Influence matrix, temporal entanglement and efficient simulation of many-body dynamics

Describing dynamics of quantum many-body systems is

challenging due to the exponential number of parameters necessary to describe highly entangled wave functions. Often, one is interested in describing time evolution of a small subsystem immersed in a larger environment. Recently, an approach based on considering the influence matrix (IM) of the environment, rather than the full wave function was introduced. IM encodes the response of the environment to any subsystem trajectory. In several dynamical regimes, IM of an infinite environment exhibits low computational complexity, linked to the area-law scaling of its temporal entanglement. Thus, IM can be efficiently compressed using matrix product states. However, direct iterative construction yields complex intermediate IMs with high temporal entanglement. We explain the physical origin of this ``temporal entanglement barrier” using a quasiparticle picture that captures the exact behavior for integrable spin chains. Based on these insights, we formulate a light-cone growth algorithm, which avoids those highly complex intermediate IM states and therefore guarantees an efficient construction of the IM. This algorithm in particular allows one to simulate transport phenomena in correlated systems.