On the Realization of quantum gates coming from the Tracy-Singh product

The Tracy-Singh product of matrices permits to construct a new gate $c⊠c^{\mathrm{\prime }}$ from two $2$-qudit gates $c$ and $c^{\mathrm{\prime }}$. If $c$ and $c^{\mathrm{\prime }}$ are both Yang-Baxter gates, then $c⊠c^{\mathrm{\prime }}$ is also a Yang-Baxter gate, and if at least one of them is entangling, then $c⊠c^{\mathrm{\prime }}$ is also entangling. A natural question arises about the realisation of these gates, $c⊠c^{\mathrm{\prime }}$, in terms of local and universal gates. In this paper, we consider this question and describe this realisation.