[Paper Review] GPU accelerated Trotter-Suzuki solver for quantum spin dynamics

Axel D. Dente, Carlos Bederián+2 more|arXiv (Cornell University)|Apr 30, 2013

Quantum many-body systems16 references6 citations

TL;DR

This paper presents a GPU-accelerated 4th-order Trotter-Suzuki algorithm for simulating out-of-equilibrium quantum spin dynamics in many-body systems. By leveraging massive parallelization of spin interactions and local rotations, the implementation achieves up to 30× speedup over CPU-based OpenMP code, enabling simulations of up to 27 spins within practical memory limits.

ABSTRACT

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).

Motivation & Objective

• Address the computational challenge of simulating out-of-equilibrium dynamics in interacting many-spin quantum systems.
• Overcome the exponential Hilbert space scaling that limits traditional CPU-based simulations.
• Develop a unitary, non-truncating numerical method to preserve quantum evolution fidelity.
• Exploit GPU parallelism to accelerate the Trotter-Suzuki decomposition for spin Hamiltonians without symmetry assumptions.
• Enable large-scale simulations of spin systems up to 27 spins, limited only by GPU memory.

Proposed method

• Implement a 4th-order Trotter-Suzuki decomposition of the time-evolution operator for spin-1/2 Hamiltonians.
• Partition the Hamiltonian into X, Y, and Z components using the XYZ decomposition, where Z terms are diagonal and X/Y terms are non-diagonal.
• Apply local rotations to transform non-diagonal X and Y terms into effective Z-like terms for efficient phase manipulation.
• Implement phase updates and local rotations via massive parallelization on GPU kernels, exploiting thread-level independence.
• Use a computational basis of $2^N$ Ising states to represent the many-body wavefunction without truncation.
• Ensure unitarity by preserving the norm of the state vector throughout time evolution, avoiding non-unitary integrators like Runge-Kutta.

Experimental results

Research questions

• RQ1 Can GPU acceleration significantly reduce execution time for time-evolving many-body quantum spin systems using the Trotter-Suzuki method?
• RQ2 To what extent can the Trotter-Suzuki algorithm be parallelized on GPU architectures for non-trivial spin Hamiltonians?
• RQ3 What is the maximum system size (in number of spins) that can be simulated efficiently using GPU-accelerated TS with full Hilbert space representation?
• RQ4 How accurate is the GPU-optimized TS implementation compared to exact or reference CPU-based simulations?
• RQ5 Can this GPU-accelerated TS solver be effectively applied to physically relevant scenarios such as thermalization in closed quantum systems or matrix product state evolution?

Key findings

• The GPU implementation achieves a maximum speedup of approximately 30 times compared to an optimized OpenMP CPU implementation on a 6-core processor.
• The method enables the simulation of up to 27 spins, constrained only by GPU memory capacity (approximately 6 GB in the tested setup).
• The relative error in time evolution, measured via the Loschmidt echo, remains within acceptable bounds for the Trotter-Suzuki approximation, validating the method's accuracy.
• The Loschmidt echo serves as a reliable witness for detecting computational errors in both GPU and CPU implementations.
• The implementation is compatible with advanced quantum state representations such as random superposition states and Matrix Product States, enabling broader applications in quantum many-body physics.
• The algorithm maintains unitarity throughout evolution, ensuring conservation of probability and physical consistency.