Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices

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.

Figure 1

Schematic picture of the phase diagram of the $\pm J$ Ising model on two-dimensional lattice (left panel) and on higher dimensions (right panel). The vertical axis expresses the temperature $T$ and the horizontal line denotes the concentration $p$ of the antiferromagnetic interactions. The multicritical point (MCP) is described by the black point. The Nishimori line is drawn by the dashed line. For higher dimensions, not only the ferromagnetic (FM) and paramagnetic phases (PM) but also the spin-glass phase (SG) exists. The phase boundary under the Nishimori line is believed to be vertical by the argument of the gauge transformation [4].

Figure 2

Triangular lattice and Union-Jack lattice. The site variable $\varphi$ is located at each site and the plaquette variable $k_I$ is introduced at each triangular face. The plaquette variables form a structure of the hexagonal and square-octagonal lattices.

Figure 3

Several clusters for the $\pm J$ three-body Ising model on the triangular lattice. These are chosen to cover the whole lattice. A cluster is the unit plaquette encircled by white spins. The spins marked black on the original lattice are traced out to yield interactions among white spins in clusters. The case denoted by $T_0$, and $U_0$ gives the principal Boltzmann factor used in the naive approach.

Figure 4

Several clusters for the $\pm J$ three-body Ising model on the Union-Jack lattice.

Figure 5

Supplementary clusters for the computation of the slope of $T_c$ at $p=1$.