A Note on Quantum Gates SWAP and iSWAP in Higher Dimensions (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. $\sqrt{-1}$), 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 $x eq ± 1$. Our explicit representation of xSWAP by a permutation matrix facilitates the proof, greatly.
Keywords: Qudits, Quantum logic gates, Universal gates, Entangling gates.