  In a topological quantum computer, universality is achieved by braiding and
quantum information is natively protected from small local errors. We address
the problem of compiling single-qubit quantum operations into braid
representations for non-abelian quasiparticles described by the Fibonacci anyon
model. We develop a probabilistically polynomial algorithm that outputs a braid
pattern to approximate a given single-qubit unitary to a desired precision. We
also classify the single-qubit unitaries that can be implemented exactly by a
Fibonacci anyon braid pattern and present an efficient algorithm to produce
their braid patterns. Our techniques produce braid patterns that meet the
uniform asymptotic lower bound on the compiled circuit depth and thus are
depth-optimal asymptotically. Our compiled circuits are significantly shorter
than those output by prior state-of-the-art methods, resulting in improvements
in depth by factors ranging from 20 to 1000 for precisions ranging between
$10^{-10}$ and $10^{-30}$.
