Simulating quantum dynamics in two-dimensional lattices with tensor network influence functional belief propagation

TL;DR

The paper tackles the challenge of simulating nonequilibrium quantum dynamics by extending tensor network influence functionals (TN-IF) to two-dimensional lattices via tree-based constructions and a belief propagation scheme (IF-BP). IF-BP provides an efficient, often accurate, means to compute local observables on loopy graphs by solving a self-consistent IF-MPS on Bethe-like structures, with exact results on trees and good accuracy on locally tree-like graphs; the authors also introduce a cluster expansion to systematically include loop correlations beyond BP. On the heavy-hex lattice, IF-BP captures long-time dynamics where traditional TN-state methods struggle, with temporal entanglement entropy (TEE) growing only logarithmically in time, enabling polynomial-cost simulations. To address loop-induced errors, a cluster expansion is developed and demonstrated in simulating 2D TFIM quench dynamics, achieving competitive or superior accuracy relative to state-of-the-art approaches, and offering a framework that can benchmark quantum devices. Overall, the work provides a scalable, hierarchy-based approach to nonequilibrium quantum dynamics in 2D, balancing exactness on trees, practical approximations on loopy graphs, and systematic loop corrections. The combination of IF-BP and cluster expansions represents a significant step toward tractable, accurate classical simulations of 2D quantum dynamics with potential practical impact on quantum hardware benchmarking and algorithm development, supported by polynomial-time scaling stemming from logarithmic TEE growth. $TEE$ grows as $O(\, ext{log}\, t)$ in the studied regimes, contributing to the method's efficiency.

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.

Figures (18)

• Figure 1: Tensor network diagram illustrating dynamics on a one-dimensional lattice. (a) Tensor network diagram for a local observable expectation value $\langle \hat{O}(m) \rangle = \mathinner{\langle{\psi(m)}|} \hat{O} \mathinner{|{\psi(m)}\rangle}$ where $m=4$ in this figure. The initial state $\mathinner{|{\psi_0}\rangle}$ is given by a matrix product state (MPS) (light gray squares), and time evolution is carried out by a matrix product operator (MPO) (light blue circles). (b) A folded tensor network (folded TN), where the folding occurs along the dashed line in (a). The darker shaded tensors denote doubly grouped tensors from (a). Tensors from the same spatial site are grouped together as $\mathcal{T}$ or $\mathcal{T}_{\hat{O}}$, shown with the light gray shading. (c) We contract the folded TN in (b) along the transverse direction around the site with $\hat{O}$. The boundary tensors are approximated by an MPS with a fixed bond dimension during the transverse contraction. The final MPS $\mathcal{I}$ (pink squares) is called an influence functional MPS (IF-MPS). (d) One-dimensional structure of the IF-MPS propagation after grouping tensors by spatial sites.
• Figure 2: Tensor network diagram illustrating dynamics on a tree lattice. (a) The initial state is a tree tensor network state, and its time evolution is carried out by a tree tensor network operator. (b) After folding the TN $\langle \hat{O}(m) \rangle$, we group tensors in the same spatial site as $\mathcal{T}$. (c) A grouped tensor $\mathcal{T}$ within a tree tensor network.
• Figure 3: (a) IF-MPS $\mathcal{I}$ for the tree subgraph at each bond. (b) TN contraction of two IF-MPSs $\mathcal{I}_{k \rightarrow j}$ and $\mathcal{I}_{l \rightarrow j}$ from sites $k$ and $l$ with the tensor $\mathcal{T}_j$ at site $j$. This creates a new IF-MPS $\mathcal{I}_{j \rightarrow i}$ directed from site $j$ to site $i$. (c) Final TN diagram after the IF-MPSs reach the site of interest with the operator $\hat{O}$. The desired expectation value is computed from the contraction of this TN.
• Figure 4: Self-consistent equation for IF-MPS in the $z=3$ Bethe lattice assuming rotational invariance.
• Figure 5: (Left) Tensor network diagram on an infinite hexagonal lattice. (Right) Within the belief propagation (BP) approximation, the fixed point BP equation is equivalent to that on the $z=3$ Bethe lattice.