# Bayesian Parameter Shift Rule in Variational Quantum Eigensolvers

arXiv: https://arxiv.org/abs/2502.02625 · HTML: https://mlsanity.com/paper/2502.02625/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 28 references that are in the map.
- First posted on arXiv 4 Feb 2025.

## Builds on (most cited first, 6 of 6)

- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Local policy search with Bayesian optimization](https://mlsanity.com/paper/2106.11899/) (2021)
- [Machine Learning for Practical Quantum Error Mitigation](https://mlsanity.com/paper/2309.17368/) (2023)
- [Stochastic Gradient Line Bayesian Optimization for Efficient Noise-Robust Optimization of Parameterized Quantum Circuits](https://mlsanity.com/paper/2111.07952/) (2021)
- [Physics-Informed Bayesian Optimization of Variational Quantum Circuits](https://mlsanity.com/paper/2406.06150/) (2024)
- [Machine-Learning-Enhanced Optimization of Noise-Resilient Variational Quantum Eigensolvers](https://mlsanity.com/paper/2501.17689/) (2025)

## Abstract (arXiv)

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.