  The Quantum Approximate Optimization Algorithm (QAOA) adopts a hybrid
quantum-classical approach to find approximate solutions to variational
optimization problems. In fact, it relies on a classical subroutine to optimize
the parameters of a quantum circuit. In this work we present a Bayesian
optimization procedure to fulfil this optimization task, and we investigate its
performance in comparison with other global optimizers. We show that our
approach allows for a significant reduction in the number of calls to the
quantum circuit, which is typically the most expensive part of the QAOA. We
demonstrate that our method works well also in the regime of slow circuit
repetition rates, and that few measurements of the quantum ansatz would already
suffice to achieve a good estimate of the energy. In addition, we study the
performance of our method in the presence of noise at gate level, and we find
that for low circuit depths it is robust against noise. Our results suggest
that the method proposed here is a promising framework to leverage the hybrid
nature of QAOA on the noisy intermediate-scale quantum devices.
