Asymptotically Optimal Topological Quantum Compiling

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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}.