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Universal approximation theorems are the foundations of classical neuralnetworks, providing theoretical guarantees that the latter are able toapproximate maps of interest. Recent results have shown that this can also beachieved in a quantum setting, whereby classical functions can be approximatedby parameterised quantum circuits. We provide here precise error bounds forspecific classes of functions and extend these results to the interesting newsetup of randomised quantum circuits, mimicking classical reservoir neuralnetworks. 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.

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