  A fundamental question in the theory of quantum computation is to understand
the ultimate space-time resource costs for performing a universal set of
logical quantum gates to arbitrary precision. Here we demonstrate that
non-Abelian anyons in Turaev-Viro quantum error correcting codes can be moved
over a distance of order the code distance, and thus braided, by a constant
depth local unitary quantum circuit followed by a permutation of qubits. Our
gates are protected in the sense that the length of error strings do not grow
by more than a constant factor. When applied to the Fibonacci code, our results
demonstrate that a universal logical gate set can be implemented on encoded
qubits through a constant depth unitary quantum circuit, and without increasing
the asymptotic scaling of the space overhead. These results also apply directly
to braiding of topological defects in surface codes. Our results reformulate
the notion of braiding in general as an effectively instantaneous process,
rather than as an adiabatic, slow process.
