In this paper we consider the compilation of quantum state permutations into quantum gates for physical quantum computers. A sequence of generic single-target gates, which realize the input permutation, are extracted using a decomposition based reversible logic synthesis algorithm. We present a compilation algorithm that translates single-target gates into a quantum circuit composed of the elementary quantum gate sets that are supported by IBM's 5-qubit and 16-qubit, and Rigetti's 8-qubit and 19-qubit superconducting transmon QPUs. Compared to generic state-of-the-art compilation techniques, our technique improves gate volume and gate depth by up to 59% and 53%, respectively.