Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirs

TL;DR

The paper addresses the problem of providing quantitative universal approximation guarantees for quantum neural networks and quantum reservoir computing. It develops two complementary architectures—trainable variational quantum circuits and randomly initialized quantum reservoirs—and proves explicit $L^2$ and $L^\infty$ error bounds rooted in Fourier-analytic representations of target functions. The main contributions include an explicit universal variational circuit with $O(\varepsilon^{-2})$ weights and $O(\lceil \log_2(\varepsilon^{-1}) \rceil)$ qubits, mean-square universal approximation for random quantum networks, and uniform universality on compact sets, all with dimension-robust error rates and clear resource trade-offs. These results provide a rigorous theoretical foundation for quantum ML architectures, showing dimension-free approximation capabilities for function classes with integrable Fourier transforms and giving concrete guidance on circuit size and qubit requirements for achieving a desired accuracy.

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 parameterised quantum circuits. We provide here precise error bounds for specific classes of functions and extend these results to the interesting new setup of randomised 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 accuracy $\varepsilon>0$ when approximating functions with integrable Fourier transform.

Figures (3)

• Figure 1: Abstract description of the quantum circuit in Definition \ref{['def:QC']}. The initial state $\ket{0}^{\otimes \mathfrak{n}}$ is processed through the quantum gates $\mathtt{V}$ and $\mathtt{U}$ and then measured.
• Figure 2: Schematic diagram of the quantum neural network $f_{n,\boldsymbol{\theta}}^{R}$ with $\mathfrak{n} = \lceil \log_2(4n) \rceil$ qubits. The initial state $\ket{0}^{\otimes \mathfrak{n}}$ is processed through the quantum gates $\mathtt{V}$ and $\mathtt{U}$ and then measured; yielding probabilities $\mathbb{P}_1^{n}$ and $\mathbb{P}_2^{n}$. These probabilities are aggregated into the network output $f_{n,\boldsymbol{\theta}}^{R}$ according to \ref{['eq:qnn']}.
• Figure 3: Schematic diagram of the quantum random neural network $F_{\boldsymbol{w}}$ with $\overline{\mathfrak{n}} = \lceil \log_2(2n) \rceil$ qubits. The initial state $\ket{0}^{\otimes \overline{\mathfrak{n}}}$ is processed through the quantum gates $\overline{\mathtt{V}}$ and $\overline{\mathtt{U}}$ and then measured, yielding probabilities $\overline{\mathbb{P}}_0, \ldots, \overline{\mathbb{P}}_{2(n-1)}$. These probabilities are aggregated into the network output $F_{\boldsymbol{w}}$ according to \ref{['eq:randomF']}.

Theorems & Definitions (36)

• Remark 2.1
• Definition 2.2
• Remark 2.3
• Theorem 2.4
• Remark 2.5
• Remark 2.6
• Corollary 2.7
• Remark 3.1
• Remark 3.2
• Remark 3.3