Optimization and experimental realization of the quantum permutation algorithm

TL;DR

This paper simplifies the quantum permutation algorithm to require only linear quantum gates, enabling more efficient implementation and higher success rates in experiments on IBM's quantum processor for 2- and 3-qubit cases.

Contribution

The authors reduce the quantum permutation algorithm's complexity from quadratic to linear gates and experimentally validate the improved scheme on a 5-qubit quantum processor.

Findings

Simplified the algorithm to require O(n) gates

Achieved higher success probability in 2-qubit experiments

First experimental verification for 3-qubit case

Abstract

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.