  The Yang-Baxter equation and it's various forms have applications in many
fields, including statistical mechanics, knot theory, and quantum information.
Unitary solutions of the braided Yang-Baxter equation are of particular
interest as quantum gates for topological quantum computers. We demonstrate a
simple construction for solutions in any dimension, which are both unitary and
universal for quantum computation. We also fully classify a family of solutions
to certain generalized Yang-Baxter equations and prove that certain instances
of the equation only have solutions that are scalar multiples of the identity.
