Error Threshold for Color Codes and Random Three-Body Ising Models

We study the error threshold of color codes, a class of topological quantum codes that allow a direct implementation of quantum Clifford gates suitable for entanglement distillation, teleportation, and fault-tolerant quantum computation. We map the error-correction process onto a statistical mechanical random three-body Ising model and study its phase diagram via Monte Carlo simulations. The obtained error threshold of $p_c=0.109(2)$ is very close to that of Kitaev’s toric code, showing that enhanced computational capabilities do not necessarily imply lower resistance to noise.

Figure 1

$p\mathrm{\text{−}}{T}_c$ phase diagram for the random three-body Ising model. For $p>p_c\approx 0.109$, the ferromagnetic order is lost. The dotted line is a guide to the eye; the black circle represents the analytically known transition temperature of the 2D Ising model. The blue (solid) line represents the $N$ line. In the regime marked by a dashed line, the exact determination of $T_c(p)$ is difficult.

Figure 2

Lattice for the TCCs with 3-colored vertices. Physical qubits of the error-correcting code correspond to triangles (stars mark the “boundaries” of the sets of triangles displayed). (a) Boundary of a vertex $v$. The stabilizer operators $X_v$ and $Z_v$ have support on the corresponding qubits. (b) Error pattern in the form of a string net. The three vertices that form its boundary are all of the information we have to correct the error. (c) Two error patterns with the same boundary. Because together they form the boundary of the three vertices marked with a circle, they are equivalent.

Figure 3

Finite-size correlation length ${\xi }_m/L$ as a function of temperature $T$ for different values of $p$. (a) $p=0$. The data cross at the critical temperature of the 2D Ising model (dashed line). (b) Finite-size scaling analysis of the data for $p=0$ using $\nu =2/3$. The scaling is very good showing that corrections to scaling are negligible. (c)–(f) For $p\lesssim p_c=0.109$ there is a signature of a transition (data for different $L$ cross), whereas for $p>p_c$ the transition vanishes [(g)–(h)].