A Heuristic for Matrix Product State Simulation of Out-of-Equilibrium Dynamics of Two-Dimensional Transverse-Field Ising Models

TL;DR

This paper introduces a heuristic method to improve Matrix Product State simulations for out-of-equilibrium dynamics in 2D quantum systems, enabling more efficient classical computations of expectation values at various energy densities.

Contribution

The paper presents a novel heuristic that accelerates MPS convergence for simulating 2D transverse-field Ising models, bridging a gap in classical simulation capabilities for intermediate energy densities.

Findings

Successfully simulated a 7x8 2D TFIM with high bond dimension on a GPU.

Achieved results comparable to digital quantum processor simulations.

Provided a scalable approach for classical out-of-equilibrium quantum dynamics simulations.

Abstract

Out-of-equilibrium dynamics of non-integrable Hamiltonian many-body quantum systems are characterized by highly entangled wave functions. Near-maximal entanglement arises in systems exhibiting thermalization or pre-thermalization, where the system converges to a steady state with a fixed energy density. Classical simulation of the time dependence of such wave functions requires exponential resources. However, typical computations aim to estimate expectation values of local operators and correlation functions to some expected precision. For thermalizing systems at sufficiently high energy densities, such computations do not require storing the full wave function. Nonetheless, constructing classical algorithms for intermediate energy densities has remained a challenge. In this paper, we propose a heuristic approach to accelerate the convergence of Matrix Product State (MPS) simulations of expectation values applicable in a broad range of energy densities. We estimate the desired observables by rescaling the MPS results at low bond dimensions with a factor that depends only on the fidelity of the MPS wave function. Using this technique, we simulated the dynamics of the two-dimensional Transverse-Field Ising Model (TFIM) on a $7\times8$ grid with periodic boundary conditions, using a maximum bond dimension of $\chi = 4096$ on a single A100 GPU. We compared our results to similar TFIM simulations on a digital quantum processor.