  We study fault-tolerant error correction in a quantum memory constructed as a
two-dimensional model of Fibonacci anyons on a torus, in the presence of
thermal noise represented by pair-creation processes and measurement errors.
The correction procedure is based on the cellular automaton decoders
originating in the works of G\'acs and Harrington. Through numerical
simulations, we observe that this code behaves fault-tolerantly and that
threshold behavior is likely present. Hence, we provide strong evidence for the
existence of a fault-tolerant universal non-Abelian topological quantum
computer.
