Performance comparison of optimization methods on variational quantum algorithms

Physical Review A, vol. 107

Abstract

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 [sequential least-squares quadratic programming, constrained optimization by linear approximations, the covariance matrix adaptation evolutionary strategy (CMA-ES), and the simultaneous perturbation stochastic approximation (SPSA)] to find 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) for 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 of 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 experiments.

Authors 8

1. Xavier Bonet-Monroig Aachen Instituut-Lorentz

   Leiden University

   Affiliation as printed

   Instituut-Lorentz, Universiteit Leiden, 2300 RA Leiden, Netherlands

2. Hao Wang Aachen Leiden Institute of Advanced Computer Science

   Leiden University

   Affiliation as printed

   Leiden Institute of Advanced Computer Science, Universiteit Leiden, 2333 CA Leiden, Netherlands

3. Diederick Vermetten Aachen Leiden Institute of Advanced Computer Science

   Leiden University

   Affiliation as printed

   Leiden Institute of Advanced Computer Science, Universiteit Leiden, 2333 CA Leiden, Netherlands

4. Bruno Senjean Aachen Instituut-Lorentz

   École Nationale Supérieure de Chimie de Montpellier · Leiden University · Centre National de la Recherche Scientifique · Université de Montpellier · Institut Charles Gerhardt Montpellier

   Affiliation as printed

   ICGM, Université Montpellier, CNRS, ENSCM, 34090 Montpellier, France

   Instituut-Lorentz, Universiteit Leiden, 2300 RA Leiden, Netherlands

5. Charles Moussa Aachen Leiden Institute of Advanced Computer Science

   Leiden University

   Affiliation as printed

   Leiden Institute of Advanced Computer Science, Universiteit Leiden, 2333 CA Leiden, Netherlands

6. Thomas Bäck Aachen Leiden Institute of Advanced Computer Science

   Leiden University

   Affiliation as printed

   Leiden Institute of Advanced Computer Science, Universiteit Leiden, 2333 CA Leiden, Netherlands

7. Vedran Dunjko Aachen Leiden Institute of Advanced Computer Science

   Leiden University

   Affiliation as printed

   Leiden Institute of Advanced Computer Science, Universiteit Leiden, 2333 CA Leiden, Netherlands

8. Thomas E. O’Brien Aachen Instituut-Lorentz

   Leiden University

   Affiliation as printed

   Google Quantum AI, 80636 Munich, Germany

   Instituut-Lorentz, Universiteit Leiden, 2300 RA Leiden, Netherlands