Efficiency of producing random unitary matrices with quantum circuits
Ludovic Arnaud, Daniel Braun · Physical Review A · 2008
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 ${n}_{q}{[\mathrm{ln}({n}_{q}∕ϵ)]}^{\ensuremath{ u}}$ with the number of qubits ${n}_{q}$ for a given fixed precision $ϵ$ and $\ensuremath{ u}>0$.