QuGStep: Refining step size selection in gradient estimation for variational quantum algorithms

Variational quantum algorithms (VQAs) offer a promising approach to solving computationally demanding problems by combining parameterized quantum circuits with classical optimization. Estimating probabilistic outcomes on quantum hardware requires repeated measurements (shots). However, in practice, the limited shot budget introduces significant noise in the evaluation of the objective function. Gradient estimation in VQAs often relies on the finite-difference, which evaluates the noisy objective function at perturbed circuit parameter values. The accuracy of this estimation is highly dependent on the choice of step size for these perturbations. An inappropriate step size can exacerbate the impact of noise, causing inaccurate gradient estimates and hindering the classical optimization in VQAs. This paper proposes QuGStep, an algorithm that addresses the challenge of determining the appropriate step size for finite-difference gradient estimation under a shot budget. QuGStep is grounded in a theorem that proves the optimal step size, which accounts for the shot budget, minimizes the error bound in gradient estimation using finite differences. Numerical experiments approximating the ground state energy of several molecules demonstrate that QuGStep can identify the appropriate step size for the given shot budget to obtain effective gradient estimation. Notably, the step size identified by QuGStep achieved convergence to the ground state energy with over 94% fewer shots compared to using a default step size (i.e., step size of 0.01). These findings highlight the potential of QuGStep to improve the practical deployment and scalability of quantum computing technologies.