The Bayesian Inversion Problem for Thermal Average Sampling of Quantum Systems

TL;DR

This paper introduces a Bayesian inversion method for sampling potential functions in quantum systems from noisy observational data, enabling accurate thermal average sampling across various observables and quantum regimes.

Contribution

The paper presents a novel Bayesian inversion framework that improves potential sampling accuracy and flexibility, applicable to multi-level quantum systems in non-adiabatic regimes.

Findings

Accurate sampling of test observables demonstrated

Method outperforms traditional local density approaches

Numerical tests confirm efficiency and robustness

Abstract

In this article, we propose a novel method for sampling potential functions based on noisy observation data of a finite number of observables in quantum canonical ensembles, which leads to the accurate sampling of a wide class of test observables. The method is based on the Bayesian inversion framework, which provides a platform for analyzing the posterior distribution and naturally leads to an efficient numerical sampling algorithm. We highlight that, the stability estimate is obtained by treating the potential functions as intermediate variables in the following way: the discrepancy between two sets of observation data of training observables can bound the distance between corresponding posterior distributions of potential functions, while the latter naturally leads to a bound of the discrepancies between corresponding thermal averages of test observables. Besides, the training observables can be more flexible than finite samples of the local density function, which are mostly used in previous researches. The method also applies to the multi-level quantum systems in the non-adiabatic regime. In addition, we provide extensive numerical tests to verify the accuracy and efficiency of the proposed algorithm.