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9th International Conference "Distributed Computing and Grid Technologies in Science and Education"

PARALLEL ALGORITHM FOR CALCULATING THE WIGNER FUNCTION FOR A QUANTUM SYSTEM WITH A POLYNOMIAL POTENTIAL

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева, Evgenii V. Burlakov, R. V. Polyakova, P. N. Sysoev, M. B. Sadovnikova

Published Dec 13, 2021• 0 citations
Publisher

Abstract

The article considers the construction of a parallel algorithm on the GPU computing architecture forfinding the Wigner function of a quantum system with a polynomial potential. A numerical-analyticalmethod for constructing the Wigner function, which is based on calculating the trace of the product ofthe density matrix and the matrix of the Weyl operator, is described. The operators are represented inthe basis of a quantum harmonic oscillator, for which the Moyal equation transforms into the Liouvilleequation. This approach enables to visually analyze the degree of anharmonicity of the system in termsof the off-diagonal elements of the density matrix. The parallel implementation on the GPU massivelyparallel architecture has reduced the calculation time by two orders of magnitude compared to thesingle-threaded version on the x86 architecture.

DOI
10.54546/mlit.2021.16.12.001

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The article considers the construction of a parallel algorithm on the GPU computing architecture forfinding the Wigner function of a quantum system with a polynomial potential. A numerical-analyticalmethod for constructing the Wigner function, which is based on calculating the trace of the product ofthe density matrix and the matrix of the Weyl operator, is described. The operators are represented inthe basis of a quantum harmonic oscillator, for which the Moyal equation transforms into the Liouvilleequation. This approach enables to visually analyze the degree of anharmonicity of the system in termsof the off-diagonal elements of the density matrix. The parallel implementation on the GPU massivelyparallel architecture has reduced the calculation time by two orders of magnitude compared to thesingle-threaded version on the x86 architecture.
Abstract The Moyal equation for the Wigner function was obtained under the assumption that the potential is an analytic function. The polynomial form of the potential is a natural approximation of the analytical potential with any necessary accuracy. The simplest quantum system with a second-order polynomial potential is a quantum harmonic oscillator. In this paper, for a quantum system with a polynomial potential of arbitrary order, explicit expressions are obtained for the matrix elements of the kernel operator in the basis of the eigenfunctions of the harmonic oscillator. Using the explicit representation for the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems with polynomial potentials. The connection of the modified Vlasov equation with the Moyal equation for the Wigner function is shown. Examples of effective numerical algorithms for finding Wigner functions with high accuracy are given.
Abstract When considering quantum systems in phase space, the Wigner function is used as a function of quasi-probability density. Finding the Wigner function is related to the calculation of a Fourier transform of a certain composition of wave functions of the corresponding quantum system. As a rule, knowledge of the Wigner function is not the final goal, and the calculation of mean values of different quantum characteristics of the system is required. An explicit solution of the Schrödinger equation can be obtained only for a narrow class of potentials, therefore, in the majority of cases, numerical methods must be used to find wave functions. As a result, finding the Wigner function is related to the numerical integration of grid wave functions. When considering a one-dimensional system, the calculation of N2 Fourier integrals of the grid wave function is needed. To provide the necessary accuracy for wave functions, corresponding to the higher states of a quantum system, a larger number of grid nodes is needed. The goal of this work was to construct the numerical-analytical method for finding the Wigner function, which allows the number of computational operations to be considerably reduced. The quantum systems with polynomial potentials were considered, for which the Wigner function is represented as a series in some known functions.
Abstract In this paper, the matrix elements explicit expressions for the kernel operator in the harmonic oscillator eigenfunctions basis are obtained. The matrix elements are expressed in terms of the two complex variables new polynomials that were constructed in this paper. In the particular case, new polynomials degenerate into Laguerre polynomials. The diagonal elements of the kernel operator matrix are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements contain frequency oscillations responsible for dissipations in the quantum systems. Using the explicit representation of the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems.
One of the most striking features of quantum mechanics is the exponential growth of resources, required to find the states of a composite system, with the size of the system. This also is the origin of the two main bottlenecks in numerical studies of complex quantum systems, that are (i) diagonalizations of big matrices and (ii) propagations of large systems of linear differential equations with global symplectic structure. Operations of the first type are purely scalable, while most of the propagation algorithms allow for the high degree of parallelism. Here we show how the workload of finding Floquet eigenstates of an ac-driven non- integrable quantum system can be shared between a general-purpose central processing unit (CPU) and a graphic processing unit (GPU), when both are working within one computing platform. Namely, diagonalization steps are delegated to the CPU, while the time propagation is performed on the GPU. This strategy led to a computational time speed-up of several order of magnitude as compared to the performance of the CPU alone.
A software package for the numerical study of quantum dissipative systems in the field of photonics and quantum optics is developed in a cluster that includes the graphical user interface. Library of C++ classes for numerical simulation of the time evolution of the density matrix, the mean values of different operators (the mean number of photons, the correlation functions of various orders, quadratic mean fluctuations, etc.), the Poincare section, and various quasi-probability distribution functions including the Wigner function in phase space is created. As an application, the results of calculations for a nonlinear oscillator in a strict quantum regime are obtained.
We present a hybrid numerical approach to simulate quantum many-body problems on two spatial dimensional quantum lattice models via the non-Abelian ab initio version of the density matrix renormalization group method on state-of-the-art high-performance computing infrastructures. We demonstrate that for the two-dimensional spinless fermion model and for the Hubbard model on torus geometry, altogether several orders of magnitude in computational time can be saved by performing calculations on an optimized basis and by utilizing hybrid CPU-multiGPU parallelization. At least an order of magnitude reduction in computational complexity results from mode optimization, while a further order of reduction in wall time is achieved by massive parallelization. Our results are measured directly in the number of floating point operations and seconds. A detailed scaling analysis of the obtained performance as a function of matrix ranks and as a function of system size up to $12\ifmmode\times\else\texttimes\fi{}12$ lattice topology is discussed. Our CPU-multiGPU model also tremendously accelerates the calculation of the one- and two-particle reduced density matrices, which can be used to construct various order parameters and trace quantum phase transitions with high fidelity.