TY - JOUR
AU - Pourkia, Arash
PY - 2023/07/30
Y2 - 2023/12/07
TI - A Note on Quantum Gates SWAP and iSWAP in Higher Dimensions
JF - European Journal of Pure and Applied Mathematics
JA - Eur. J. Pure Appl. Math.
VL - 16
IS - 3
SE -
DO - 10.29020/nybg.ejpam.v16i3.4824
UR - https://ejpam.com/index.php/ejpam/article/view/4824
SP - 1695-1704
AB - <p>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 $x
eq \pm 1$. Our explicit representation of xSWAP by a permutation matrix facilitates the proof, greatly.</p>
ER -