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Mixing and localisation in random time-periodic quantum circuits of Clifford unitaries

How much does local and time-periodic dynamics resemble a random unitary? In the present work we address this question by using the Clifford formalism from quantum computation. We analyse a Floquet model with disorder, characterised by a family of local, time-periodic, random quantum circuits in one spatial dimension. We observe that the evolution operator enjoys an extra symmetry at times that are a half-integer multiple of the period. With this we prove that after the scrambling time, namely when any initial perturbation has propagated throughout the system, the evolution operator cannot be distinguished from a (Haar) random unitary when all qubits are measured with Pauli operators. This indistinguishability decreases as time goes on, which is in high contrast to the more studied case of (time-dependent) random circuits. We also prove that the evolution of Pauli operators displays a form of mixing. These results require the dimension of the local subsystem to be large. In the opposite regime our system displays a novel form of localisation, produced by the appearance of effective one-sided walls, which prevent perturbations from crossing the wall in one direction but not the other.