Simulating quantum dynamics in two-dimensional lattices with tensor network influence functional belief propagation (2504.07344v2)

Abstract: Describing nonequilibrium quantum dynamics remains a significant computational challenge due to the growth of spatial entanglement. The tensor network influence functional (TN-IF) approach mitigates this problem for computing the time evolution of local observables by encoding the subsystem's influence functional path integral as a matrix product state (MPS), thereby shifting the resource governing computational cost from spatial entanglement to temporal entanglement. We extend the applicability of the TN-IF method to two-dimensional lattices by demonstrating its construction on tree lattices and proposing a belief propagation (BP) algorithm for the TN-IF, termed influence functional BP (IF-BP), to simulate local observable dynamics on arbitrary graphs. Even though the BP algorithm introduces uncontrolled approximation errors on arbitrary graphs, it provides an accurate description for locally tree-like lattices. Numerical simulations of the kicked Ising model on a heavy-hex lattice, motivated by a recent quantum experiment, highlight the effectiveness of the IF-BP method, which demonstrates superior performance in capturing long-time dynamics where traditional tensor network state-based methods struggle. Our results further reveal that the temporal entanglement entropy (TEE) only grows logarithmically with time for this model, resulting in a polynomial computational cost for the whole method. We further construct a cluster expansion of IF-BP to introduce loop correlations beyond the BP approximation, providing a systematic correction to the IF-BP estimate. We demonstrate the power of the cluster expansion of the IF-BP in simulating the quantum quench dynamics of the 2D transverse field Ising model, obtaining numerical results that improve on the state-of-the-art.

Overview of Tensor Network Influence Functional Belief Propagation for Quantum Dynamics

The paper "Simulating quantum dynamics in two-dimensional lattices with tensor network influence functional belief propagation" addresses the challenges of simulating nonequilibrium quantum dynamics in higher-dimensional systems, which are notoriously difficult due to the rapid growth of spatial entanglement. The authors propose a novel approach that leverages the concept of tensor network influence functional (TN-IF) and belief propagation (BP), allowing for more efficient simulation of such systems.

Background and Methodology

The paper builds upon existing methods for simulating quantum dynamics in one-dimensional systems, where tensor network states benefit from area-law scaling of spatial entanglement. However, these methods struggle in nonequilibrium settings due to temporal entanglement growth. The TN-IF approach circumvents the exponential cost by representing the system's influence functional as a matrix product state (MPS), focusing computational resources on temporal rather than spatial entanglement.

To extend TN-IF to two-dimensional lattice models, the authors initially construct the TN-IF for tree lattices, exploiting their loop-less topology. This forms the foundation for introducing the belief propagation algorithm to simulate dynamics on arbitrary graphs. While introducing approximation errors, BP delivers accurate descriptions for locally tree-like lattices due to its ability to handle large spatial loops effectively.

A key numerical enhancement proposed is the cluster expansion of TN-IF, which incorporates loop correlations beyond the BP approximation, addressing shortcomings when dealing with loopy graphs such as square lattices. Through systematic corrections afforded by cluster expansion, the method achieves high fidelity results over longer simulation times and complex interactions, improving upon current state-of-the-art techniques.

Numerical Results and Implications

The authors demonstrate the efficacy of their approach with simulations of the kicked Ising model on a heavy-hex lattice, inspired by recent quantum experiments. The IF-BP approach showed superior performance in capturing long-time dynamics, revealing a logarithmic relationship between the temporal entanglement entropy (TEE) and time for the model. This results in a polynomial computational cost, enhancing scalability for classical simulations of quantum systems.

Further extending the TN-IF framework, the cluster expansion introduced in the paper successfully accounts for loop effects in loopy graphs. Applied to 2D transverse field Ising models, it delivered enhanced accuracy to numerical results compared to various other methods, underlining its potential as a robust tool for simulating complex quantum dynamics.

Future Directions

This work opens several avenues for further research and exploration. Extending the TN-IF and BP framework to three-dimensional systems could push the boundaries of what can be efficiently simulated classically. Additionally, integrating machine learning techniques with TN-IF could enhance the optimization processes involved in MPS compression and cluster expansion. Exploring the applications of TN-IF in non-Hamiltonian or open quantum systems could also be fruitful, expanding theoretical and practical understandings of quantum dynamics in a broader context.

The TN-IF BP method described not only provides a significant advancement in computational techniques for simulating quantum dynamics but also contributes to the fundamental theoretical understanding of entanglement growth and influence functionals, supporting the non-equilibrium quantum systems exploration.