The complexity of estimating order parameters in quench dynamics of 1D Ising models

In nonequilibrium dynamics, a common objective is estimating a dynamical order parameter, specified by the expectation value of local (few-body) observables. It is commonly accepted that if the full many-body state cannot be efficiently simulated, classical simulation of such order parameters is intractable. Here we demonstrate that the common lore is not always correct. We study this in the case of quench dynamics of 1D Ising spin chains in both integrable and chaotic models and explore the complexity of simulation based on a truncated matrix product state representation. We study dynamical quantum phase transitions and the growth of correlations using the Time Evolving Block Decimation (TEBD) algorithm. We show that even for models where exact simulations of the full many-body state are intractable due to volume-law growth of entanglement, the order parameters can be efficiently simulated. We quantify this based on the Hilbert-Schmidt distance between the exact and approximate reduced density matrices for different levels of truncation, which upper-bounds the error in estimating the expectation values of local observables.