Tensor networks for out-of-equilibrium dynamics: transverse contraction and temporal entanglement

Out-of-equilibrium dynamics of quantum many-body systems constitutes an intriguing problem because of the rich phenomenology it contains. However, it constitutes, in general, an exponentially-hard task. In this seminar, we face this problem within the tensor network formalism. In particular, we focus on 1D quantum systems, for which the Matrix Product State (MPS) ansatz is typically used. Thus, we introduce the basics of MPS and formulate the conventional algorithms for this problem, which are usually restricted to short times because of the entanglement spreading. In this context, transverse contraction algorithms are proposed, which based on the concept of temporal entanglement may represent an advantage in terms of efficiency compared to the standard approaches. We review these algorithms and show the conjectured complexity scales associated with the temporal entanglement.