  We discuss how to significantly reduce leakage errors in topological quantum
computation by introducing an irrelevant error in phase, using the construction
of a CNOT gate in the Fibonacci anyon model as a concrete example. To be
specific, we construct a functional braid in a six-anyon Hilbert space that
exchanges two neighboring anyons while conserving the encoded quantum
information. The leakage error is $\sim$$10^{-10}$ for a braid of $\sim$100
interchanges of anyons. Applying the braid greatly reduces the leakage error in
the construction of generic controlled-rotation gates.
