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arXiv:1811.08044 · Communications on Pure and Applied Mathematics

Inchworm Monte Carlo Method for Open Quantum Systems

Zhenning Cai, Jianfeng Lu, Siyao Yang

Published Mar 6, 2020• 26 citations
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Abstract

Abstract We investigate in this work a recently proposed diagrammatic quantum Monte Carlo method—the inchworm Monte Carlo method—for open quantum systems. We establish its validity rigorously based on resummation of Dyson series. Moreover, we introduce an integro‐differential equation formulation for open quantum systems, which illuminates the mathematical structure of the inchworm algorithm. This new formulation leads to an improvement of the inchworm algorithm by introducing classical deterministic time‐integration schemes. The numerical method is validated by applications to the spin‐boson model. © 2020 Wiley Periodicals, Inc.

DOI
10.1002/cpa.21888

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Abstract We investigate in this work a recently proposed diagrammatic quantum Monte Carlo method—the inchworm Monte Carlo method—for open quantum systems. We establish its validity rigorously based on resummation of Dyson series. Moreover, we introduce an integro‐differential equation formulation for open quantum systems, which illuminates the mathematical structure of the inchworm algorithm. This new formulation leads to an improvement of the inchworm algorithm by introducing classical deterministic time‐integration schemes. The numerical method is validated by applications to the spin‐boson model. © 2020 Wiley Periodicals, Inc.
We consider the numerical analysis of the inchworm Monte Carlo method, which is proposed recently to tackle the numerical sign problem for open quantum systems. We focus on the growth of the numerical error with respect to the simulation time, for which the inchworm Monte Carlo method shows a flatter curve than the direct application of Monte Carlo method to the classical Dyson series. To better understand the underlying mechanism of the inchworm Monte Carlo method, we distinguish two types of exponential error growth, which are known as the numerical sign problem and the error amplification. The former is due to the fast growth of variance in the stochastic method, which can be observed from the Dyson series, and the latter comes from the evolution of the numerical solution. Our analysis demonstrates that the technique of partial resummation can be considered as a tool to balance these two types of error, and the inchworm Monte Carlo method is a successful case where the numerical sign problem is effectively suppressed by such means. We first demonstrate our idea in the context of ordinary differential equations, and then provide complete analysis for the inchworm Monte Carlo method. Several numerical experiments are carried out to verify our theoretical results.
Abstract. We present two diagrammatic Monte Carlo methods for quantum systems coupled with harmonic baths, whose dynamics are described by integro-differential equations. The first approach can be considered as a reformulation of the Dyson series, and the second one, called “bold-thin-bold diagrammatic Monte Carlo,” is based on resummation of the diagrams in the Dyson series to accelerate its convergence. The underlying mechanism of the governing equations associated with the two methods lies in the recurrence relation of the path integrals, which is the most costly part in the numerical methods. The proposed algorithms give an extension to the work of Cai, Lu, and Yang [ Comput. Phys. Commun. 278 (2022), 108417], where the algorithms are designed based on reusing the previous calculations of bath influence functionals. Compared with the algorithms therein, our methods further include the reuse of system-associated functionals and show better performance in terms of computational efficiency and memory cost. We demonstrate the two methods in the framework of spin-boson model, and numerical experiments are carried out to verify the validity of the methods.
We study the real-time simulation of open quantum systems, where the system is modeled by a spin chain with each spin associated with its own harmonic bath. Our method couples the inchworm method for the spin-boson model and the modular path integral methodology for spin systems. In particular, the introduction of the inchworm method can significantly suppress the numerical sign problem. Both methods are tweaked to make them work seamlessly with each other. We represent our approach in the language of diagrammatic methods and analyze the asymptotic behavior of the computational cost. Extensive numerical experiments are performed to validate our method.
In this paper, we provide a detailed description of the inchworm Monte Carlo formalism for the exact study of real-time non-adiabatic dynamics. This method optimally recycles Monte Carlo information from earlier times to greatly suppress the dynamical sign problem. Using the example of the spin-boson model, we formulate the inchworm expansion in two distinct ways: The first with respect to an expansion in the system-bath coupling and the second as an expansion in the diabatic coupling. The latter approach motivates the development of a cumulant version of the inchworm Monte Carlo method, which has the benefit of improved scaling. This paper deals completely with methodology, while Paper II provides a comprehensive comparison of the performance of the inchworm Monte Carlo algorithms to other exact methodologies as well as a discussion of the relative advantages and disadvantages of each.
In this second paper of a two part series, we present extensive benchmark results for two different inchworm Monte Carlo expansions for the spin-boson model. Our results are compared to previously developed numerically exact approaches for this problem. A detailed discussion of convergence and error propagation is presented. Our results and analysis allow for an understanding of the benefits and drawbacks of inchworm Monte Carlo compared to other approaches for exact real-time non-adiabatic quantum dynamics.
Abstract We present two fast algorithms which apply inclusion–exclusion principle to sum over the bosonic diagrams in bare diagrammatic quantum Monte Carlo and inchworm Monte Carlo method, respectively. In the case of inchworm Monte Carlo, the proposed fast algorithm gives an extension to the work [2018 Inclusion–exclusion principle for many-body diagrammatics Phys. Rev. B 98 115152] from fermionic to bosonic systems. We prove that the proposed fast algorithms reduce the computational complexity from double factorial to exponential. Numerical experiments are carried out to verify the theoretical results and to compare the efficiency of the methods.