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We study the scaling of the convergence of several statistical properties ofa recently introduced random unitary circuit ensemble towards their limitsgiven by the circular unitary ensemble (CUE). Our study includes the fulldistribution of the absolute square of a matrix element, moments of thatdistribution up to order eight, as well as correlators containing up to 16matrix elements in a given column of the unitary matrices. Our numericalscaling analysis shows that all of these quantities can be reproducedefficiently, with a number of random gates which scales at most as $n_q\log(n_q/\epsilon)$ with the number of qubits $n_q$ for a given fixed precision$\epsilon$. This suggests that quantities which require an exponentially largenumber of gates are of more complex nature.

Efficiency of Producing Random Unitary Matrices with Quantum Circuits