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Large-scale Classical Simulation of Quantum Systems Using the Trotter-Suzuki Decomposition

Luca Calderaro

Published Sep 1, 2015• 0 citations
OpenAlex

Abstract

Many theoretical studies and experimental results rely on the use of numerical analysis for the solution of the Schrodinger equation. Indeed, for nontrivial quantum systems, a complete solution of the dynamics is difficult to achieve analytically. We extended the implementation of a highly optimized solver to simulate the evolution of a wave function on a 2D lattice. We also implemented the imaginary time evolution to approximate the ground state. The dynamics of the system is now described by a Hamiltonian that includes an external potential and a contact interaction term. The algorithm is based on the second-order Trotter-Suzuki approximation and it is implemented on CPU and GPU kernels that run efficiently on a cluster. We proved the accuracy of the code solving the Gross-Pitaevskii equation for a Bose-Einstein condensate and reproducing the experimental results, obtained at NIST, of the soliton dynamics in a cloud of sodium atoms. The code is available under an open source license, and it is exposed as an application program interface and a command-line interface. The code is also accessible in Python and MATLAB. Future development of the code include the extension to a 3D lattice, whereas the actual implementation can already find applications in ultracold atom physics.

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Many theoretical studies and experimental results rely on the use of numerical analysis for the solution of the Schrodinger equation. Indeed, for
nontrivial quantum systems, a complete solution of the dynamics is difficult to achieve analytically.
We extended the implementation of a highly optimized solver to simulate the evolution of a wave function on a 2D lattice. We also implemented the imaginary time evolution to approximate the ground state. The dynamics of the system is now described by a Hamiltonian that includes an external potential and a contact interaction term. The algorithm is based on the second-order Trotter-Suzuki approximation and it is implemented on CPU and GPU kernels that run efficiently on a cluster.
We proved the accuracy of the code solving the Gross-Pitaevskii equation
for a Bose-Einstein condensate and reproducing the experimental results,
obtained at NIST, of the soliton dynamics in a cloud of sodium atoms. The
code is available under an open source license, and it is exposed as an application program interface and a command-line interface. The code is also accessible in Python and MATLAB.
Future development of the code include the extension to a 3D lattice, whereas the actual implementation can already find applications in ultracold atom physics.
The resolution of dynamics in out of equilibrium quantum spin systems lies at the heart of fundamental questions among Quantum Information Processing, Statistical Mechanics and Nano-Technologies. Efficient computational simulations of interacting many-spin systems are extremely valuable tools for tackling such questions. Here, we use the Trotter-Suzuki (TS) algorithm, a well-known strategy that provides the evolution of quantum systems, to address the spin dynamics. We present a GPU implementation of a particular TS version, which has been previously implemented on single cores in CPUs. We develop a massive parallel version of this algorithm and compare the efficiency between CPU and GPU implementations. This method reduces the execution time considerably and is capable of dealing with systems of up to 27 spins (only limited by GPU memory).
Hamiltonian formulations of lattice field theories provide access to real-time dynamics, but their simulation is difficult to implement efficiently. Trotter-Suzuki decompositions are at the center of time evolution computation, either on quantum hardware or classically, for instance with the use of tensor networks. While low-order Trotterizations remain the standard choice due to their simplicity, higher-order schemes offer the potential for improved efficiency. In this work we outline a short guide to Trotter-Suzuki schemes and their implementations in general. To help with this, we highlight new efficient schemes found by our optimization framework, and demonstrate their performance on the Heisenberg model.
(Suzuki-)Trotter decompositions, splitting methods, (Lie) product formulae... The most common numerical methods for the time evolution of quantum systems come with many names. And they are used practically everywhere with applications ranging from the solution of classical equations of motion and various Monte Carlo simulations to the real and imaginary time evolution on classical as well as quantum computers. Here we review the state of the art of said methods, focussing especially on the progress made over the last few years. We highlight recently discovered efficient time evolution algorithms and explain how best to use them in practice. A central part of this work is the estimation of error bounds that has improved greatly within the past decade. The relevance of time evolution methods for quantum computing is discussed with a focus on noisy hardware. Finally, a comprehensive overview of generalisations, related methods and alternatives to Trotterization is provided. This includes time-dependent Hamiltonian dynamics, processed methods, multi-product formulae, symplectic integrators, TDVP for tensor networks, quantum signal processing, Crouch-Grossman methods and more. The overall perspective in this work is that of a theoretical physicist. All mathematical proofs as well as some technical details are omitted for easier readability. Instead, this review serves as a hands-on guide and, of course, as a starting point for references that provide further details.
Hamiltonian formulations of lattice field theories provide access to real-time dynamics, but their simulation is difficult to implement efficiently. Trotter-Suzuki decompositions are at the center of time evolution computation, either on quantum hardware or classically, for instance with the use of tensor networks. While low-order Trotterizations remain the standard choice due to their simplicity, higher-order schemes offer the potential for improved efficiency. In this work we outline a short guide to Trotter-Suzuki schemes and their implementations in general. To help with this, we highlight new efficient schemes found by our optimization framework, and demonstrate their performance on the Heisenberg model.
Accurately simulating long-time dynamics of many-body systems is a challenge in both classical and quantum computing due to the accumulation of Trotter errors. While low-order Trotter-Suzuki decompositions are straightforward to implement, their rapidly growing error limits access to long-time observables. We present a framework for constructing efficient high-order Trotter-Suzuki schemes by identifying their structure and directly optimizing their parameters over a high-dimensional space. This method enables the discovery of new schemes with significantly improved efficiency compared to traditional constructions, such as those by Suzuki and Yoshida. Based on the theoretical efficiency and practical performance, we recommend two novel highly efficient schemes at 4th4^{\textrm{th}}4th and 6th6^{\textrm{th}}6th order. We also demonstrate the effectiveness of these decompositions on the Heisenberg model and the quantum harmonic oscillator, and find that for a fixed final time they perform better across the computational cost. Even when using large time steps, they surpass established low-order schemes like the Leapfrog. Finally, we investigate the in-practice performance of different Trotter schemes and find the decompositions with more uniform coefficients tend to feature improved error accumulation over long times. We have included this observation into our choice of recommended schemes.