  We study the scaling of the convergence of several statistical properties of
a recently introduced random unitary circuit ensemble towards their limits
given by the circular unitary ensemble (CUE). Our study includes the full
distribution of the absolute square of a matrix element, moments of that
distribution up to order eight, as well as correlators containing up to 16
matrix elements in a given column of the unitary matrices. Our numerical
scaling analysis shows that all of these quantities can be reproduced
efficiently, 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 large
number of gates are of more complex nature.
