  We present a formula for an infinite number of universal quantum logic gates,
which are $4$ by $4$ unitary solutions to the Yang-Baxter (Y-B) equation. We
obtain this family from a certain representation of the cyclic group of order
$n$. We then show that this {\it discrete} family, parametrized by integers
$n$, is in fact, a small sub-class of a larger {\it continuous} family,
parametrized by real numbers $\theta$, of universal quantum gates. We discuss
the corresponding Yang-Baxterization and related symmetries in the concomitant
Hamiltonian.
