Performance comparison of optimization methods on variational quantum algorithms

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-theshelf 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.

Resource frugal optimizer for quantum machine learning

“…A potential future research direction is to extend our analysis to more complicated, non-linear loss functions such as the log-likelihood, exploring in more detail how to introduce shot frugality to gradient-free optimizers [50][51][52]. Furthermore, applying our optimizer to a problem of interest on a real device is an interesting and logical next step.…”

“…It would also be interesting to introduce different techniques to reduce the number of shots spent such as Bayesian optimization [27]. Finally, other empirical studies related to algorithm selection and configuration [51,55,[61][62][63][64][65] can be applied to Refoqus on different datasets [63,66,67].…”

Programming Quantum Neural Networks on NISQ Systems: An Overview of Technologies and Methodologies

“…For this reason, they were widely used in developing the first QNNs. This optimizer class includes the Nelder–Mead [ 70 ] and COBYLA algorithms [ 71 ]. These gradient-free optimizer methods are often provided within the QNN frameworks (e.g., they are readily available in ) or available via external packages, such as [ 72 ].…”

A Vision for the Future of Multiscale Modeling

“…Given the resemblance of these algorithms to reinforcement learning and optimal control theory, , various strategies have been borrowed from these domains to enhance both the performance and the range of applicability of these methods. , Typically, these methods have an asymptotic runtime of scriptO ( K / ϵ 2 ) where K is a constant depending on the particular method (and system) considered and ϵ is the accuracy threshold we impose when solving the given task. Due to this asymptotic scaling, whether it is possible to demonstrate quantum advantage with these methodologies is still an open question .…”