Bayesian Parameter Shift Rule in Variational Quantum Eigensolvers

TL;DR

This paper introduces a Bayesian variant of parameter shift rules for variational quantum eigensolvers, utilizing Gaussian processes to improve gradient estimation, accelerate optimization, and reduce observation costs.

Contribution

The paper proposes a Bayesian parameter shift rule using Gaussian processes, enabling flexible, uncertainty-aware gradient estimation and more efficient VQE optimization.

Findings

Bayesian PSR accelerates VQE optimization compared to traditional methods.

The approach reduces observation costs through the GradCoRe concept.

Numerical experiments demonstrate superior performance over state-of-the-art methods.

Abstract

Parameter shift rules (PSRs) are key techniques for efficient gradient estimation in variational quantum eigensolvers (VQEs). In this paper, we propose its Bayesian variant, where Gaussian processes with appropriate kernels are used to estimate the gradient of the VQE objective. Our Bayesian PSR offers flexible gradient estimation from observations at arbitrary locations with uncertainty information and reduces to the generalized PSR in special cases. In stochastic gradient descent (SGD), the flexibility of Bayesian PSR allows the reuse of observations in previous steps, which accelerates the optimization process. Furthermore, the accessibility to the posterior uncertainty, along with our proposed notion of gradient confident region (GradCoRe), enables us to minimize the observation costs in each SGD step. Our numerical experiments show that the VQE optimization with Bayesian PSR and GradCoRe significantly accelerates SGD and outperforms the state-of-the-art methods, including sequential minimal optimization.