Inchworm Monte Carlo Method for Open Quantum Systems

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. Keywords: quantumMonte Carlo, open quantum system, diagrammatic methods, spin-boson model to obtain a Markovian approximation, known as the Lindblad equation [12,13], which is the quantum analog of the Langevin dynamics. The Markovian approximation however breaks down for open systems with stronger coupling, so that one has to keep track of the non-Markovian dynamics of the projected density matrix describing the system of interests [3]. For classical systems, such dynamics is often modeled and studied as generalized Langevin equations [24,63], while for quantum systems, the non-Markovian dynamics is much more complicated, and many works have been devoted to this topic. Analogously to the generalized Langevin equations, generalized quantum master equation with memory kernel has been used to model non-Markovian open quantum dynamics [29,42,43,52], though it is often complicated to come up with a good estimate of the memory kernel, especially when the memory is long range.Besides the quantum master equation framework, the non-Markovian evolution of the system can be also directly modeled and simulated using the path-integral approaches such as the QuAPI (quasi-adiabatic propagator path integral) methods [36,41] and the HEOM (hierarchical equations of motion) technique [56]. These methods yield accurate numerical results (often referred as "numerically exact" in the literature), while the computational cost is extremely huge, often unaffordable. To reduce the computational cost, one common strategy is to replace the exact summation or numerical integration in these methods by Monte Carlo methods. In this paper, we are going to study a specific type of path-integral methods called the diagrammatic quantum Monte Carlo method [60] to solve the time-dependent open quantum systems. In particular, our study is largely motivated by the inchworm Monte Carlo method recently proposed in [4,10] to reduce the variance in quantum Monte Carlo by diagrammatic resummation.The basis of the diagrammatic quantum Monte Carlo method has been established as early as 1960s [28]. However, as other quantum Monte Carlo methods, this type of methods also suffer from the notorious dynamical sign problem, meaning that the number of Monte Carlo samples is required to grow at least exponentially in time in order to keep the accuracy of the simulation. To relieve the dynamical sign problem, Stockburger and Grabert introduced stochastic unraveling of influence...

Inclusion-Exclusion Principle for Open Quantum Systems with Bosonic Bath

“…The bare dQMC method can be only applied to short-time simulation, since the variance of the integrand in the Monte Carlo method grows exponentially with simulation time, which is known as the dynamical sign problem [21,22,28]. One approach to alleviate the sign problem is the inchworm Monte Carlo method proposed in [9], which introduces the full propagator G(s i , s f ) defined by (see [8] for a derivation)…”

“…While these deterministic methods require some additional modeling of the open quantum system, the bare diagrammatic quantum Monte Carlo (dQMC) method [17] applies Monte Carlo sampling to directly compute the summations and high-dimensional integrals in the Dyson series expansion of the quantum observable [30], and after applying Wick's theorem [23], this approach can be represented as the summation of all possible diagrams, each of which is determined by a finite time sequences and a partition of them into pairs. However, such technique may encounter the notorious numerical sign problem [7][8][9], meaning that the number of Monte Carlo samples is required to grow at least exponentially (with respect to physical time) in order to keep the accuracy of the simulation.…”

“…It introduces bold lines as partial resummations of bare dQMC, so that the total number of diagrams can be reduced, and the sign problem is hence suppressed. This approach is further improved in [8] by writing the evolution of the bold lines as an integrodifferential equation, which only requires to sum over "linked" diagrams, so that the computational cost can be further reduced. Even after such reductions, however, as the number of points in the time sequence m increases, the total number of diagrams still grows as a double factorial O((m − 1)!!).…”

Differential Equation Based Path Integral for System-Bath Dynamics

“…In order to check the validity of our method, we first study the following parameters, which have been considered in [7,1]:…”

“…Other approaches based on this idea include the blip-summed method [16,17], small matrix decomposition of the path integral (SMatPI) method [18], and important path sampling [23] design algorithms, which explore extra the numerical sparsity of the problem to further reduce the memory cost. Some stochastic methods, such as diagrammatic quantum Monte Carlo (dQMC) method [28] and inchworm Monte Carlo method [4,1,29,2], apply stochastic approaches to estimate path integrals. These methods no longer suffer from the curse of memory cost.…”