Time evolution methods for matrix-product states*

Matrix-product states (MPS) have become the de facto standard for the investigation of one-dimensional quantum many body systems, also out-of-equilibrium.
Various approaches have been introduced for computing the time evolution of MPS, e.g., a time-dependent variational principle (TDVP) for MPS as well as matrix product operator (MPOs) representations of the time evolution operator.
In this talk I review important developments and compare four commonly used methods applied to five representative examples, including systems with long-ranged interactions or in 2D.
These results give insights to the state-of-the-art treatment of MPS out-of-equilibrium and a guideline for which method to choose for a problem at hand.