  Three-dimensional (3D) color codes have advantages for fault-tolerant quantum
computing, such as protected quantum gates with relatively low overhead and
robustness against imperfect measurement of error syndromes. Here we
investigate the storage threshold error rates for bit-flip and phase-flip noise
in the 3D color code on the body-centererd cubic lattice, assuming perfect
syndrome measurements. In particular, by exploiting a connection between error
correction and statistical mechanics, we estimate the threshold for 1D
string-like and 2D sheet-like logical operators to be $p^{(1)}_\mathrm{3DCC}
\simeq 1.9\%$ and $p^{(2)}_\mathrm{3DCC} \simeq 27.6\%$. We obtain these
results by using parallel tempering Monte Carlo simulations to study the
disorder-temperature phase diagrams of two new 3D statistical-mechanical
models: the 4- and 6-body random coupling Ising models.
