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Computational Science & Discovery

Numerical ansatz for solving integro-differential equations with increasingly smooth memory kernels: spin-boson model and beyond

Michael P. Zwolak

Published Nov 20, 2008• 9 citations
Publisher

Abstract

We present an efficient and stable numerical ansatz for solving a class of integro-differential equations. We define the class as integro-differential equations with increasingly smooth memory kernels. The resulting algorithm reduces the computational cost from the usual T 2 to T C ( T ), where T is the total simulation time and C ( T ) is some function. For instance, C ( T ) is equal to ln T for polynomially decaying memory kernels. Due to the common occurrence of increasingly smooth memory kernels in physical, chemical and biological systems, the algorithm can be applied in quite a wide variety of situations. We demonstrate the performance of the algorithm by examining two cases. Firstly, we compare the algorithm to a typical numerical procedure for a simple integro-differential equation. Secondly, we solve the non-interacting blip approximation equations for the spin-boson model in real time.

DOI
10.1088/1749-4699/1/1/015002

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Numerical ansatz for solving integro-differential equations with increasingly smooth memory kernels: spin-boson model and beyond

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We present an efficient and stable numerical ansatz for solving a class of integro-differential equations. We define the class as integro-differential equations with increasingly smooth memory kernels. The resulting algorithm reduces the computational cost from the usual T 2 to T C ( T ), where T is the total simulation time and C ( T ) is some function. For instance, C ( T ) is equal to ln T for polynomially decaying memory kernels. Due to the common occurrence of increasingly smooth memory kernels in physical, chemical and biological systems, the algorithm can be applied in quite a wide variety of situations. We demonstrate the performance of the algorithm by examining two cases. Firstly, we compare the algorithm to a typical numerical procedure for a simple integro-differential equation. Secondly, we solve the non-interacting blip approximation equations for the spin-boson model in real time.
We present an efficient and stable numerical ansatz for solving a class of integro-differential equations. We define the class as integro-differential equations with increasingly smooth memory kernels. The resulting algorithm reduces the computational cost from the usual T^2 to T*C(T), where T is the total simulation time and C(T) is some function. For instance, C(T) is equal to lnT for polynomially decaying memory kernels. Due to the common occurrence of increasingly smooth memory kernels in physical, chemical, and biological systems, the algorithm can be applied in quite a wide variety of situations. We demonstrate the performance of the algorithm by examining two cases. First, we compare the algorithm to a typical numerical procedure for a simple integro-differential equation. Second, we solve the NIBA equations for the spin-boson model in real time.
Existing quantum algorithms for simulating dynamical systems -- from Hamiltonian simulation to linear and nonlinear differential equations solvers -- simulate Markovian dynamics, in which the system's future evolution depends solely on its current state. We turn our attention to developing quantum algorithms for non-Markovian dynamical systems where the system's future evolution depends on its past history and thus has memory. Specifically, we develop efficient algorithms for linear Volterra integro-differential equations (VIDEs) with a convolution memory kernel that output a quantum state encoding the state description over a time interval or at a particular time. Given efficient circuits for the problem inputs, our algorithms achieve an exponential speedup in system size over existing classical algorithms. We develop an algorithm for general kernels assuming that M<1\textsf{M} < 1M<1, where M\textsf{M}M characterizes the strength of the memory term relative to the dissipation of the Markovian part of the dynamics. We complement this with lower bounds for general-kernel VIDEs when M≥1\textsf{M} \geq 1M≥1, showing that the problem becomes intractable for a family of systems. However, by specializing to structured kernels which admit concise decompositions over exponentials, we develop efficient quantum algorithms even when M≥1\textsf M \geq 1M≥1 by converting the VIDE into a larger set of ODEs, a procedure which we call Markovianization. As an application of the overall framework, we discuss the Mori-Zwanzig formalism used in open quantum systems and fluid dynamics. Overall, our results expand the range of dynamical systems that quantum computers can simulate efficiently.
Abstract We present a general numerical simulation method to solve non-Markovian dynamics of an open quantum system influenced by quantum Brownian motion. Based on the determined memory kernel function, this method enables the resolution of non-Markovian dynamics for a wide range of system Hamiltonians and spectral densities. The system dynamics are described by exact integro-differential operator equations without any common approximations and they are simulated in this work by definite-number equations with stochastic initial conditions. This approach ensures the applicability of mature numerical methods and maintains computational complexity that remains largely invariant, even when dealing with more complex models. The high accuracy of our simulation is evident from a comparison with the results obtained from corresponding exact master equations, underscoring the reliability and precision of our method.
Generalized master equations provide a concise formalism for studying reduced population dynamics. Usually, these master equations require a perturbative expansion of the memory kernels governing the dynamics; in order to prevent divergences, these expansions must be resummed. Resummation techniques of perturbation series are ubiquitous in physics, but they have not been readily studied for the time-dependent memory kernels used in generalized master equations. In this paper, we present a comparison of different resummation techniques for such memory kernels up to fourth order. We study specifically the spin-boson Hamiltonian as a model system bath Hamiltonian, treating the diabatic coupling between the two states as a perturbation. A novel derivation of the fourth-order memory kernel for the spin-boson problem is presented; then, the second- and fourth-order kernels are evaluated numerically for a variety of spin-boson parameter regimes. We find that resumming the kernels through fourth order using a Padé approximant results in divergent populations in the strong electronic coupling regime due to a singularity introduced by the nature of the resummation, and thus recommend a non-divergent exponential resummation (the "Landau-Zener resummation" of previous work). The inclusion of fourth-order effects in a Landau-Zener-resummed kernel is shown to improve both the dephasing rate and the obedience of detailed balance over simpler prescriptions like the non-interacting blip approximation, showing a relatively quick convergence on the exact answer. The results suggest that including higher-order contributions to the memory kernel of a generalized master equation and performing an appropriate resummation can provide a numerically-exact solution to system-bath dynamics for a general spectral density, opening the way to a new class of methods for treating system-bath dynamics.
Strong coupling between a system and its environment leads to the emergence of non-Markovian dynamics, which cannot be described by a time-local master equation. One way to capture such dynamics is to use numerical real-time path integrals, where assuming a finite bath memory time enables manageable simulation scaling. However, by comparing to the exactly soluble independent boson model, we show that the presence of transient negative decay rates in the exact dynamics can result in simulations with unphysical exponential growth of density matrix elements when the finite memory approximation is used. We therefore reformulate this approximation in such a way that the exact dynamics are reproduced identically and then apply our new method to the spin-boson model with superohmic environmental coupling, commonly used to model phonon environments, but which cannot be solved exactly. Our new method allows us to easily access parameter regimes where we find revivals in population dynamics which are due to non-Markovian backflow of information from the bath to the system.
We present a projection-based, stability-preserving methodology for computing time correlation functions in open quantum systems governed by generalized quantum master equations with non-Markovian effects. Building upon the memory kernel coupling theory framework, our approach transforms the memory kernel hierarchy into a system of coupled linear differential equations through Mori-Zwanzig projection, followed by spectral projection onto stable eigenmodes to ensure numerical stability. By systematically eliminating unstable modes while preserving the physically relevant dynamics, our method guaranties long-time convergence without introducing artificial damping or ad hoc modifications. The theoretical framework maintains mathematical rigor through orthogonal projection operators and spectral decomposition. Benchmark calculations on the spin-boson model show excellent agreement with exact hierarchical equations of motion results while achieving significant computational efficiency. This approach provides a versatile and reliable framework for simulating non-Markovian dynamics in complex systems.