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

Fault-tolerant error correction for a universal non-Abelian topological quantum computer at finite temperature