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

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