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The quantum permutation algorithm provides computational speed-up overclassical algorithms in determining the parity of a given cyclic permutation.For its $n$-qubit implementations, the number of required quantum gates scalesquadratically with $n$ due to the quantum Fourier transforms included. We showhere for the $n$-qubit case that the algorithm can be simplified so that itrequires only $O(n)$ quantum gates, which theoretically reduces the complexityof the implementation. In order to test our results experimentally, we utilizeIBM's $5$-qubit quantum processor to realize the algorithm by using theoriginal and simplified recipes for the $2$-qubit case. It turns out that thelatter results in a significantly higher success probability which allows us toverify the algorithm more precisely than the previous experimentalrealizations. We also verify the algorithm for the first time for the $3$-qubitcase with a considerable success probability by taking the advantage of oursimplified scheme.

Optimization and experimental realization of the quantum permutation algorithm