Quantum nature of order-parameter transitions in adaptive random circuits with feedback

Controlling chaotic dynamics is a fundamental problem in the dynamics of classical and quantum systems. Recently, this problem has been studied in the context of monitored quantum circuits with feedback, debuting as adaptive circuit that can steer the dynamics to a pre-determined dark state. Prior studies on this transition, including control-induced and absorbing phase transitions, have suggested a classical nature for the underlying universality class. In this work, we reveal the quantum aspects of this seemingly "classical" transition by disentangling the classical and quantum fluctuations of observables and introducing a novel critical exponent that is unique to the quantum contribution. This quantum critical exponent characterizes the transition within the area-law entangled phase from classically chaotic but quantum coherent dynamics to classical fixed-point behavior, and can be probed experimentally without the need for postselection.