State-of-the-art quantum simulators permit local temporal control of interactions and midcircuit readout. These capabilities open the way toward the exploration of intriguing nonequilibrium phenomena. We illustrate this with a kinetically constrained many-body quantum system that has a natural implementation on Rydberg quantum simulators. The evolution proceeds in discrete time and is generated by repeatedly entangling the system with an auxiliary environment that is monitored and reset after each time step. Despite featuring an uncorrelated infinite-temperature average stationary state, the dynamics displays coexistence of fast and slow space-time regions in stochastic realizations of the system state. The time record of measurement outcomes on the environment serves as natural probe for such dynamical heterogeneity, which we characterize using tools from large deviation theory. Our Letter establishes the large deviation framework for discrete-time open quantum many-body systems as a means to characterize complex dynamics and collective phenomena in quantum processors and simulators.

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• Open quantum systems
• Quantum simulation
• Stochastic dynamical systems
• Kinetically constrained models
• Large deviation & rare event statistics
• Stochastic processes & statistics