Hydrodynamics in random unitary circuits with and without conservation laws

The scrambling of quantum information in closed many-body systems has received considerable recent attention. Two useful measures of scrambling have emerged: the spreading of initially-local operators, and the related concept of out-of-time-ordered correlation functions (OTOCs). We tackle this problem by considering 1D spin-chains evolving under random local unitary circuits and prove a number of exact results on the behavior of OTOCs. These results follow from the observation that the spreading of operators in random circuits is described by a ``hydrodynamical’’ equation of motion. Moreover, we also consider local random unitary circuits that explicitly conserve a U(1) charge and argue, with numerical and analytical evidence, that the presence of a conservation law slows relaxation in both time ordered and time-out-of ordered correlation functions. We conjecture that the hydrodynamical description applies to more generic ergodic systems and support this numerically.