Topological subsystem codes from graphs and hypergraphs

Topological subsystem codes were proposed by Bombin based on 3-face-colorable cubic graphs. Suchara, Bravyi, and Terhal generalized this construction and proposed a method to construct topological subsystem codes using 3-valent hypergraphs that satisfy certain constraints. Finding such hypergraphs and computing their parameters, however, is a nontrivial task. We propose families of topological subsystem codes that were previously not known. In particular, our constructions give codes which cannot be derived from Bombin's construction. We also study the error recovery schemes for the proposed subsystem codes and give detailed schedules for the syndrome measurement that take advantage of the 2-locality of the gauge group. The study also leads to a new and general construction for color codes.

Figure 1

Inserting rank-3 edges in the faces of ${\Gamma }_2$ to obtain the hypergraph ${\Gamma }_h$. The rank-3 edges correspond to triangles. (a) A face $f$ in $\mathrm{F}$. (b) Inserting rank-3 edges in $f$ by promoting the $b$ edges to rank-3 edges. (c) Inserting rank-3 edges in $f$ by promoting the $g$ edges to rank-3 edges.

Figure 2

Illustrating Construction 5 from a 2-colex on a torus. Opposite sides are identified. (a) 2-colex. (b) Subsystem code. (c) Encoder $X$ operator (in bold). (d) Encoder $Z$ operator (in bold).

Figure 3

Stabilizer generators from a face in $\mathrm{F}$ for the subsystem codes of Construction 4; one of them is dependent. (a) A hypercycle ${\sigma }_1$ in $f$ (shown in bold edges) consisting of only rank-2 edges. (b) A hypercycle ${\sigma }_2$ in $f$ (shown in bold edges) with both rank-2 and rank-3 edges. (c) A dependent hypercycle ${\sigma }_3$ which is a combination of ${\sigma }_1$ and ${\sigma }_2$ over ${\mathbb{F}}_2$. We shall view ${\sigma }_1$ and ${\sigma }_2$ as the two independent hypercycles associated with $f$.

Figure 4

Decomposing ${\sigma }_i$ in Fig. 3 so that $W\left({\sigma }_i\right)$ can be generated using the elements of $\mathcal{G}$. (a) Decomposing the hypercycle ${\sigma }_1$. (b) Decomposing the hypercycle ${\sigma }_2$. In each of the above $W\left({\sigma }_i\right)$ can be generated as the product of link operators corresponding to the bold edges. Note that these decompositions are with respect to the link operators of the derived graph ${\overline{\Gamma }}_h$ while the cycles are defined with respect to the hypergraph ${\Gamma }_h$.

Figure 5

A cycle ${\sigma }_1$ of rank-2 edges in the boundary of $f$, shown in bold, when $f$ has no rank-3 edges in its boundary. Some of the edges incident on $f$ may be rank-3 but none in the boundary are.

Figure 6

Stabilizer generators for a face which has no rank-3 edges in its boundary when $\mathrm{F}={\mathrm{F}}_{\mathrm{r}}$ and $f\notin \mathrm{F}$. (a) A cycle ${\sigma }_1$ of rank-2 edges in the boundary of $f$, shown in bold; note that a rank-3 edge is incident on every vertex of $f$ unlike Fig. 5. (b) A cycle ${\sigma }_2$ of rank-2 and rank-3, shown in bold; ${\sigma }_2$ differs from the cycle in Fig. 3b, in that the “outer” rank-2 edges may be either $r$ or $g$. (c) Decomposing the hypercycle ${\sigma }_2$ so that $W\left({\sigma }_2\right)$ can be generated using the elements of $\mathcal{G}$. Note the decomposition refers to ${\overline{\Gamma }}_h$.

Figure 7

Stabilizer generators for a $v$ face in $\mathrm{F}$, for the subsystem codes in Theorem 6. (a) A hypercycle ${\sigma }_1$ for a $v$ face in $\mathrm{F}$. The decomposition for $W\left({\sigma }_1\right)$ is same as in Fig. 7a. (b) A cycle ${\sigma }_2$ of rank-2 and rank-3 edges, shown in bold. (c) Decomposing ${\sigma }_2$ so that $W\left({\sigma }_2\right)$ can be generated using the elements of $\mathcal{G}$.

Figure 8

Stabilizer generators for a $v$ face in ${\mathrm{F}}_{\mathrm{r}}\setminus \mathrm{F}$, for the subsystem codes in Theorem 6. (i) ${\sigma }_1=\partial \left(f\right)$ (not shown) and (ii) ${\sigma }_2$ (in bold) consists of the rank-3 edges of all the adjacent $f$ faces in $\mathrm{F}$ adjacent through an $e$ face and the rank-2 edges connecting them. The decomposition for $W\left({\sigma }_2\right)$ is shown in Fig. 9.

Figure 9

Decomposition for $W\left({\sigma }_2\right)$. The product of the link operators shown in bold edges gives $W\left({\sigma }_2\right)$.

Figure 10

If $b$ defined in Eq. (23) has support on one rank-3 edge of a $v$ face, then it has support on all the rank-3 edges of the $v$ face. Furthermore, $\{a_{w_0},a_{w_2},a_{w_4},...\}\cup \{a_{v_1},a_{v_3},...\}$ are all nonzero or $\{a_{w_1},a_{w_3},...\}\cup \{a_{v_0},a_{v_2},...\}$ are all nonzero.

Figure 11

For the hypergraph in Theorem 5, if $b$ has support on one rank-3 edge, then it has support on all rank-3 edges in ${\Gamma }_h$.

Figure 12

For the hypergraph in Theorem 6, if $b$ has support on one rank-3 edge, then it has support on all rank-3 edges in ${\Gamma }_h$.

Figure 13

If $u\notin O_{\sigma }$, and $X_u{X}_i$, $Y_u{Y}_j$, and $Z_u{Z}_v$ act on $u$, then $v\notin \mathrm{supp}\left(O_{\sigma }\right)$. Suppose now that $u$ is in the support of a rank-3 edge $e=(u,v,w)\notin \mathrm{supp}\left(O_{\sigma }\right)$. The solid edges indicate the link operators which are in the support of $K^{(x,y)}{K}^{\left(z\right)}$, while the dashed edges do not. Then $e$ must occur in two cycles, namely, ${\sigma }_2^{f_a}$ and ${\sigma }_2^{f_b}$, one which encloses $f_a$, and another which encloses $f_b$. Note that $W\left({\sigma }_2^{f_a}\right)W\left({\sigma }_2^{f_b}\right)$ is in the stabilizer and it has no support on $e$. If the same cycle encloses both $f_a$ and $f_b$, then the edge occurs twice in that cycle and $W\left({\sigma }_2^{f_a}\right)$ has no support on $e$.

Figure 14

Noninterference of syndrome measurement. The faces $f_a$, $f_b$, $f_c$ have stabilizer generators that have overlapping support. The edges are labeled with the round in which they are measured; the subscripts indicate the faces with which the edge is associated. Thus $3_a$ indicates that this edge should be measured in the third round and it is used in the stabilizer generator $W\left({\sigma }_2\right)$ of the face $f_a$.

Figure 15

A subsystem code in which some of the stabilizer generators are nonlocal. This is derived from the color code on a torus from a square octagon lattice. Opposite sides are identified. (a) 2-colex. (b) Subsystem code.

Figure 16

The Petersen graph although cubic and satisfying (H1)–(H4), does not lead to a subsystem code via Construction 3; it is not 3-edge colorable.