  The quantum permutation algorithm provides computational speed-up over
classical algorithms in determining the parity of a given cyclic permutation.
For its $n$-qubit implementations, the number of required quantum gates scales
quadratically with $n$ due to the quantum Fourier transforms included. We show
here for the $n$-qubit case that the algorithm can be simplified so that it
requires only $O(n)$ quantum gates, which theoretically reduces the complexity
of the implementation. In order to test our results experimentally, we utilize
IBM's $5$-qubit quantum processor to realize the algorithm by using the
original and simplified recipes for the $2$-qubit case. It turns out that the
latter results in a significantly higher success probability which allows us to
verify the algorithm more precisely than the previous experimental
realizations. We also verify the algorithm for the first time for the $3$-qubit
case with a considerable success probability by taking the advantage of our
simplified scheme.
