Decoding color codes by projection onto surface codes

We propose a general strategy to decode color codes, which is based on the projection of the error onto three surface codes. This provides a method to transform every decoding algorithm of surface codes into a decoding algorithm of color codes. Applying this idea to a family of hexagonal color codes, with the perfect matching decoding algorithm for the three corresponding surface codes, we find a phase error threshold of approximately $8.7\%$. Finally, our approach enables us to establish a general lower bound on the error threshold of a family of color codes depending on the threshold of the three corresponding surface codes. These results are based on a chain complex interpretation of surface codes and color codes.

Figure 1

Block error rate after decoding as a function of the phase error percentage, for hexagonal toric codes of length $9.2^{2m}$.

Figure 2

A tiling of the torus with 3-colorable faces and its dual. The vertices of the dual inherit the 3-coloration.

Figure 3

A triangle hyperedge and an hexagonal hyperface of the hypergraph $\mathcal{H}$ associated with a hexagonal tiling of the torus.

Figure 4

A triangular tiling of the torus $G^{*}$ with 3-colored vertices and its subtiling $G^{*}\left(R\right)$ induced by the dotted (red) edges.

Figure 5

An example of decoding of a color code. (a) An error $x$ whose support is the set of gray hyperedges. Its syndrome $s$ is composed of the vertices supporting a symbol 1. (b) We restrict the syndrome to the dashed (blue) graph $G^{*}\left(B\right)$. (c) We decode the projection on the surface code $G^{*}\left(B\right)$. This defines a set ${\tilde{b}}_B$ of dashed (blue) edges. (d) We consider the union $\tilde{b}$ of the three sets ${\tilde{b}}_{\mathbf{c}}$ of edges computed in the three surface codes associated with the graph $G^{*}\left(\mathbf{c}\right)$. (e) The estimation $\tilde{x}$ of the error is the set of gray edges obtained by filling the cycle $\tilde{b}$. (f) The difference $x-\tilde{x}$ corresponds to the gray hyperedges. It is a sum of two hyperfaces, therefore the vector $\tilde{x}$ is a good estimation of $x$.

Figure 6

An example of an error acting on a color code and its boundary estimation $\tilde{b}$, which is not a boundary. In that case, Algorithm 1 returns NOLIFTING.

Figure 7

Block error rate after decoding as a function of the phase (or bit-flip) error rate, for hexagonal color codes of parameters $\left[[18.4^m,4,4.2^m]\right]$.