  Parameterized quantum circuits can be used as quantum neural networks and
have the potential to outperform their classical counterparts when trained for
addressing learning problems. To date, much of the results on their performance
on practical problems are heuristic in nature. In particular, the convergence
rate for the training of quantum neural networks is not fully understood. Here,
we analyze the dynamics of gradient descent for the training error of a class
of variational quantum machine learning models. We define wide quantum neural
networks as parameterized quantum circuits in the limit of a large number of
qubits and variational parameters. We then find a simple analytic formula that
captures the average behavior of their loss function and discuss the
consequences of our findings. For example, for random quantum circuits, we
predict and characterize an exponential decay of the residual training error as
a function of the parameters of the system. We finally validate our analytic
results with numerical experiments.
