Tricolored lattice gauge theory with randomness: fault tolerance in topological color codes

We compute the error threshold of color codes-a class of topological quantum codes that allow a direct implementation of quantum Clifford gates-when both qubit and measurement errors are present. By mapping the problem onto a statistical-mechanical three-dimensional disordered Ising lattice gauge theory, we estimate via large-scale Monte Carlo simulations that color codes are stable against 4.8(2)% errors. Furthermore, by evaluating the skewness of the Wilson loop distributions, we introduce a very sensitive probe to locate first-order phase transitions in lattice gauge theories.