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

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 $χ= 4096$ on a single A100 GPU. We compared our results to similar TFIM simulations on a digital quantum processor.