We analyze surface codes, the topological quantum error-correcting codes introduced by
Kitaev. In these codes, qubits are arranged in a two-dimensional array on a surface of
nontrivial topology, and encoded quantum operations are associated with nontrivial
homology cycles of the surface. We formulate protocols for error recovery, and study the
efficacy of these protocols. An order-disorder phase transition occurs in this system at a
nonzero critical value of the error rate; if the error rate is below the critical value (the …