A Berry-Esseen theorem for wide quantum neural networks

Quantum neural networks are the quantum counterpart of deep neural networks and generate model functions given by the expectation value of a quantum observable measured on the state generated by a parametric quantum circuit. Parametric quantum circuits are made by the composition of elementary parametric quantum operations (gates) and are considered prime candidates for practical applications of quantum computing with the noisy intermediate-scale quantum devices that will be available in the forthcoming years. Quantum neural networks have wide applications both in machine learning problems, such as supervised learning, and in optimization problems. The recent work [Girardi et al., arXiv:2402.08726] has proven that the law of the model function generated by an untrained quantum neural network with random parameters converges in distribution to a Gaussian process in the limit of infinite width of the circuit. In this thesis we establish a quantitative version of this result. We consider randomly initialized quantum neural networks of finite width and a single input, and we establish an upper bound on the Kolmogorov distance between the law of the random variable generated by the network and the Gaussian law with the same mean and variance. Our proof is based on the method of cumulants to derive an upper bound on the absolute difference between the characteristic functions of the two random variables.