Optimization and experimental realization of the quantum permutation algorithm

Abstract

The quantum permutation algorithm provides computational speed-up over classical algorithms for 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. 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.