REVIEW article

Front. Quantum Sci. Technol.

Sec. Quantum Information Theory

Volume 4 - 2025 | doi: 10.3389/frqst.2025.1568471

This article is part of the Research TopicOpen Quantum Systems in Quantum TechnologiesView all articles

Tensor-network approaches to out-of-equilibrium dynamics bridging between open and closed systems

Provisionally accepted
Sergio  Cerezo-RoquebrúnSergio Cerezo-Roquebrún1Aleix  Bou-ComasAleix Bou-Comas2Jan  T SchneiderJan T Schneider2Esperanza  LópezEsperanza López1Luca  TagliacozzoLuca Tagliacozzo2*Stefano  CarignanoStefano Carignano3
  • 1Institute of Theoretical Physics, Faculty of Science, Autonomous University of Madrid, Madrid, Madrid, Spain
  • 2Spanish National Research Council (CSIC), Madrid, Spain
  • 3Barcelona Supercomputing Center, Barcelona, Catalonia, Spain

The final, formatted version of the article will be published soon.

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 spatio-temporal 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.

Keywords: keyword, open quantum systems, closed quantum dynamics, tensor networks, Influence functionals, process tensor, temporal entropies, reduced transition matrices

Received: 29 Jan 2025; Accepted: 06 Mar 2025.

Copyright: © 2025 Cerezo-Roquebrún, Bou-Comas, Schneider, López, Tagliacozzo and Carignano. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Luca Tagliacozzo, Spanish National Research Council (CSIC), Madrid, Spain

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

Research integrity at Frontiers

94% of researchers rate our articles as excellent or good

Learn more about the work of our research integrity team to safeguard the quality of each article we publish.


Find out more