(untitled) (European Journal of Pure and Applied Mathematics, 2023)
We present explicit descriptions for the swap gate and the iswap gate in any arbitrary dimension d ≥ 2, in terms of permutation matrices. Moreover, we unify these gates by introducing a more general gate xSWAP which includes SWAP and iSWAP for x=1 and x=i (i.e. -1), respectively. The higher dimensional xSWAP e.g., the swap and iswap gates for d > 2 serve as quantum logic gates that operate on two d-level qudits. For d=2, it is well known that iSWAP unlike SWAP is universal for quantum computing. We will prove this fact for xSWAP in any dimension d, when x eq ± 1. Our explicit representation of xSWAP by a permutation matrix facilitates the proof, greatly.