Numerically efficient unitary evolution for Hamiltonians beyond nearest-neighbors

TL;DR

The authors address the challenge of simulating real-time dynamics in quantum spin systems with long-range or nonlocal interactions by introducing a direct exponentiation method for Pauli strings to construct compact MPOs for the time-evolution operator. This approach yields a maximum MPO bond dimension scaling as $w=2^r$ for interactions of range $r$, significantly reducing overhead compared to swap-gate constructions and enabling efficient TEBD simulations for periodic geometries and cluster terms. They demonstrate the method on nonintegrable models (ANNNI chain and 2D quantum Ising cylinder), showing accurate reproduction of entanglement growth and Loschmidt-echo dynamics, with improved efficiency and flexibility relative to TDVP and other schemes. The work highlights the method's versatility, robustness to alternative truncations, and potential applicability to Rydberg systems, thermal states, and quantum circuits, marking a practical advance in tensor-network simulations of complex, nonlocal Hamiltonians.

Abstract

Matrix product states (MPSs) and matrix product operators (MPOs) are fundamental tools in the study of quantum many-body systems, particularly in the context of tensor network methods such as Time-Evolving Block Decimation (TEBD). However, constructing compact MPO representations for Hamiltonians with interactions beyond nearest-neighbors, such as those arising in atomic, molecular, and optical (AMO) systems or in systems with ring geometry, remains a challenge. In this paper, we propose a novel approach for the direct construction of compact MPOs tailored specifically for the exponential of spin Hamiltonians. This approach allows for a more efficient time evolution, using TEBD, of spin systems with interactions beyond nearest-neighbors, such as long-range spin-chains, periodic systems and more complex cluster model, with interactions involving more than two spins.

Figures (4)

• Figure 1: a) Time evolution of the half chain bipartite entanglement after global quantum quenches of the external magnetic field from the disordered phase ($h_0=1.25$) into the ordered one ($h_1=0.90$). Blue markers represent simulations carried out with our TEBD approach, while the red stars correspond to single site TDVP simulations. The data are collected for $J_1=-1$, $N=60$, $\delta=0.05$ and different values of $J_2\in[0,0.15]$. Increasing $J_2$ reduces entanglement growth, signaling that frustrated competing interactions slow the spread of correlations in the system. b) Evolution of the Loschmidt echo rate function. The data are collected for $J_1=-1$, $N=100$, $\delta=0.01$ and $J_2\in[0,0.15]$ after a global quantum quench from $h_0=2.00$ to $h_1=0.01$. The fact that $J_2$ breaks integrability is reflected in the loss of exact periodicity of $r(t)$. Moreover, the number of peaks and their sharpness is reduced with increasing frustration strength.
• Figure 2: a) Scaling of the truncation errors with the bond dimension $\chi$ for the entanglement entropy shown in Fig. \ref{['fig:annni_obs']}. b) Scaling of the numerical simulation time with $\chi$. Blue markers represent simulations carried out with our TEBD approach, while the red ones correspond to single site TDVP simulations. We can observe that, for the case studied, the performances of our approach are better than those of single site TDVP simulations.
• Figure 3: a) Evolution of the rate function of the Loschmidt echo after a global quantum quench in the external magnetic field from $h_0=0$ to $h_1=1.3$. We considered $J_x=J_y=1$, $L_x=6$, $L_y=50$ and a time step of $\delta=0.01$. b) The right panel, instead, shows the scaling of the truncation error with the bond dimension $\chi$. c) Zigzag mapping from two-dimensional to one-dimensional lattice.
• Figure 4: Error in the Loschmidt echo using different truncation schemes, measured with respect to exact diagonalization. Data are obtained for Hamiltonian \ref{['eq:model']} with $r=3$, $N=15$, $J=1$ and $h=0.3$, with a quench amplitude of $\Delta h=0.5$. The time step used for the time evolution is $\delta=0.01$. Blue dots represent error obtained with TEBD2 while the green line corresponds to exact that obtained with $U^{Z2}$.