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arXiv:2505.10211 · Computer Physics Communications

PyLIT: Reformulation and implementation of the analytic continuation problem using kernel representation methods

Alexander Benedix Robles, Phil-Alexander Hofmann, Thomas Chuna, Tobias Dornheim, Michael Hecht

Published Oct 24, 2025• 5 citations
arXivHTMLPDF

Abstract

Path integral Monte Carlo (PIMC) simulations are a cornerstone method for studying quantum many-body systems, such as warm dense matter and ultracold atoms. The analytic continuation needed to estimate dynamic quantities from these simulations amounts to an inverse Laplace transform, which is an ill-conditioned problem. If this challenging problem were surmounted, dynamical observables such as the dynamic structure factor (DSF) S ( q , ω ) —a key property e.g. in x-ray and neutron scattering experiments—could be extracted from the imaginary-time correlation functions F ( q , τ ) estimates. Although of fundamental importance, the analytic continuation problem remains challenging due to its ill-posedness, and state-of-the-art techniques continue to deliver unsatisfactory results. To address this challenge, we express the DSF as a linear combination of kernel functions with known Laplace transforms that have been tailored to satisfy its physical constraints, e.g. , detailed balance. Then we employ least-squares optimization regularized with a Bayesian prior estimate to determine the coefficients of this linear combination. We explore various regularization term, such as the commonly used entropic regularizer, as well as the uncommon L 2 -distance and CDF L 2 -distance. We also explore techniques for setting the regularization weight. A key outcome and contribution is the open-source package PyLIT ( Py thon L aplace I nverse T ransform), which leverages Numba for C-level performance, unifying the presented formulations. PyLIT’s core functionality is kernel construction and optimization. In our applications, we find that PyLIT’s DSF estimates share many qualitative features with other more established methods. Drawing from our insights, we identify three key findings. Firstly, independent of the regularization choice, utilizing non-uniform grid point distributions reduced the number of unknowns and thus reduced our space of possible solutions. Secondly, the L 2 -distance and CDF L 2 -distance, previously unexplored regularizers, benefit from their linear gradients and perform about as well as entriopic regularization. Thirdly, future work can meaningfully combine regularized and stochastic optimization.

DOI
10.1016/j.cpc.2025.109904

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Path integral Monte Carlo (PIMC) simulations are a cornerstone method for studying quantum many-body systems, such as warm dense matter and ultracold atoms. The analytic continuation needed to estimate dynamic quantities from these simulations amounts to an inverse Laplace transform, which is an ill-conditioned problem. If this challenging problem were surmounted, dynamical observables such as the dynamic structure factor (DSF) S ( q , ω ) —a key property e.g. in x-ray and neutron scattering experiments—could be extracted from the imaginary-time correlation functions F ( q , τ ) estimates. Although of fundamental importance, the analytic continuation problem remains challenging due to its ill-posedness, and state-of-the-art techniques continue to deliver unsatisfactory results. To address this challenge, we express the DSF as a linear combination of kernel functions with known Laplace transforms that have been tailored to satisfy its physical constraints, e.g. , detailed balance. Then we employ least-squares optimization regularized with a Bayesian prior estimate to determine the coefficients of this linear combination. We explore various regularization term, such as the commonly used entropic regularizer, as well as the uncommon L 2 -distance and CDF L 2 -distance. We also explore techniques for setting the regularization weight. A key outcome and contribution is the open-source package PyLIT ( Py thon L aplace I nverse T ransform), which leverages Numba for C-level performance, unifying the presented formulations. PyLIT’s core functionality is kernel construction and optimization. In our applications, we find that PyLIT’s DSF estimates share many qualitative features with other more established methods. Drawing from our insights, we identify three key findings. Firstly, independent of the regularization choice, utilizing non-uniform grid point distributions reduced the number of unknowns and thus reduced our space of possible solutions. Secondly, the L 2 -distance and CDF L 2 -distance, previously unexplored regularizers, benefit from their linear gradients and perform about as well as entriopic regularization. Thirdly, future work can meaningfully combine regularized and stochastic optimization.
An accurate theoretical description of the dynamic properties of correlated quantum many-body systems, such as the dynamic structure factor S(q, ω), is important in many fields. Unfortunately, highly accurate quantum Monte Carlo methods are usually restricted to the imaginary time domain, and the analytic continuation of the imaginary-time density–density correlation function F(q, τ) to real frequencies is a notoriously hard problem. Here, it is argued that often no such analytic continuation is required because by definition, F(q, τ) contains the same physical information as does S(q, ω), only represented unfamiliarly. Specifically, it is shown how one can directly extract key information such as the temperature or quasi-particle excitation energies from the τ domain, which is highly relevant for equation-of-state measurements of matter under extreme conditions [T. Dornheim et al., Nat. Commun. 13, 7911 (2022)]. As a practical example, ab initio path-integral Monte Carlo results for the uniform electron gas (UEG) are considered, and it is shown that even nontrivial processes such as the roton feature of the UEG at low density [T. Dornheim et al., Commun. Phys. 5, 304 (2022)] are manifested straightforwardly in F(q, τ). A comprehensive overview is given of various useful properties of F(q, τ) and how it relates to the usual dynamic structure factor. In fact, working directly in the τ domain is advantageous for many reasons and opens up multiple avenues for future applications.
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.
Many fields of physics use quantum Monte Carlo (QMC) simulations to simulate quantum systems in imaginary-time τττ and estimate imaginary-time correlation functions (ITCF). However, extracting dynamic ωωω-dependent quantities from ITCFs is a notoriously difficult task, known as analytic continuation (AC), that amounts to solving an exponentially ill-conditioned inverse problem. Within the AC literature, there are competing stochastic and regularized approaches, as well as an emerging collection of works using parameterized models like neural networks. Here we transcend the traditional divides between the communities, introducing the regularized stochastic optimization method (RSOM). This method reformulates AC as a dictionary learning problem, discovering a sparse dictionary to represent the solution. Our approach is motivated by the astounding results dictionary learning has produced in many scientific fields. Remarkably, RSOM discovers a sparse dictionary that maps an ill-conditioned inverse problem to a low-dimensional problem that is well-conditioned. We demonstrate that the method yields competitive results for common synthetic test problems as well as for authentic QMC data from the finite temperature electron gas. This work exposes that a dictionary exists within all stochastic and regularized methods and that dictionary learning provides a new angle of attack for future AC methods.
We present a machine learning approach to the inversion of Fredholm integrals of the first kind. The approach provides a natural regularization in cases where the inverse of the Fredholm kernel is ill-conditioned. It also provides an efficient and stable treatment of constraints. The key observation is that the stability of the forward problem permits the construction of a large database of outputs for physically meaningful inputs. We apply machine learning to this database to generate a regression function of controlled complexity, which returns approximate solutions for previously unseen inputs; the approximate solutions are then projected onto the subspace of functions satisfying relevant constraints. We also derive and present uncertainty estimates. We illustrate the approach by applying it to the analytical continuation problem of quantum many-body physics, which involves reconstructing the frequency dependence of physical excitation spectra from data obtained at specific points in the complex frequency plane. Under standard error metrics the method performs as well or better than the Maximum Entropy method for low input noise and is substantially more robust to increased input noise. We expect the methodology to be similarly effective for any problem involving a formally ill-conditioned inversion, provided that the forward problem can be efficiently solved.
We present a machine-learning approach to a long-standing issue in quantum many-body physics, namely, analytic continuation. This notorious ill-conditioned problem of obtaining spectral function from an imaginary time Green's function has been a focus of new method developments for past decades. Here we demonstrate the usefulness of modern machine-learning techniques including convolutional neural networks and the variants of a stochastic gradient descent optimizer. The machine-learning continuation kernel is successfully realized without any ``domain knowledge,'' which means that any physical ``prior'' is not utilized in the kernel construction and the neural networks ``learn'' the knowledge solely from ``training.'' The outstanding performance is achieved for both insulating and metallic band structure. Our machine-learning-based approach not only provides the more accurate spectrum than the conventional methods in terms of peak positions and heights, but is also more robust against the noise which is the required key feature for any continuation technique to be successful. Furthermore, its computation speed is 104–105{10}^{4}\text{--}{10}^{5}104–105 times faster than the maximum entropy method.
The ab initio path integral Monte Carlo (PIMC) approach is one of the most successful methods in quantum many-body theory. A particular strength of this method is its straightforward access to imaginary-time correlation functions (ITCFs). For example, the well-known density–density ITCF F(q, τ) allows one to estimate the linear response of a given system for all wave vectors q from a single simulation of the unperturbed system. Moreover, it constitutes the basis for the reconstruction of the dynamic structure factor S(q, ω)—a key quantity in state-of-the-art scattering experiments. In this work, we present analogous relations between the nonlinear density response in the quadratic and cubic order of the perturbation strength and generalized ITCFs measuring correlations between up to four imaginary-time arguments. As a practical demonstration of our new approach, we carry out simulations of the warm dense electron gas and find excellent agreement with previous PIMC results that had been obtained with substantially larger computational effort. In addition, we give a relation between a cubic ITCF and the triple dynamic structure factor S(q1, ω1; q2, ω2), which evokes the enticing possibility to study dynamic three-body effects on an ab initio level.