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arXiv:cond-mat/0611412

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

Michael P. Zwolak

Published Nov 15, 2006• 0 citations
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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 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.

DOI
10.48550/arxiv.cond-mat/0611412

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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 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.
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.
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.
Memory effects in a quantum system coupled to an environment are one of the central features in the theory of open quantum systems. The dynamics of such quantum systems are typically governed by an equation of motion with a time-convolution integral of the memory kernel. However, solving such integro-differential equations is challenging, especially when the memory kernel is nonstationary (not time-translation invariant). In this paper, we analytically and nonperturbatively solve such integro-differential equations with a nonstationary memory kernel by employing a train of Dirac-delta switchings. We then apply this method to the damped Jaynes-Cummings model and the damped harmonic oscillator model to demonstrate that (i) our solution asymptotes to the well-known exact solution in the continuum limit, and that (ii) our method also enables us to visualize the memory effect in the environment.
We use the assumption that the potential for the AAA-boson system can be written as a sum of pairwise acting forces to decompose the wave function into Faddeev components that fulfill a Faddeev type equation. Expanding these components in terms of potential harmonic (PH) polynomials and projecting on the potential basis for a specific pair of particles results in a two-variable integro-differential equations suitable for AAA-boson bound-state studies. The solution of the equation requires the evaluation of Jacobi polynomials ${P}_{K}^{\ensuremath{\alpha},\ensuremath{\beta}}(x)$ and of the weight function W(z)W(z)W(z) which give severe numerical problems for very large AAA. However, using appropriate limits for $A\ensuremath{\rightarrow}\ensuremath{\infty}$ we obtain a variant equation which depends only on the input two-body interaction, and the kernel in the integral part has a simple analytic form. This equation can be readily applied to a variety of bosonic systems such as microclusters of noble gasses. We employ it to obtain results for $A\ensuremath{\in}(10\ensuremath{-}100)$ 87Rb^{87}\mathrm{Rb}87Rb atoms interacting via interatomic interactions and confined by an externally applied trapping potential Vtrap(r){V}_{\mathrm{trap}}(r)Vtrap​(r). Our results are in excellent agreement with those previously obtained using the potential harmonic expansion method (PHEM) and the diffusion Monte Carlo (DMC) method.
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.