Propagation of correlations in Local Random Quantum Circuits

We derive a dynamical bound on the propagation of correlations in local random quantum circuits - lattice spin systems where piecewise quantum operations - in space and time - occur with classical probabilities. Correlations are quantified by the Frobenius norm of the commutator of two positive operators acting on space-like separated local Hilbert spaces. For times $t=O(L)$ correlations spread to distances $\mathcal{D}=t$ growing, at best, diffusively for any distance within that radius with extensively suppressed distance dependent corrections whereas for $t=o(L^2)$ all parts of the system get almost equally correlated with exponentially suppressed distance dependent corrections and approach the maximum amount of correlations that may be established asymptotically.