An unknown unitary gates, which is secretly chosen from several known ones, can always be distinguished perfectly. In this paper, we implement such a task on IBM's quantum processor. More precisely, we experimentally demonstrate the discrimination of two qubit unitary gates, the identity gate and the 23π-phase shift gate, using two discrimination schemes -- the parallel scheme and the sequential scheme. We program these two schemes on the \emph{ibmqx4}, a 5-qubit superconducting quantum processor via IBM cloud, with the help of the QSI modules [S. Liu et al.,~arXiv:1710.09500, 2017]. We report that both discrimination schemes achieve success probabilities at least 85%.