Connecting Entanglement in Time and Space: Improving the Folding Algorithm

TL;DR

The paper addresses the challenge of simulating real-time dynamics in 1D quantum systems where entanglement grows rapidly during evolution. It develops a continuum-transverse framework that equates temporal entanglement with the spatial entanglement of a modified (often non-Hermitian) Hamiltonian $H_L$, enabling a CMPS-based description of the left/right transverse states. Building on this, the authors introduce the hybrid folding algorithm, which combines folding with flexible gauge transformations to diagonalize transfer matrices and reduce truncation error, achieving higher accuracy at the same bond dimension. They apply the method to a transverse-plus-parallel field Ising chain, observing long-time quasi-periodic oscillations for certain initial states and providing stronger numerical evidence that isolated quantum systems may fail to thermalize universally. This approach improves classical simulations of non-equilibrium 1D quantum dynamics and informs future developments in continuous-MPS techniques and gauge-stable tensor-network algorithms.

Abstract

The "folding algorithm"\cite{fold1} is a matrix product state algorithm for simulating quantum systems that involves a spatial evolution of a matrix product state. Hence, the computational effort of this algorithm is controlled by the temporal entanglement. We show that this temporal entanglement is, in many cases, equal to the spatial entanglement of a modified Hamiltonian. This inspires a modification to the folding algorithm, that we call the "hybrid algorithm". We find that this leads to improved accuracy for the same numerical effort. We then use these algorithms to study relaxation in a transverse plus parallel field Ising model, finding persistent quasi-periodic oscillations for certain choices of initial conditions.

Figures (11)

• Figure 1: Figure showing tensor network. Observable is at middle of tensor network. Bottom half is forward evolution in time, called "forward contour", while top half is backwards evolution in time, called "return contour". States at top and bottom are matrix product states. We show two Trotter-Suzuki steps on each contour. There are a total of $7$ sites. The operator at the middle of the network represents an observable on the fourth site.
• Figure 2: Entanglement entropy as a function of time for free fermi system. Top line is entropy using Hamiltonian $\tilde{H}$, while bottom two lines corresponding to using a Hermitian Hamiltonian (either the original Hamiltonian with couplings having changed sign to the right of the middle, or the original Hamiltonian). Inset shows logarithmic scale for time axis, showing logarithmic growth of entropy for the latter two cases.
• Figure 3: Error in expectation value of $X$ operator, for times $t=2,4,6$ and for normal and hybrid algorithms. Initial conditions are in $|X+\rangle$ state.
• Figure 4: Error (per column) in expectation value of identity, for times $t=2,4,6$ and for normal and hybrid algorithms. Initial conditions are in $|X+\rangle$ state.
• Figure 5: Error in expectation value of $X$ operator, for times $t=2,4,6$ and for normal and hybrid algorithms. Initial conditions are in $|X-\rangle$ state.