  Variational Hybrid Quantum Classical Algorithms (VHQCAs) are a class of
quantum algorithms intended to run on noisy intermediate-scale quantum (NISQ)
devices. These algorithms employ a parameterized quantum circuit (ansatz) and a
quantum-classical feedback loop. A classical device is used to optimize the
parameters in order to minimize a cost function that can be computed far more
efficiently on a quantum device. The cost function is constructed such that
finding the ansatz parameters that minimize its value, solves some problem of
interest. We focus specifically on the Variational Quantum Linear Solver
(VQLS), and examine the effect of several gradient-free and gradient-based
classical optimizers on performance. We focus on both the average rate of
convergence of the classical optimizers studied, as well as the distribution of
their average termination cost values, and how these are affected by noise. Our
work demonstrates that realistic noise levels on NISQ devices present a
challenge to the optimization process. All classical optimizers appear to be
very negatively affected by the presence of realistic noise. If noise levels
are significantly improved, there may be a good reason for preferring
gradient-based methods in the future, which performed better than the
gradient-free methods with the only shot-noise present. The gradient-free
optimizers, Simultaneous Perturbation Stochastic Approximation (SPSA) and
Powell's method, and the gradient-based optimizers, AMSGrad and BFGS performed
the best in the noisy simulation, and appear to be less affected by noise than
the rest of the methods. SPSA appears to be the best performing method. COBYLA,
Nelder-Mead and Conjugate-Gradient methods appear to be the most heavily
affected by noise, with even slight noise levels significantly impacting their
performance.
