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.

翻译：通用逼近定理是经典神经网络的理论基础，为后者能够逼近目标映射提供了理论保证。近期研究结果表明，这一目标在量子框架下同样可以实现——经典函数可通过参数化量子电路进行逼近。本文针对特定函数类给出了精确的误差界，并将这些结果拓展到随机量子电路这一模拟经典储层神经网络的新兴框架。我们的研究特别证明：当逼近具有可积傅里叶变换的函数时，仅需具有$\mathcal{O}(\varepsilon^{-2})$个权重参数和$\mathcal{O} (\lceil \log_2(\varepsilon^{-1}) \rceil)$个量子比特的量子神经网络，即可达到$\varepsilon>0$的逼近精度。