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

TL;DR

This paper establishes theoretical error bounds for quantum neural networks and reservoirs, demonstrating their capacity to approximate functions with quantifiable accuracy and resource requirements.

Contribution

It provides precise error bounds for quantum neural networks and extends these results to randomized quantum circuits resembling classical reservoirs.

Findings

Quantum neural networks can approximate functions with accuracy ε using O(ε^{-2}) weights.

The number of qubits needed scales with the logarithm of 1/ε.

Error bounds are derived for functions with integrable Fourier transforms.

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.