The dynamics of quantum many-body systems out-of-equilibrium are pivotal in various fields, ranging from quantum information and the theory of thermalization to impurity physics. Fundamentally, the numerical study of larger quantum systems is challenging due to the exponential number of parameters necessary to describe the wavefunction. If their entanglement is low, wavefunctions can be approximated with relatively few parameters using tensor networks. Since equilibrium wavefunctions have low entanglement, this makes computations viable. However, when simulating dynamics, entanglement grows rapidly with the evolution time. Here we discuss a new approach to many-body dynamics, by using insights from the field of open quantum systems. We consider dynamics of a subsystem, and view the rest of the many-body system as a bath. The bath's properties are encoded in the influence functional (IF) on the space of Schwinger-Keldysh trajectories. Treating the IF as a "wave function" in the temporal domain, we introduced the concept of Temporal Entanglement (TE) which can be interpreted as the "quantum memory" of the bath. We show that in several broad and relevant classes of systems, such as in proximity to dissipative, integrable, many-body localized phases or dual unitary points, TE exhibits favorable scaling. This allows the IF to be efficiently compressed as tensor network, opening the door to a new family of computational methods based on low temporal rather than spatial entanglement. For baths consisting of free fermions, we introduce a procedure to obtain the IF directly from the spectral density. This allows us to compute dynamics of impurity problems, such as the Single Impurity Anderson model, achieving state of the art accuracy and time scales. Finally I will show how the IF can also be reconstructed from physical measurements, which could be performed on a quantum computer. This leads to the formulation of a quantum-classical algorithm for computing the time evolution of observables in complicated quantum systems.

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