  This paper explores of the role of unitary braiding operators in quantum
computing. We show that a single specific solution R (the Bell basis change
matrix) of the Yang-Baxter Equation is a universal gate for quantum computing,
in the presence of local unitary transformations. We show that this same R
generates a new non-trivial invariant of braids, knots, and links. Other
solutions of the Yang-Baxter Equation are also shown to be universal for
quantum computation. The paper discusses these results in the context of
comparing quantum and topological points of view. In particular, we discuss
quantum computation of link invariants, the relationship between quantum
entanglement and topological entanglement, teleportation, and the structure of
braiding in a topological quantum field theory.
