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We study a model of free fermions on a chain with dynamics generated byrandom unitary gates acting on nearest neighbor bonds and present an exactcalculation of time-ordered and out-of-time-ordered correlators. We considerthree distinct cases: the random circuit with spatio-temporal disorder (i) withand (ii) without particle number conservation and (iii) the particlenon-conserving case with purely temporal disorder. In all three cases, temporaldisorder causes diffusive operator spreading and $\sim\sqrt{t}$ entanglementgrowth. This is in sharp contrast to Anderson localization for the case ofstatic disorder and with the ballistic behavior observed in both the clean caseof Hamiltonian evolution and in fully random unitary quantum circuits.

Diffusive Operator Spreading for Random Unitary Free Fermion Circuits