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

TL;DR

This work tackles the challenge of classically simulating out-of-equilibrium dynamics in highly entangled 2D quantum systems by introducing a fidelity-based rescaling heuristic for Matrix Product State (MPS) simulations. The method rescales MPS-derived observables using the MPS fidelity $F_{MPS}$, enabling accurate estimates at bond dimensions much smaller than those required for full state fidelity, with a specialized exponent $\gamma(n,χ)$ for two-body observables. Calibrated on grids up to $6×8$, the approach is demonstrated on the 2D Transverse-Field Ising Model (TFIM) up to $11×11$, achieving rapid convergence for $\langle Z_{tot}^2 \rangle$ through a scaling law $\gamma_{fit} = a \log_2(χ)/n + b$, and validated against a 7×8 quantum-device experiment, albeit with some discrepancies. The findings offer a practical path to scalable classical simulations of non-equilibrium quantum dynamics and indicate directions for refining error bounds and applicability to other 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.

Figures (10)

• Figure 1: (Left) Example of qubit grouping for a $\boldsymbol{7\times8}$ grid. (Right) Final structure of the MPS. Qubits in each row are split into two groups, and qubits in the same group are fused together. The dashed lines correspond to the boundary conditions, while the arrows represent the order in which blocks in the MPS are ordered.
• Figure 2: MPS computation of magnetization The figures show the convergence of $\langle Z^{}_{\rm tot}\rangle_{\rm MPS}$ without any rescaling factor (left) and by rescaling it using the fidelity $F_{\rm MPS}$, see Eq. (\ref{['eq:Z_tot_via_F']}).
• Figure 3: Comparison of the single-body observable $\boldsymbol{\langle Z^{}_{\rm tot}\rangle}$ with $\boldsymbol{\langle Z^{2}_{\rm tot}\rangle}$, at different temperatures. Here, the $\gamma_{\rm fit}$ values are $0.723$, $0.477$, and $0.279$ for low-, mid-, and high-temperature respectively.
• Figure 4: Fit for the $\boldsymbol{\gamma^*}$ exponent, for three different initial states. Correlation between the optimal exponent $\gamma^*$ and the rescaled bond dimension $\log_2\chi / n$, varying the system size and shape. The fit is obtained by using systems with at least $30$ qubits. The fitting parameters are reported in \ref{['tab:gamma_scaling']}.
• Figure 5: Comparison of the Ising order parameter with and without rescaling. The left and right figures show the convergence of $\langle Z^{2}_{\rm tot}\rangle_{\rm MPS}$ and $\langle Z^{2}_{\rm tot}\rangle_{\gamma_{\rm fit}}$, respectively, with increasing bond dimension.