Simulating dynamics of the two-dimensional transverse-field Ising model: a comparative study of large-scale classical numerics

TL;DR

This study benchmarks state-of-the-art classical numerical methods for simulating the real-time dynamics of the two-dimensional transverse-field Ising model on a square lattice, focusing on quantum annealing and post-quench protocols relevant to Rydberg-atom QPUs. By employing MPS-TDVP, TTN-TDVP, 2DTN-BP, and NQS-tVMC in a cross-benchmark against equal hardware, the work reveals how these methods complement each other across quasi-adiabatic, Kibble-Zurek, and post-quench regimes. A key contribution is the introduction of symmetry-based convergence criteria to diagnose observable reliability, showing that 2DTN-BP captures short-time 2D dynamics well while tensor-network methods struggle with long-range correlations and near-critical dynamics, and that NQS performs best in quasi-adiabatic annealing but faces challenges near non-adiabatic transitions. The findings illuminate the practical boundaries between classical simulability and quantum advantage for 2D non-equilibrium quantum matter and provide actionable benchmarks for both future numerical method development and experimental validation with Rydberg arrays.

Abstract

The quantum dynamics of many-qubit systems is an outstanding problem that has recently driven significant advances in both numerical methods and programmable quantum processing units. In this work, we employ a comprehensive toolbox of state-of-the-art numerical approaches to classically simulate the dynamics of the two-dimensional transverse field Ising model. Our methods include three different tensor network techniques -- matrix product states, tree-tensor networks, and two-dimensional tensor-networks under the belief propagation approximation -- as well as time-dependent variational Monte Carlo with Neural Quantum States. We focus on two paradigmatic dynamical protocols: (i) quantum annealing through a critical point and (ii) post-quench dynamics. Our extensive results show the quantitative predictions of various state-of-the-art numerical methods providing a benchmark for future numerical investigations and experimental studies with the aim to push the limitations on classical and QPUs. In particular, our work connects classical simulability to different regimes associated with quantum dynamics in Rydberg arrays - namely, quasi-adiabatic dynamics, the Kibble-Zurek mechanism, and quantum quenches.

Figures (20)

• Figure 1: We consider a square layout of atoms and study the Transverse Field Ising model with two types of protocols: annealing and quench. We then use a toolbox of state-of-the-art classical solvers to replicate two types of observables: magnetization and two-point correlations.
• Figure 2: Panel (a) shows a schematic phase diagram for the antiferromagnetic transverse field Ising model on the square lattice, while panel (b) displays the sweep profiles characterized by three time scales: $t_{\mathrm{rise}}$, $t_{\mathrm{sweep}}$, and $t_{\mathrm{fall}}$. The corresponding trajectories in the phase diagram for the two annealing schedules discussed in Sec. \ref{['sec4.1']} are shown as blue dotted and red dashed lines, which we denote as annealing (I) and (II), respectively. In the two cases, we consider that the initial value for the longitudinal field is $h^0_z = -8J$, while the one at the end of the sweep is $h^f_z = 0$. Furthermore, we consider for (I) $t_{\mathrm{rise}}J=1.5$, $t_{\mathrm{sweep}}J=1.5$, and different values of $t_{\mathrm{fall}}$ and for (II) $t_{\mathrm{rise}}J=1.5$, $t_{\mathrm{fall}}J=1.5$, and different values of $t_{\mathrm{sweep}}$; more details about the annealing sweeps are presented in the main text.
• Figure 3: Panel (a) shows a schematic of the zero-temperature ferromagnetic phase diagram of the transverse-field Ising model, $H_\text{FM}$, on the square lattice for $h_z/J = 0$. We highlight the three values of the transverse field $h_x/J$ considered for post-quench dynamics in this work: one at, and two below, the critical field $h_c/J \approx 3$. Panel (b) displays the three quench protocols employed in our analysis, corresponding to the field values of $h_x/J$ indicated in panel (a) and $h_z/J = 0$.
• Figure 4: Annealing (magnetization). Cross-benchmarks of local magnetization, $\langle \sigma_{r_i}^z \rangle$, obtained with different numerical methods (MPS, TTN, NQS and 2DTN) during the two annealing sweeps described in Fig. \ref{['fig2']}. The set of panels labeled (a) and (b) correspond to sweeps (I) and (II), respectively. In panels (a1) and (b1), we present the time dependence of $\langle \sigma_{r_0}^z \rangle$. Panels (a2) and (a3) [(b2) and (b3)] show the spatial distribution of $\langle \sigma_{r_i}^z \rangle$ along a horizontal line at two representative times, $t_1$ and $t_f$. Panels (a4) and (a5) [(b4) and (b5)] display $\epsilon_z$ between each method and MPS results at $t_1$ [$t_f$] for different system sizes.
• Figure 5: Annealing (correlations). Cross-benchmarks of connected correlations, $C(\delta_i)$, obtained with different numerical methods (MPS, TTN, NQS and 2DTN) during the two annealing sweeps described in Fig. \ref{['fig2']}. The set of panels labeled (a) and (b) correspond to sweeps (I) and (II), respectively. In panels (a1) and (b1), we present the time-dependence of nearest-neighbor correlations, $C(\delta_1)$. Panels (a2) and (a3) [(b2) and (b3)] show the spatial distribution of $C(\delta_i)$ along a horizontal line at two representative times, $t_1$ and $t_f$. Finally, panels (a4) and (a5) [(b4) and (b5)] display the discrepancies between the results obtained by the different methods, $\epsilon_{zz}$, at $t_1$ and $t_f$ in relation to MPS results for different system sizes.