  The development of powerful numerical techniques has drastically improved our
understanding of quantum matter out of equilibrium. Inspired by recent progress
in the area of noisy intermediate-scale quantum devices, this paper highlights
hybrid Schr\"odinger-Feynman techniques as an innovative approach to
efficiently simulate certain aspects of many-body quantum dynamics on classical
computers. To this end, we explore the nonequilibrium dynamics of two large
subsystems, which interact sporadically in time, but otherwise evolve
independently from each other. We consider subsystems with tunable disorder
strength, relevant in the context of many-body localization, where one
subsystem can act as a bath for the other. Importantly, studying the full
interacting system, we observe that signatures of thermalization are enhanced
compared to the reference case of having two independent subsystems. Notably,
with the here proposed Schr\"odinger-Feynman method, we are able to simulate
the pure-state survival probability in systems significantly larger than
accessible by standard sparse-matrix techniques.
