  Variational hybrid quantum-classical optimization represents one of the most
promising avenue to show the advantage of nowadays noisy intermediate-scale
quantum computers in solving hard problems, such as finding the minimum-energy
state of a Hamiltonian or solving some machine-learning tasks. In these devices
noise is unavoidable and impossible to error-correct, yet its role in the
optimization process is not well understood, especially from the theoretical
viewpoint. Here we consider a minimization problem with respect to a
variational state, iteratively obtained via a parametric quantum circuit,
taking into account both the role of noise and the stochastic nature of quantum
measurement outcomes. We show that the accuracy of the result obtained for a
fixed number of iterations is bounded by a quantity related to the Quantum
Fisher Information of the variational state. Using this bound, we study the
convergence property of the quantum approximate optimization algorithm under
realistic noise models, showing the robustness of the algorithm against
different noise strengths.
