A topological quantum computer encodes quantum information in a topologically protected Hilbert space of non-Abelian anyons. Braiding non-Abelian anyons in space-time leads to robust quantum information processing. In this talk, I will first give a brief overview of the emergence and braiding of anyons in fractional quantum Hall systems. Then, I will use Fibonacci anyons to discuss braid (or quantum gate) construction toward universal topological quantum computation, with focus on avoiding leakage errors in braid construction. Such considerations result in high-accuracy low-leakage quantum gates for topological quantum computation.