AUTHOR=Cerezo-Roquebrún Sergio , Bou-Comas Aleix , Schneider Jan T. , López Esperanza , Tagliacozzo Luca , Carignano Stefano 

TITLE=Spatio-temporal tensor-network approaches to out-of-equilibrium dynamics bridging open and closed systems

JOURNAL=Frontiers in Quantum Science and Technology

VOLUME=Volume 4 - 2025

YEAR=2025

URL=https://www.frontiersin.org/journals/quantum-science-and-technology/articles/10.3389/frqst.2025.1568471

DOI=10.3389/frqst.2025.1568471

ISSN=2813-2181

ABSTRACT=The study of many-body quantum systems out of equilibrium remains a significant challenge, with complexity barriers arising in both state- and operator-based representations. Here, we review the recent approaches based on finding better contraction strategies for the full spatiotemporal tensor networks that encode the path integral of the dynamics, as well as the conceptual integration of influence functionals, process tensors, and transfer matrices within the tensor network formalism. We discuss recent algorithmic developments, highlight the complexity of influence functionals in various dynamical regimes, and present consistent results of different communities, showing how ergodic dynamics render these functionals exponentially difficult to compress. Finally, we provide an outlook on strategies to encode complementary influence functional overlaps, paving the way for accurate descriptions of open and closed quantum systems with tensor networks.