Quantum simulations with graphics processing units

Abstract

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.