Constructing functional braids for low-leakage topological quantum computing

We discuss how to significantly reduce leakage errors in topological quantum computation by introducing an irrelevant error in phase, using the construction of a controlled-NOT (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 similar to 10(-10) for a braid of similar to 100 interchanges of anyons. Applying the braid greatly reduces the leakage error in the construction of generic controlled-rotation gates.