Universal Approximation Theorem and error bounds for quantum neural
  networks and quantum reservoirs
Lukas Gonon Antoine Jacquier
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.