Three-dimensional color code thresholds via statistical-mechanical mapping

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 mea-
surement of error syndromes. Here we investigate the storage threshold error rates for bit-
ip and
phase-
ip 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)
3DCC ‘ 1:9% and p(2)
3DCC ‘ 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.