String Theory Journal Club

Quantum complexity is a measure of the minimal number ofelementary operations required to approximately prepare a given state orunitary channel. Recently, this concept has found applications beyondquantum computing—in studying the dynamics of quantum many-body systemsand the long-time properties of AdS black holes. In this context Brownand Susskind conjectured that the complexity of a chaotic quantum systemgrows linearly in time up to times exponential in the system size,saturating at a maximal value, and remaining maximally complex untilundergoing recurrences at doubly-exponential times. In this work weprove the saturation and recurrence of the complexity of quantum statesand unitaries in a model of chaotic time-evolution based on randomquantum circuits, in which a local random unitary transformation isapplied to the system at every time step. Importantly, our findings holdfor quite general random circuit models, irrespective of the gate setand geometry of qubit interactions. Our results advance an understandingof the long-time behaviour of chaotic quantum systems and could shedlight on the physics of black hole interiors. From a technicalperspective our results are based on establishing new quantitativeconnections between the Haar measure and high-degree approximatedesigns, as well as the fact that random quantum circuits ofsufficiently high depth converge to approximate designs.