Relation between surface codes and hypermap-homology quantum codes

Recently, a new class of quantum codes based on hypermaps has been proposed. These codes are obtained from embeddings of hypergraphs as opposed to surface codes which are obtained from the embeddings of graphs. It is natural to compare these two classes of codes and their relation to each other. In this context two related questions are addressed in this paper: Can the parameters of hypermap-homology codes be superior to those of surface codes, and what is precisely the relation between these two classes of quantum codes? We show that a canonical hypermap code is identical to a surface code, whereas a noncanonical hypermap code can be transformed to a surface code by cnot gates alone. Our approach is constructive; we construct the related surface code and the transformation involving cnot gates.

Figure 1

A hypergraph embedded on a torus; opposite ends are to be identified. The circles are the vertices of the hypergraph. The square vertices are the hyperedges of the hypergraph. The labels are to the left as one moves from the squares to the circles.

Figure 2

Maps relating to the embedding of the hypergraph.

Figure 3

A typical face in the embedding of a hypergraph $\Gamma$. It has an even number of darts and exactly $\left|f\right|/2$ darts have their labels inside the face. The darts $l_{\alpha }$ and $i_{\alpha }$ are related as $i_{\alpha }=\tau \left(l_{\alpha }\right)$. If (i) the darts $i_1$ and $i_2$ are adjacent in $f$, i.e., $\sigma \left[{\tau }^{-1}\left(i_1\right)\right]=i_2$ and (ii) $\tau \left(i_2\right)$ is a special dart, the effect of $R_{i_2}^{i_1}$ is to transform the hypermap as in Fig. 4.

Figure 4

The effect of $R_{i_2}^{i_1}$ on the hypermap in Fig. 3. The resulting code is still canonical with respect to the modified hypermap. A cnot gate between qubits not satisfying the conditions in Fig. 3 leads to a nontrivial modification of the hypermap in general.

Figure 5

Illustrating the addition of edges between $v_{\ni i}$ and $v_{\ni {\tau }^{-1}\left(i\right)}$. The additional edges are shown in color. We call them edges to distinguish them from the darts (half-edges) of the hypermap. Observe that $\partial \left(f^{\prime }\right)=d_2\left(f\right)$.

Figure 6

A hyperedge in the embedding of the hypergraph $\Gamma$. The addition of edges leads to a creation of triangular faces each containing exactly one label.

Figure 7

The deletion of the darts and the hyperedge creates a new face with exactly one special edge. On deletion of the special edge, say $i_j$, it gets merged with exactly one face $f$. The boundary of $f$ now includes the rest of the edges of $e$, i.e., excepting the special edge $i_j$.

Figure 8

The graph ${\mathsf{G}}_{\Gamma }^{\prime }$ for the hypermap in Fig. 1. It is obtained by the addition of new edges to the hypermap. Any two adjacent darts and a newly added edge form a triangle which contains exactly one label; therefore the newly added edge can be uniquely identified by a label.

Figure 9

(a) The graph ${\mathsf{G}}_{\Gamma }^{\prime \prime }$ obtained by the removal of the darts and hyperedge vertices in ${\mathsf{G}}_{\Gamma }^{\prime }$; the hyperedges are transformed to faces in ${\mathsf{G}}_{\Gamma }^{\prime \prime }$. (b) The graph ${\overline{\mathsf{G}}}_{\Gamma }$ obtained by removing the special edges, i.e., $\{$3,7$\}$; each hyperedge face $e_i$ is merged with exactly one face. ${\overline{\mathsf{G}}}_{\Gamma }$ has the same stabilizer as the hypermap-homology code of Fig. 1.