A Note on Quantum Gates SWAP and iSWAP in Higher Dimensions

We present explicit descriptions for the swap gate and the iswap gate in any arbitrary dimension $d \geq 2$, 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. $\sqrt{-1}$), 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 $x \neq \pm 1$. Our explicit representation of xSWAP by a permutation matrix facilitates the proof, greatly.