@article{Pourkia_2023, title={A Note on Quantum Gates SWAP and iSWAP in Higher Dimensions}, volume={16}, url={https://ejpam.com/index.php/ejpam/article/view/4824}, DOI={10.29020/nybg.ejpam.v16i3.4824}, abstractNote={<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 $d &gt; 2$ 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>}, number={3}, journal={European Journal of Pure and Applied Mathematics}, author={Pourkia, Arash}, year={2023}, month={Jul.}, pages={1695–1704} }