  Accuracy thresholds of quantum error correcting codes, which exploit
topological properties of systems, defined on two different arrangements of
qubits are predicted. We study the topological color codes on the hexagonal
lattice and on the square-octagonal lattice by the use of mapping into the spin
glass systems. The analysis for the corresponding spin glass systems consists
of the duality, and the gauge symmetry, which has succeeded in deriving
locations of special points, which are deeply related with the accuracy
thresholds of topological error correcting codes. We predict that the accuracy
thresholds for the topological color codes would be $1-p_c = 0.1096-8 $ for the
hexagonal lattice and $1-p_c = 0.1092-3$ for the square-octagonal lattice,
where $1-p$ denotes the error probability on each qubit. Hence both of them are
expected to be slightly lower than the probability $1-p_c = 0.110028$ for the
quantum Gilbert-Varshamov bound with a zero encoding rate.
