Cyclic groups and quantum logic gates

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 discrete family, parametrized by integers n, is in fact, a small sub-class of a larger continuous family, parametrized by real numbers θ, of universal quantum gates. We discuss the corresponding Yang-Baxterization and related symmetries in the concomitant Hamiltonian.