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arXiv:1810.03047 · PLoS ONE

QTM: Computational package using MPI protocol for Quantum Trajectories Method

Marek Sawerwain, Joanna Wiśniewska

Published Dec 10, 2018• 1 citation
arXivHTMLPDF

Abstract

The Quantum Trajectories Method (QTM) is one of the frequently used methods for studying open quantum systems. The main idea of this method is the evolution of wave functions which describe the system (as functions of time). Then, so-called quantum jumps are applied at a randomly selected point in time. The obtained system state is called as a trajectory. After averaging many single trajectories, we obtain the approximation of the behavior of a quantum system. This fact also allows us to use parallel computation methods. In the article, we discuss the QTM package which is supported by the MPI technology. Using MPI allowed utilizing the parallel computing for calculating the trajectories and averaging them-as the effect of these actions, the time taken by calculations is shorter. In spite of using the C++ programming language, the presented solution is easy to utilize and does not need any advanced programming techniques. At the same time, it offers a higher performance than other packages realizing the QTM. It is especially important in the case of harder computational tasks, and the use of MPI allows improving the performance of particular problems which can be solved in the field of open quantum systems.

DOI
10.1371/journal.pone.0208263

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The Quantum Trajectories Method (QTM) is one of the frequently used methods for studying open quantum systems. The main idea of this method is the evolution of wave functions which describe the system (as functions of time). Then, so-called quantum jumps are applied at a randomly selected point in time. The obtained system state is called as a trajectory. After averaging many single trajectories, we obtain the approximation of the behavior of a quantum system. This fact also allows us to use parallel computation methods. In the article, we discuss the QTM package which is supported by the MPI technology. Using MPI allowed utilizing the parallel computing for calculating the trajectories and averaging them-as the effect of these actions, the time taken by calculations is shorter. In spite of using the C++ programming language, the presented solution is easy to utilize and does not need any advanced programming techniques. At the same time, it offers a higher performance than other packages realizing the QTM. It is especially important in the case of harder computational tasks, and the use of MPI allows improving the performance of particular problems which can be solved in the field of open quantum systems.
Nowadays, a dynamic progress in computational techniques allows for development of various methods, which offer significant speed-up of computations, especially those related to the problems of quantum optics and quantum computing. In this work, we propose computational solutions which re-implement the quantum trajectory method (QTM) algorithm in modern parallel computation environments in which multi-core CPUs and modern many-core GPUs can be used. In consequence, new computational routines are developed in more effective way than those applied in other commonly used packages, such as Quantum Optics Toolbox (QOT) for Matlab or QuTIP for Python.
Time-dependent wavepackets are widely used to model various phenomena in physics. One approach in simulating the wavepacket dynamics is the quantum trajectory method (QTM). Based on the hydrodynamic formulation of quantum mechanics, the QTM represents the wavepacket by an unstructured set of pseudoparticles whose trajectories are coupled by the quantum potential. The governing equations for the pseudoparticle trajectories are solved using a computationally-intensive moving weighted least squares (MWLS) algorithm, and the trajectories can be computed in parallel. This paper contributes a strategy for improving the performance of wavepacket simulations using the QTM. Specifically, adaptivity is incorporated into the MWLS algorithm, and loop scheduling techniques are employed to dynamically load balance the parallel computation of the trajectories. The adaptive MWLS algorithm reduces the amount of computations without sacrificing accuracy, while adaptive loop scheduling addresses the load imbalance introduced by the algorithm and the runtime system. Results of experiments on a Linux cluster are presented to confirm that the adaptive MWLS reduces the trajectory computation time by up to 24%, and adaptive loop scheduling achieves parallel efficiencies of up to 85% when simulating a free particle.
Time-dependent wavepackets are widely used to model various phenomena in physics. One approach in simulating the wavepacket dynamics is the quantum trajectory method (QTM). Based on the hydrodynamic formulation of quantum mechanics, the QTM represents the wavepacket by an unstructured set of pseudoparticles whose trajectories are coupled by the quantum potential. The governing equations for the pseudoparticle trajectories are solved using a computationally intensive moving weighted least squares (MWLS) algorithm, and the trajectories can be computed in parallel. This paper contributes a strategy for improving the performance of wavepacket simulations using the QTM. Specifically, adaptivity is incorporated into the MWLS algorithm, and loop scheduling techniques are employed to dynamically load balance the parallel computation of the trajectories. The adaptive MWLS algorithm reduces the amount of computations without sacrificing accuracy, while adaptive loop scheduling addresses the load imbalance introduced by the algorithm and the runtime system. Results of experiments on a Linux cluster are presented to confirm that the adaptive MWLS reduces the trajectory computation time by up to 24%, and adaptive loop scheduling achieves parallel efficiencies of up to 85% when simulating a free particle.
Recently, in an effort to solve more realistic problems in quantum dynamics, much attention has been directed into numerically integrating the quantum hydrodynamic equations of motions (QHEM), as opposed to directly solving the time-dependent Schrödinger equation (TDSE). Such efforts have been provoked by the many numerical drawbacks encountered when solving the TDSE on a fixed-grid. In this dissertation, one trajectory method for integrating the QHEM is reviewed, and two novel trajectories methods are described. The first of these, the quantum trajectory method (QTM), was introduced in 1999 and has been used to solve many problems in quantum dynamics since then. However, severe numerical problems are encountered when this method is applied to problems that form wave function nodes. To get around this problem, new methods for numerically integrating the QHEM are needed. In the first novel method described, the arbitrary Lagrangian-Eulerian (ALE) method, particle trajectories are governed by a predetermined equation of motion that is user-supplied. The ALE method remedies inflation and compression problems encountered in the pure Lagrangian QTM. In the second new method discussed, the derivative propagating method (DPM), single quantum trajectories can be calculated one at a time, as opposed to the ensemble propagation of the QTM and ALE method. Using these two methods, new solutions to the QHEM are obtained where the QTM fails. In addition to solving the QHEM, the DPM is also used to solve the classical Klein-Kramers equation in this dissertation. This equation governs the Markovian phase space evolution of a system coupled to an environment such as a heat bath. This marks the first time single trajectories have been used to solve both the QHEM and the Klein-Kramers equations.
We provide a software package for numerical simulations and modeling of complex quantum systems in the presence of dissipation and decoherence for a wider class of problems in the field of quantum technologies. This software is based on the method of quantum trajectories usually used for calculations of the density matrix. An important part of this Toolkit is the universal user interface, which is based on Tool Command Language (TCL) scripting language. It is elaborated in such a manner that the system description and system parameters should not be included in the source code. The core is implemented as a generic set of C++ classes, which can be efficiently reused for modeling of a wide range of photonic systems. The code has been written so as to facilitate optimization of the performance without breaking the object-orientedness of the design. We demonstrate that this software package is very useful for high performance parallel calculations on the Cluster.
The quantum trajectory method (QTM) was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for N fluid elements (particles) moving under the influence of both the force from the potential surface and from the quantum potential. In this study, distributed approximating functionals (DAFs) are used on a uniform grid to compute the necessary derivatives in the equations of motion. Transformations between the physical grid where the particle coordinates are defined and the uniform grid are handled through a Jacobian, which is also computed using DAFs. A difficult problem associated with computing derivatives on finite grids is the edge problem. This is handled effectively by using DAFs within a least squares approach to extrapolate from the known function region into the neighboring regions. The QTM–DAF is then applied to wave packet transmission through a one-dimensional Eckart potential. Emphasis is placed upon computation of the transmitted density and wave function. A problem that develops when part of the wave packet reflects back into the reactant region is avoided in this study by introducing a potential ramp to sweep the reflected particles away from the barrier region.