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arXiv:2511.06915 · Physical Review Research

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

Thomas Chuna, Nicholas Barnfield, Paul Hamann, Sebastian Schwalbe, Michael P. Friedlander, Tobias Dornheim

Published May 11, 2026• 1 citation
arXivHTMLPDF

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 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, for entropy maximization (sometimes known as the maximum entropy method—MEM) appropriate to use. We show that Bryan’s algorithm is appropriate when the noise is near zero or when the Bayesian prior is near 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. Together, these three results indicate that enhancing the quality of the Bayesian prior enhances the validity of the MEM and has better return on investment than reducing the noise. 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 , .

DOI
10.1103/yqjh-ycbh

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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 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, for entropy maximization (sometimes known as the maximum entropy method—MEM) appropriate to use. We show that Bryan’s algorithm is appropriate when the noise is near zero or when the Bayesian prior is near 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. Together, these three results indicate that enhancing the quality of the Bayesian prior enhances the validity of the MEM and has better return on investment than reducing the noise. 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 , .
Abstract Many fields of physics use quantum Monte Carlo techniques, but struggle to estimate dynamic spectra via the analytic continuation of imaginary-time quantum Monte Carlo data. One of the most ubiquitous approaches to analytic continuation is the maximum entropy method (MEM). We supply a dual Newton optimization algorithm to be used within the MEM and provide analytic bounds for the algorithm’s error. The optimization algorithm is freely availible on github ( repository ). The MEM is typically used with Bryan’s controversial algorithm (Rothkopf 2020 Data 5 55). We present new theoretical issues that are not yet in the literature. Our algorithm has all the theoretical benefits of Bryan’s algorithm without these theoretical issues the implementation of the dual Newton optimizer within the MEM is freely available on github ( repository ). We compare the MEM with Bryan’s optimization to the MEM with our dual Newton optimization on test problems from lattice quantum chromodynamics and plasma physics. These comparisons show that in the presence of noise the dual Newton algorithm produces better estimates and error bars; this indicates the limits of Bryan’s algorithm’s applicability. We use the MEM to investigate authentic quantum Monte Carlo data for the uniform electron gas at warm dense matter conditions and further substantiate the roton-type feature in the dispersion relation.
The presence of dynamical information is a feature distinguishing a finite-temperature quantum Monte Carlo simulation from a classical one. We now discuss numerical methods for extracting this information that use techniques and concepts borrowed from an area of probability theory called Bayesian statistical inference. The use of these techniques and concepts provided a solution to the very difficult problem of analytically continuing imaginary-time Green's functions, estimated by a quantum Monte Carlo simulation, to the real-time axis. Baym and Mermin (1961) proved that a unique mapping between these functions exists. However, executing this mapping numerically, with a simulation's incomplete and noisy data, transforms the problem into one without a unique solution and thus into a problem of finding a “best” solution according to some reasonable criterion. Instead of executing the analytic continuation between imaginary- and real-time Green's functions, thereby obtaining real-time dynamics, we instead estimate the experimentally relevant spectral density function these Green's functions share. We present three “best” solutions and emphasize that making the simulation data consistent with the assumptions of the numerical approach is a key step toward finding any of these best solutions. Preliminary comments The title of this chapter, “Analytic Continuation,” is unusual in the sense that it describes the task we wish to accomplish instead of the method we use to accomplish it. If we used the name of the method, the title would be something like “Bayesian Statistical Inference Using an Entropic Prior.” A shorter title would be “The Maximum Entropy Method.”We hope by the end of the chapter the reader will agree that using the short title is perhaps too glib and the longer one has meaningful content.
We report multipronged progress on the stochastic averaging approach to numerical analytic continuation of imaginary-time correlation functions computed by quantum Monte Carlo simulations. After reviewing the conventional maximum-entropy approach and established stochastic analytic continuation methods, we present several new developments in which the configurational entropy of the sampled spectrum plays a key role. Parametrizing the spectrum as a large number of δ-functions in continuous frequency space, an exact calculation of the entropy lends support to a simple goodness-of-fit criterion for the optimal sampling temperature. We also compare spectra sampled in continuous frequency with those from amplitudes sampled on a fixed frequency grid. Insights into the functional form of the entropy in different cases allow us to demonstrate equivalence in a generalized thermodynamic limit (large number of degrees of freedom) of the average spectrum and the maximum-entropy solution, with different parametrizations corresponding to different forms of the entropy in the prior probability. These results revise prevailing notions of the maximum-entropy method and its relationship to stochastic analytic continuation. In further developments of the sampling approach, we explore various adjustable (optimized) constraints that allow sharp low-temperature spectral features to be resolved, in particular at the lower frequency edge. The constraints, e.g., the location of the edge or the spectral weight of a quasi-particle peak, are optimized using a statistical criterion based on entropy minimization under the condition of optimal fit. We show with several examples that this method can correctly reproduce both narrow and broad quasi-particle peaks. We next introduce a parametrization for more intricate spectral functions with sharp edges, e.g., power-law singularities. We present tests with synthetic data as well as with real simulation data for the spin-1/2 Heisenberg chain, where a divergent edge of the dynamic structure factor is due to deconfined spinon excitations. Our results demonstrate that distortions of sharp edges or quasi-particle peaks, which arise with other analytic continuation methods, propagate and cause artificial spectral features also at higher energies. The constrained sampling methods overcome this problem and allow analytic continuation of spectra with sharp edge features at unprecedented fidelity. We present results for S=1/2 Heisenberg 2-leg and 3-leg ladders to illustrate the ability of the methods to resolve spectral features arising from both elementary and composite excitations. Finally, we also propose how the methods developed here could be used as “pre processors” for analytic continuation by machine learning. Edge singularities and narrow quasi-particle peaks being ubiquitous in quantum many-body systems, we expect the new methods to be broadly useful and take numerical analytic continuation to a new quantitative level in many applications.
An outstanding problem in the simulation of condensed-matter phenomena is how to obtain dynamical information. We consider the numerical analytic continuation of imaginary-time quantum Monte Carlo data to obtain real-frequency spectral functions. This is an extremely ill-posed problem similar to the inversion of a Laplace transform. We suggest an image-reconstruction approach, which has been widely applied to data analysis in experimental research. Specifically, we apply the maximum-entropy method (ME) to the analytic continuation of quantum Monte Carlo data. We report encouraging preliminary results for the Fano-Anderson model of an impurity state in a continuum. The incorporation of additional prior information, such as sum rules and asymptotic behavior, can be expected to significantly improve results. We compare (ME) to alternative methods. We also discuss statistical error propagation for the analytic continuation problem via the likelihood function, which is independent of the choice of image-reconstruction method. This includes the sensitivity of the data to structure in the spectral function, the optimization of Monte Carlo simulations, and how to incorporate covariance in the statistical errors of the Monte Carlo method.
Bayesian parametric analytic continuation (BPAC) is proposed for the analytic continuation of noisy imaginary-time Green's function data as, e.g., obtained by continuous-time quantum Monte Carlo simulations (CTQMC). Within BPAC, the spectral function is inferred from a suitable set of parametrized basis functions. Bayesian model comparison then allows to assess the reliability of different parametrizations. The required evidence integrals of such a model comparison are determined by nested sampling. Compared to the maximum entropy method (MEM), routinely used for the analytic continuation of CTQMC data, the presented approach allows to infer whether the data support specific structures of the spectral function. We demonstrate the capability of BPAC in terms of CTQMC data for an Anderson impurity model (AIM) that shows a generalized Kondo scenario and compare the BPAC reconstruction to the MEM as well as to the spectral function obtained from the real-time fork tensor product state impurity solver where no analytic continuation is required. Furthermore, we present a combination of MEM and BPAC and its application to an AIM arising from the ab initio treatment of SrVO3{\mathrm{SrVO}}_{3}SrVO3​.