  Variational quantum algorithms (VQAs) offer a promising path toward using
near-term quantum hardware for applications in academic and industrial
research. These algorithms aim to find approximate solutions to quantum
problems by optimizing a parametrized quantum circuit using a classical
optimization algorithm. A successful VQA requires fast and reliable classical
optimization algorithms. Understanding and optimizing how off-the-shelf
optimization methods perform in this context is important for the future of the
field. In this work, we study the performance of four commonly used
gradient-free optimization methods: SLSQP, COBYLA, CMA-ES, and SPSA, at finding
ground-state energies of a range of small chemistry and material science
problems. We test a telescoping sampling scheme (where the accuracy of the
cost-function estimate provided to the optimizer is increased as the
optimization converges) on all methods, demonstrating mixed results across our
range of optimizers and problems chosen. We further hyperparameter tune two of
the four optimizers (CMA-ES and SPSA) across a large range of models and
demonstrate that with appropriate hyperparameter tuning, CMA-ES is competitive
with and sometimes outperforms SPSA (which is not observed in the absence of
hyperparameter tuning). Finally, we investigate the ability of an optimizer to
beat the `sampling noise floor' given by the sampling noise on each
cost-function estimate provided to the optimizer. Our results demonstrate the
necessity for tailoring and hyperparameter-tuning known optimization techniques
for inherently-noisy variational quantum algorithms and that the variational
landscape that one finds in a VQA is highly problem- and system-dependent. This
provides guidance for future implementations of these algorithms in the
experiment.
