Quantum neural networks (QNNs) constitute the quantum version of deep neural models, where the generated functions are defined by the expectation values of quantum observables measured on the output of parametric circuits. A fundamental breakthrough in the theory of classical deep learning has been the proof that, in the limit of infinite width, the probability distribution of the function generated by a neural network converges to a Gaussian process.

In this presentation, I will explore the extension of these properties to the quantum domain. While recent advancements have established this convergence qualitatively, we provide a rigorous quantitative proof. Using Stein's method for normal approximation, we establish explicit upper bounds on the Wasserstein distance of order 1 between the distribution of a finite-width QNN and the limiting Gaussian process. Furthermore, I will analyze the training dynamics under gradient flow, proving that these quantitative bounds remain valid throughout the optimization process and are uniform in time. This analysis confirms that large-width QNNs preserve their Gaussian characteristics even for infinite training time, providing a solid theoretical foundation for understanding the behavior and stability of overparameterized quantum machine learning models.\\ \noindent \textit{This talk is based on joint works with F. Girardi, D. Pastorello, and G. De Palma.}