Numerical Feynman integrals for density operator dynamics using master equation interpolants: faster convergence and significant reduction of computational cost
The Feynman integral is one of the most accurate methods for calculating density operator dynamics in open quantum systems. However, the number of time steps that can realistically be used is always limited, therefore one often obtains an approximation of the density operator at a sparse grid of points in time. Instead of relying only on \textit{ad hoc} interpolation methods such as splines to estimate the system density operator in between these points, I propose a method that uses physical information to assist with this interpolation. This method is tested on a physically significant system, on which its use allows important qualitative features of the density operator dynamics to be captured with as little as 2 time steps in the Feynman integral. This method allows for an enormous reduction in the amount of memory and CPU time required for approximating density operator dynamics within a desired accuracy. Since this method does not change the way the Feynman integral itself is calculated, the value of the density operator approximation at the points in time used to discretize the Feynamn integral will be the same whether or not this method is used, but its approximation in between these points in time is considerably improved by this method.
The Feynman integral is one of the most accurate methods for calculating density operator dynamics in open quantum systems. However, the number of time steps that can realistically be used is always limited, therefore one often obtains an approximation of the density operator at a sparse grid of points in time. Instead of relying only on \textit{ad hoc} interpolation methods such as splines to estimate the system density operator in between these points, I propose a method that uses physical information to assist with this interpolation. This method is tested on a physically significant system, on which its use allows important qualitative features of the density operator dynamics to be captured with as little as 2 time steps in the Feynman integral. This method allows for an enormous reduction in the amount of memory and CPU time required for approximating density operator dynamics within a desired accuracy. Since this method does not change the way the Feynman integral itself is calculated, the value of the density operator approximation at the points in time used to discretize the Feynamn integral will be the same whether or not this method is used, but its approximation in between these points in time is considerably improved by this method.
One of the most successful methods for calculating reduced density operator dynamics in open quantum systems, that can give numerically exact results, uses Feynman integrals. However, when simulating the dynamics for a given amount of time, the number of time steps that can realistically be used with this method is always limited, therefore one often obtains an approximation of the reduced density operator at a sparse grid of points in time. Instead of relying only on ad hoc interpolation methods (such as splines) to estimate the system density operator in between these points, I propose a method that uses physical information to assist with this interpolation. This method is tested on a physically significant system, on which its use allows important qualitative features of the density operator dynamics to be captured with as little as two time steps in the Feynman integral. This method allows for an enormous reduction in the amount of memory and CPU time required for approximating density operator dynamics within a desired accuracy. Since this method does not change the way the Feynman integral itself is calculated, the value of the density operator approximation at the points in time used to discretize the Feynamn integral will be the same whether or not this method is used, but its approximation in between these points in time is considerably improved by this method. A list of ways in which this proposed method can be further improved is presented in the last section of the article.
The time evolution of Markovian open quantum systems is governed by Lindblad master equations, whose solution can be formally written as the Lindbladian exponential acting on the initial density matrix. By expanding this Lindbladian exponential into the Taylor series, we propose a generic method for integrating Lindblad master equations. In this method, the series is truncated, retaining a finite number of terms, and the iterative actions of Lindbladian on the density matrix follow the corresponding master equation. Our method offers significant improvements in numerical efficiencies both in memory cost and computation time, especially for systems with many degrees of freedom. Moreover, our proposed method can be integrated seamlessly with tensor networks. Two illustrative examples, a two-level system exhibiting damped Rabi oscillations and a driven dissipative Heisenberg chain, are used to demonstrate the validity of our method. The superiority of our method is benchmarked with detailed performance tests.
We describe an efficient path integral scheme for calculating the propagator of an arbitrary quantum system, as well as that of a stochastic system in special cases where the Fokker–Planck equation obeys strict detailed balance. The basic idea is to split the respective Hamiltonian into two exactly solvable parts and then to employ a symmetric decomposition of the time evolution operator, which is exact up to a high order in the time step. The resulting single step propagator allows rather large time steps in a path integral and leads to convergence with fewer time slices. Because it involves no system-specific reference system, the algorithm is amenable to all known numerical schemes available for evaluating quantum path integrals. In this way one obtains a highly accurate method, which is simultaneously fast, stable, and computationally simple. Numerical applications to the real time quantum dynamics in a double well and to the stochastic dynamics of a bistable system coupled to a harmonic mode show our method to be superior over the approach developed by the Makri group in their quasiadiabatic propagator representation, to say nothing about the propagation scheme based on the standard Trotter splitting.
A technique for simulating quantum dynamics in phase space i discussed. It makes use of ensembles of classical trajectories to approximate the distribution functions and their derivatives by implementing Adaptive Kernel Density Estimation. It is found to improve the accuracy and stability of the simulations compared to more conventional particle methods. Formulation of the method in higher dimensions is straightforward. Complete analytical description of physical systems exist only for the most simple ideal cases. The vast majority of physical problems require the solution of approximate equations most fre- quently by using numerical calculations. To this end, much effort has been devoted to the devel- opment of suitable techniques to be used in computer simulations. The increasing computational power makes possible the description of more complicated, that is, less ideal systems allowing for more precise predictions of their properties as well as for a better understanding of the underlying phenomena. Among the wealth of numerical approaches to solve dynamical equations by means of discretization, particle methods have gained attention in the last few decades as a conceptu- ally appealing alternative. Conventional numerical solvers rely on a regular discretization of the independent variables to facilitate the evaluation of derivatives to a given order of accuracy. As a consequence, a mesh needs to be defined as the base of the simulations, parts of which remain empty at any given simulation time even with the most sophisticated mesh schemes. The cen- tral idea of particle methods consists of representing functions by sets of particles conveniently located and weighted such that the function is appropriately sampled by the particles. Evolution of the functions is then accomplished by an equivalent evolution of the particles. This meshless or Lagrangian type of discretization has some intrinsic computational efficiency advantages because there is no wasted space. The advantage, however, comes at the expense of having to evaluate derivatives of functions on unstructured arrays of sampling points. The issues about convergence and stability of the resulting algorithms have been the subject of active and fruitfull research. Currently there are some well established techniques to solve dynamical equations with particle methods. Our interest is to provide an alternative ingredient to further improve the performance of numerical solutions of dynamical problems of physical relevance. In the present work we discuss, in the context of particle methods, the implementation of a technique for representing functions and derivatives with adaptive parameters. Adaptiveness require the parameters to be automatically ad- justed to optimal values according to the location of the particles. As shown below, the proposed scheme is conceptually simple and has been tested in a number of different situations. In some simple test models the present work focuses on quantum dynamics, previously treated with more
We are interested in the simulation of open quantum systems governed by the Lindblad master equation in an infinite-dimensional Hilbert space. To simulate the solution of this equation, the standard approach involves two sequential approximations: first, we truncate the Hilbert space to derive a differential equation in a finite-dimensional subspace. Then, we use discrete time-step to obtain a numerical solution to the finite-dimensional evolution. In this paper, we establish bounds for these two approximations that can be explicitly computed to guarantee the accuracy of the numerical results. Through numerical examples, we demonstrate the efficiency of our method, empirically highlighting the tightness of the upper bound. While adaptive time-stepping is already a common practice in the time discretization of the Lindblad equation, we extend this approach by showing how to dynamically adjust the truncation of the Hilbert space. This enables fully adaptive simulations of the density matrix. For large-scale simulations, this approach can significantly reduce computational time and relieves users of the challenge of selecting an appropriate truncation. Furthermore, as a special case, our method naturally applies to Hamiltonian (unitary) dynamics.
We propose a computationally efficient method to solve the dynamics of operators of bosonic quantum systems coupled to their environments. The method maps the operator under interest to a set of complex-valued functions, and its adjoint master equation to a set of partial differential equations for these functions, which are subsequently solved numerically. In the limit of weak coupling to the environment, the mapping of the operator enables storing the operator efficiently during the simulation, leading to approximately quadratic improvement in the memory consumption compared with the direct approach of solving the adjoint master equation in the number basis, while retaining the computation time comparable. Moreover, the method enables efficient parallelization which allows to optimize for the actual computational time to reach an approximately quadratic speed up, while retaining the memory consumption comparable to the direct approach. We foresee the method to prove useful, e.g., for the verification of the operation of superconducting quantum processors.