  Fibonacci anyons are non-Abelian particles for which braiding is universal
for quantum computation. Reichardt has shown how to systematically generate
nontrivial braids for three Fibonacci anyons which yield unitary operations
with off-diagonal matrix elements that can be made arbitrarily small in a
particular natural basis through a simple and efficient iterative procedure.
This procedure does not require brute force search, the Solovay-Kitaev method,
or any other numerical technique, but the phases of the resulting diagonal
matrix elements cannot be directly controlled. We show that despite this lack
of control the resulting braids can be used to systematically construct
entangling gates for two qubits encoded by Fibonacci anyons.
