notably, we demonstrate that for approximating functions that satisfy specific smoothness criteria, the quantum circuit size and number of trainable parameters of our proposed PQCs can be smaller than those of deep ReLU neural networks. We further validate the approximation capability of PQCs through numerical experiments. Our results provide a theoretical foundation for designing practical PQCs and quantum neural networks for machine learning tasks that can be implemented on near-term quantum devices, paving the way for the advancement of quantum machine learning.
With the continuously increasing number of quantum bits in quantum computers, there are growing interests in exploring applications that can harvest the power of them. Recently, several attempts were made to implement neural networks, known to be computationally intensive, in hybrid quantum-classical scheme computing. While encouraging results are shown, two fundamental questions need to be answered: (1) whether neural networks in hybrid quantum-classical computing can leverage quantum power and meanwhile approximate any function within a given error bound, i.e., universal approximability; (2) how do these neural networks compare with ones on a classical computer in terms of representation power? This work sheds light on these two questions from a theoretical perspective.
Quantum neural networks (QNNs) are an analog of classical neural networks in the world of quantum computing, which are represented by a unitary matrix with trainable parameters. Inspired by the universal approximation property of classical neural networks, ensuring that every continuous function can be arbitrarily well approximated uniformly on a compact set of a Euclidean space, some recent works have established analogous results for QNNs, ranging from single-qubit to multi-qubit QNNs, and even hybrid classical-quantum models. In this paper, we study the approximation capabilities of QNNs for periodic functions with respect to the supremum norm. We use the Jackson inequality to approximate a given function by implementing its approximating trigonometric polynomial via a suitable QNN. In particular, we see that by restricting to the class of periodic functions, one can achieve a quadratic reduction of the number of parameters, producing better approximation results than in the literature. Moreover, the smoother the function, the fewer parameters are needed to construct a QNN to approximate the function.
In this work, we address the question whether a sufficiently deep quantum neural network can approximate a target function as accurate as possible. We start with simple but typical physical situations that the target functions are physical observables, and then we extend our discussion to situations that the learning targets are not directly physical observables, but can be expressed as physical observables in an enlarged Hilbert space with multiple replicas, such as the Loshimidt echo and the Renyi entropy. The main finding is that an accurate approximation is possible only when the input wave functions in the dataset do not exhaust the entire Hilbert space that the quantum circuit acts on, and more precisely, the Hilbert space dimension of the former has to be less than half of the Hilbert space dimension of the latter. In some cases, this requirement can be satisfied automatically because of the intrinsic properties of the dataset, for instance, when the input wave function has to be symmetric between different replicas. And if this requirement cannot be satisfied by the dataset, we show that the expressivity capabilities can be restored by adding one ancillary qubit where the wave function is always fixed at input. Our studies point toward establishing a quantum neural network analogy of the universal approximation theorem that lays the foundation for expressivity of classical neural networks.
Quantum neural networks (QNN) have been proposed as a promising architecture for quantum machine learning. There exist a number of different quantum circuit designs being branded as QNNs, however no clear candidate has presented itself as more suitable than the others. Rather, the search for a ``quantum perceptron" -- the fundamental building block of a QNN -- is still underway. One candidate is quantum perceptrons designed to emulate the nonlinear activation functions of classical perceptrons. Such sigmoid quantum perceptrons (SQPs) inherit the universal approximation property that guarantees that classical neural networks can approximate any function. However, this does not guarantee that QNNs built from SQPs will have any quantum advantage over their classical counterparts. Here we critically investigate both the capabilities and performance of SQP networks by computing their effective dimension and effective capacity, as well as examining their performance on real learning problems. The results are compared to those obtained for other candidate networks which lack activation functions. It is found that simpler, and apparently easier-to-implement parametric quantum circuits actually perform better than SQPs. This indicates that the universal approximation theorem, which a cornerstone of the theory of classical neural networks, is not a relevant criterion for QNNs.