Almost-Optimal Computational Basis State Transpositions

We give an explicit construction to perform any $n$-qubit computational basis state transposition using $\mathrm{\Theta }(n)$ gates. This nearly coincides with the lower bound $\mathrm{\Omega }(n\mathrm{/}\log (nd))$ on worst-case and average-case gate complexity to perform transpositions using a $d$-element gate-set, which we also prove.