Projecting three-dimensional color codes onto three-dimensional toric codes

Toric codes and color codes are two important classes of topological codes. Kubica et al. [A. Kubica et al., New J. Phys. 17, 083026 (2015)] showed that any $D$-dimensional color code can be mapped to a finite number of toric codes in $D$ dimensions. We propose an alternate map of three-dimensional (3D) color codes to 3D toric codes with a view to decoding 3D color codes. Our approach builds on Delfosse's result [N. Delfosse, Phys. Rev. A 89, 012317 (2014)] for 2D color codes and exploits the topological properties of these codes. Our result reduces the decoding of 3D color codes to that of 3D toric codes. Bit-flip errors are decoded by projecting on one set of 3D toric codes, while phase-flip errors are decoded by projecting onto another set of 3D toric codes. We use these projections to study the performance of a class of 3D color codes called stacked codes.

Figure 1

We can recover the boundary of a tetrahedron from the edges by reconstructing the faces and then combining the faces. From each projection ${\pi }_c$, we recover a face. The faces from all the projections give the boundary of the tetrahedron.

Figure 2

We can recover the boundary of a tetrahedron from the vertices by first recovering the edges in the boundary of the tetrahedron. From each projection ${\pi }_{c{c}^{\prime }}$, we recover an edge. With the edges recovered, we can proceed, as illustrated in Fig. 1, to recover the boundary of the tetrahedron.

Figure 3

Consider the $by$-edge (in bold) and the tetrahedra (qubits) incident on it. Since only $r$-, $g$-vertices (i.e., red and green) will survive in ${\mathrm{\Gamma }}^{*\setminus by}$, each of these surviving vertices will have exactly two $rg$-edges (of the tetrahedra incident on the $by$-edge) incident on them. Therefore, these $rg$-edges form a cycle, as shown on the right. These edges form the boundary of a face in ${\mathrm{\Gamma }}^{*\setminus by}$. The faces of other complexes ${\mathrm{\Gamma }}^{*\setminus c{c}^{\prime }}$ are similarly formed.

Figure 4

A 3-colex $\mathrm{\Gamma }$ and its dual ${\mathrm{\Gamma }}^{*}$; $i$-cells of ${\mathrm{\Gamma }}^{*}$ correspond to the ($3-i$)-cells of $\mathrm{\Gamma }$. The minor complexes ${\mathrm{\Gamma }}^{*\setminus r}$ and ${\mathrm{\Gamma }}^{*\setminus ry}$ are also shown. (In $\mathrm{\Gamma }$, we use smooth lines for green edges, dotted lines for red edges, dashed lines for blue edges, and thick lines for yellow edges.)

Figure 5

Performance of the stacked color codes over the bit-flip channel.

Figure 6

Performance of the stacked color codes over the phase-flip channel.