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GPU-accelerated solutions of the nonlinear Schrödinger equation

Benjamin D. Smith, Logan Cooke, Lindsay J. LeBlanc

Published Oct 28, 2020• 0 citations
OpenAlex

Abstract

As a first approximation beyond linearity, the nonlinear Schrodinger equation reliably describes a broad class of physical systems. Though numerical solutions of this model are well-established, these methods can be computationally complex, especially when system-specific details are incorporated. In this paper, we demonstrate how numerical computations that exploit the features of a graphics processing unit (GPU) result in 40-70$\times$ reduction in the time for solutions (depending on hardware details). As a specific case study, we investigate the Gross-Pitaevskii equation, a specific version of the nonlinear Schrodinger model, as it describes a trapped, interacting two-component Bose-Einstein condensate subject to spatially dependent interspin coupling, resulting in an analog to a spin-Hall system. This computational method allows us to probe high-resolution spatial features -- revealing an interaction dependent phase transition -- all in a reasonable amount of time with readily available hardware.

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As a first approximation beyond linearity, the nonlinear Schrodinger equation reliably describes a broad class of physical systems. Though numerical solutions of this model are well-established, these methods can be computationally complex, especially when system-specific details are incorporated. In this paper, we demonstrate how numerical computations that exploit the features of a graphics processing unit (GPU) result in 40-70×\times× reduction in the time for solutions (depending on hardware details). As a specific case study, we investigate the Gross-Pitaevskii equation, a specific version of the nonlinear Schrodinger model, as it describes a trapped, interacting two-component Bose-Einstein condensate subject to spatially dependent interspin coupling, resulting in an analog to a spin-Hall system. This computational method allows us to probe high-resolution spatial features -- revealing an interaction dependent phase transition -- all in a reasonable amount of time with readily available hardware.
This document aims to describe the methodology of quantum lattice gas algorithm based quantum physics simulations with details on implementation using graphics processing units. A variety of examples illustrating the general application of these methods are included. These examples are used to highlight topics such as scaling, potentials, momentums, multiple particle and dimensions, and various Hamiltonians in quantum simulations. Although intended to fullfill the doctor of philosophy degree requirement, this document is well suited as an introduction to this approach to modeling physics. In recent times parallelization has significantly improved the available computational power. Graphics processing units (GPUs) are a shining example of this, with thousands of compute cores and continually improved general computational capabilities, these processors can outshine their central processing unit (CPU) counterparts by orders of magnitude. Presented here are a number of examples of efficient computational methods for simulating physics using GPUs. Our quantum simulation method involves the quantum lattice gas model of quantum computation [59, 68, 64], however much of the parallel technology implementation is applicable to other numerical approaches. The quantum lattice gas model of computation is based on a unitary stream-collide quantum algorithm applied repeatedly over time upon an ordered array of qubits [53, 56, 68, 61, 63, 67]. The stream-collide quantum algorithm represents local quantum evolution of the system of quantum particles whose motion is constrained to reside on a spacetime lattice. Each particle in the system undergoes a quantum walk, interacting locally with other particles in the system that may reside on the same point. We will first mathematically describe the basic idea of quantum lattice gas algorithms. Next we will discuss GPUs and the basic features needed for implementation. We will then describe several examples of quantum lattice gas algorithm approach to illustrate the various applications and implementation challenges. Of particular interest is a look at modeling time-dependent many-body quantum mechanics using quantum lattice gas methods. For practical purposes we focus on the case of two fermions governed by the nonrelativistic Schrödinger equation. Numerical comparisons to analytic prediction show the high accuracy of these methods. Quantum lattice gas methods are chosen as a unitary algorithm for time dependent quantum simulation as they are efficient, accurate, and avoid the Fermi sign problem. Our findings from two-particle simulations under various conditions include the physics of the one and two particle systems diverging in their respective time-dependent behavior with increasing entanglement. We also confirm the multiple particle extension of the Ehrenfest theorem. Using the examples discussed throughout this document, we will then describe our specific computational implementation utilizing the computational power of GPUs through Nvidia’s compute unified device architecture (CUDA) language. Here we will discuss the open source software our group is developing based on these methods, along with some implementation verification and debugging concepts. Education is another application peripherally connected to this work and our software. The applications of this type of modeling as a learning tool is not fully explored yet, but the opportunities are promising.
Although quantum mechanics is linear, there are nevertheless quantum systems with multiple interacting particles in which the effective evolution of a single particle is governed by a nonlinear equation. This includes Bose-Einstein condensates, which are governed by the Gross-Pitaevskii equation, which is a cubic nonlinear Schrodinger equation with a term proportional to /[psi]/ ²[psi] scaling of Grover's algorithm. Since the Gross- Pitaevskii equation effectively approximates the multi- particle Schrodinger equation, for which Grover's algorithm is optimal, our result leads to a quantum information-theoretic bound on the number of particles needed for this approximation to hold, asymptotically. The Gross-Pitaevskii equation is not the only nonlinearity of the form f(/[psi]/²)[psi] that arises in effective equations for the evolution of real quantum physical systems, however : The cubic-quintic nonlinear Schrodinger equation describes light propagation in nonlinear Kerr media with defocusing corrections, and the logarithmic nonlinear Schrodinger equation describes Bose liquids under certain conditions. Analysis of computation with such systems yields some surprising results; for example, when time-measurement precision is included in the resource accounting, searching a database'' when there is a single correct answer may be easier than searching when there are multiple correct answers. These results lead to quantum information-theoretic bounds on the physical resources required for these effective nonlinear theories to hold, asymptotically. Furthermore, strongly regular graphs, which have no global symmetry, are sufficiently complete for quantum search on them to asymptotically behave like unstructured search. Certain sufficiently complete graphs retain the improved runtime and resource scalings for some nonlinearities, so our scheme for nonlinear, analog quantum computation retains its benefits even when some structure is introduced
We present an approach for integrating the time evolution of quantum systems. We leverage the computation power of graphics processing units (GPUs) to perform the integration of all time steps in parallel. The performance boost is especially prominent for small to medium-sized quantum systems. The devised algorithm can largely be implemented using the recently-specified batched versions of the BLAS routines, and can therefore be easily ported to a variety of platforms. Our PARAllelized Matrix Exponentiation for Numerical Time evolution (PARAMENT) implementation runs on CUDA-enabled graphics processing units. Program Title: PARAMENT CPC Library link to program files: https://doi.org/10.17632/zy5v4xs89d.1 Developer's repository link: https://github.com/parament-integrator/parament Licensing provisions: Apache 2.0 Programming language: C / CUDA / Python Nature of problem: Time-integration of the Schrödinger equation with a time-dependent Hamiltonian for quantum systems with a small Hilbert space but many time-steps. Solution method: A 4th order Magnus integrator, highly parallelized on a GPU, implemented using a small subset of BLAS functions for improved portability.
The homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions for the Gross-Pitaevskii equations, a set of nonlinear Schrödinger equation used in simulation of spin-1 Bose-Einstein condensates trapped in a harmonic potential. We investigate the one-dimensional case and get the approximate analytical solutions successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are in agreement well with each other when the atomic interaction is weakly. We also find a class of exact solutions for the stationary states of the spin-1 system with harmonic potential for a special case.