The noiseless limit and improved-prior limit of the maximum entropy method and their implications for the analytic continuation problem

TL;DR

This paper analyzes the conditions under which the maximum entropy method is effective for the analytic continuation problem in quantum Monte Carlo simulations, highlighting when Bryan's algorithm is appropriate and proposing improvements.

Contribution

The study clarifies when entropy maximization is suitable, compares Bryan's algorithm with a new dual approach, and explores the impact of noise and priors on the method's accuracy.

Findings

Bryan's algorithm is suitable with low noise or accurate priors.

Stochastic sampling reduces to entropy maximization near the true solution.

Improved Bayesian priors enhance the analytic continuation accuracy.

Abstract

Quantum Monte Carlo (QMC) methods are uniquely capable of providing exact simulations of quantum many-body systems. Unfortunately, the applications of a QMC simulation are limited because extracting dynamic properties requires solving the analytic continuation (AC) problem. Across the many fields that use QMC methods, there is no universally accepted analytic continuation algorithm for extracting dynamic properties, but many publications compare to the maximum entropy method. We investigate when entropy maximization is an acceptable approach. We show that stochastic sampling algorithms reduce to entropy maximization when the Bayesian prior is near to the true solution. We investigate when is Bryan's controversial optimization algorithm [Bryan, Eur. Biophys. J. 18, 165-174 (1990)] for entropy maximization (sometimes known as the maximum entropy method) appropriate to use. We show that Bryan's algorithm is appropriate when the noise is near zero or when the Bayesian prior is near to the true solution. We also investigate the mean squared error, finding a better scaling when the Bayesian prior is near the true solution than when the noise is near zero. We point to examples of improved data-driven Bayesian priors that have already leveraged this advantage. We support these results by solving the double Gaussian problem using both Bryan's algorithm and the newly formulated dual approach to entropy maximization [Chuna et al., J. Phys. A: Math. Theor. 58, 335203 (2025)].