Quantum complexity is a measure of the minimal number of elementary operations required to approximately prepare a given state or unitary channel. Recently this concept has found applications beyond quantum computing---in the classification of topological phases of matter and in the description of chaotic many-body systems. Furthermore, within the context of the AdS/CFT correspondence, it has been postulated that the complexity of a specific time-evolved many-body quantum state is sensitive to the long-time properties of AdS-black hole interiors. In this context Brown and Susskind conjectured that the complexity of a chaotic quantum system grows linearly in time up to times exponential in the system size, saturating at a maximal value, and remaining maximally complex until undergoing recurrences at the doubly-exponential times. In this work we prove the saturation and recurrence of complexity of quantum states and unitaries in a model of chaotic time-evolution based on local random quantum circuits, in which a local random unitary transformation is applied to the system at every time step. Importantly, our findings do not depend on details of the model of random circuits, such as geometry of interactions between the qubits. Our results advance an understanding of the long-time behaviour of chaotic quantum systems and could shed light on the physics of black hole interiors. From a technical perspective our results are based on establishing new quantitative connections between the Haar measure and high-degree approximate designs, as well as the fact that random quantum circuits of sufficiently high depth converge to approximate designs.
The talk is based on joint work with Nicolas Hunter-Jones and Michal Horodecki.
(Please note the later start time of 12:30 p.m. for this seminar.)
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