  A novel class of hybrid quantum-classical algorithms based on the variational
approach have recently emerged from separate proposals addressing, for example,
quantum chemistry and combinatorial problems. These algorithms provide an
approximate solution to the problem at hand by encoding it in the state of a
quantum computer. The operations used to prepare the state are not a priori
fixed but, quite the opposite, are subjected to a classical optimization
procedure that modifies the quantum gates and improves the quality of the
approximate solution. While the quantum hardware determines the size of the
problem and what states are achievable (limited, respectively, by the number of
qubits and by the kind and number of possible quantum gates), it is the
classical optimization procedure that determines the way in which the quantum
states are explored and whether the best available solution is actually
reached. In addition, the quantities required in the optimization, for example
the objective function itself, have to be estimated with finite precision in
any experimental implementation. While it is desirable to have very precise
estimates, this comes at the cost of repeating the state preparation and
measurement multiple times. Here we analyze the competing requirements of high
precision and low number of repetitions and study how the overall performance
of the variational algorithm is affected by the precision level and the choice
of the optimization method. Finally, this study introduces quasi-Newton
optimization methods in the general context of hybrid variational algorithms
and presents quantitative results for the Quantum Approximate Optimization
Algorithm.
