# Graph quantum code[1]

## Description

A stabilizer code on tensor products of \(G\)-valued qudits for Abelian \(G\) whose encoding isometry is defined using a graph [1; Eqs. (4-5)]. An analytical form of the codewords exists in terms of the adjacency matrix of the graph and bicharacters of the Abelian group [1]; see [2; Eq. (1)]. A graph quantum code for \(G=\mathbb{Z}_2\) contains a cluster state as one of its codewords and reduces to a cluster state when its logical dimension is one [3].

## Protection

## Parents

- Group-based quantum code
- Stabilizer code — Graph quantum codes are a subset of stabilizer codes over \(G\)-valued qudits for Abelian \(G\) [5]. Any stabilizer code over Abelian \(G\) is locally equivalent to a graph quantum code [5] (see also [2,6]).

## Children

- \([[10,1,4]]_{G}\) tenfold code — The \([[10,1,4]]_{G}\) group code is defined using a graph [1; Prop. V.1].
- \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code — The \([[5,1,3]]_{\mathbb{Z}_q}\) code is equivalent via a single-modular-qudit Clifford circuit to a graph quantum code for the group \(G=Z_q\) [1].
- \([[5,1,3]]_q\) Galois-qudit code — The \([[5,1,3]]_q\) code is equivalent via a single-Galois-qudit Clifford circuit to a graph quantum code for the group \(G=GF(q)\) [1].

## Cousins

- Qubit stabilizer code — Graph quantum codes for \(G=\mathbb{Z}_2\) are a subset of qubit stabilizer codes [5]. Any qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_2\) via a single-qubit Clifford circuit [5] (see also [2,6]).
- Modular-qudit stabilizer code — Graph quantum codes for \(G=\mathbb{Z}_q\) are a subset of modular-qudit stabilizer codes [5]. Any modular-qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_q\) via a single-modular-qudit Clifford circuit [5] (see also [2,6]).
- Galois-qudit stabilizer code — Graph quantum codes for \(G=GF(q)\) are a subset of Galois-qudit stabilizer codes [5]. Any Galois-qubit stabilizer code is equivalent to a graph quantum code for \(G=GF(q)\) via a single-Galois-qudit Clifford circuit [5] (see also [2,6]).
- Cluster-state code — A graph quantum code for \(G=\mathbb{Z}_2\) contains a cluster state as one of its codewords and reduces to a cluster state when its logical dimension is one [3].
- Modular-qudit cluster-state code — A graph quantum code for \(G=\mathbb{Z}_q\) reduces to a modular-qudit cluster state when its logical dimension is one [3].
- Group-based cluster-state code — A graph quantum code for Abelian \(G\) reduces to a group-based cluster state when its logical dimension is one [3].
- \([[7,1,3]]\) Steane code — Four non-isomorphic graphs yield graph quantum codes that are equivalent to the Steane code under a single-qubit-Clifford circuit [2].
- \([[11,1,5]]_3\) qutrit Golay code — The qutrit Golay code can be realized as a graph quantum code [7; Fig. 2].
- \([[6,2,3]]_{q}\) code — The \([[6,2,3]]_{q}\) code family contains examples of graph quantum codes [8].
- \([[7,3,3]]_{q}\) code — The \([[7,3,3]]_{q}\) code family contains examples of graph quantum codes [8].

## References

- [1]
- D. Schlingemann and R. F. Werner, “Quantum error-correcting codes associated with graphs”, Physical Review A 65, (2001) arXiv:quant-ph/0012111 DOI
- [2]
- M. Grassl, A. Klappenecker, and M. Rotteler, “Graphs, quadratic forms, and quantum codes”, Proceedings IEEE International Symposium on Information Theory, arXiv:quant-ph/0703112 DOI
- [3]
- Y. Hwang and J. Heo, “On the relation between a graph code and a graph state”, (2015) arXiv:1511.05647
- [4]
- C. Cafaro, D. Markham, and P. van Loock, “Scheme for constructing graphs associated with stabilizer quantum codes”, (2014) arXiv:1407.2777
- [5]
- D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
- [6]
- M. Van den Nest, J. Dehaene, and B. De Moor, “Graphical description of the action of local Clifford transformations on graph states”, Physical Review A 69, (2004) arXiv:quant-ph/0308151 DOI
- [7]
- S. Prakash, “Magic state distillation with the ternary Golay code”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, (2020) arXiv:2003.02717 DOI
- [8]
- Keqin Feng, “Quantum codes [[6, 2, 3]]/sub p/ and [[7, 3, 3]]/sub p/ (p ≥ 3) exist”, IEEE Transactions on Information Theory 48, 2384 (2002) DOI

## Page edit log

- Victor V. Albert (2024-06-10) — most recent

## Cite as:

“Graph quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/graph_quantum

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/groups/graph_quantum.yml.