Efficiency of producing random unitary matrices with quantum circuits

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. 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 nₐln ({nₐ∕ϵ) }^ with the number of qubits nₐ for a given fixed precision ϵ and >0.