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

TL;DR

This paper reviews recent tensor-network methods for simulating out-of-equilibrium quantum many-body systems, focusing on contraction strategies, influence functionals, and the challenges posed by ergodic dynamics.

Contribution

It introduces a comprehensive review of tensor-network approaches that unify open and closed quantum system dynamics through advanced contraction and influence functional techniques.

Findings

Ergodic dynamics make influence functionals exponentially hard to compress.

Recent algorithmic developments improve tensor network contraction strategies.

Unified framework for open and closed systems via influence functionals and transfer matrices.

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. In this work, we review 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.