Graph quantum code

Graph quantum code[1]

Alternative names: Abelian group-qudit CWS code.

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

The Knill-Laflamme conditions have a graph-based analogue [1]; see Ref. [4; Sec. III].

Cousins

Galois-qudit stabilizer code— Graph quantum codes for \(G=\mathbb{F}_q\) are a subset of Galois-qudit stabilizer codes [5]. Any Galois-qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{F}_q\) via a single-Galois-qudit Clifford circuit [5] (see also [2,6]).
Galois-qudit CWS code— A single Galois-qudit cluster state is used to construct a modular-qudit CWS code.
\([[5,1,3]]_q\) 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 \(G=\mathbb{F}_q\) [1].
\([[6,2,3]]_{q}\) code— The \([[6,2,3]]_{q}\) code family contains examples of graph quantum codes [7].
\([[7,3,3]]_{q}\) code— The \([[7,3,3]]_{q}\) code family contains examples of graph quantum codes [7].

Member of code lists

Primary Hierarchy

Quantum Domain

Group quantum Kingdom

Group-based quantum codeQECC Quantum

Stabilizer codeHamiltonian-based QECC Quantum

Parents

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]).

Group-based cluster-state codeQECC Quantum

Group-based cluster-state codes reduce to graph codes for Abelian \(G\).

Graph quantum code

Children

Rotor cluster-state code

Graph quantum codes for \(G=\mathbb{Z}\) reduce to rotor cluster-state codes.

\([[10,1,4]]_{G}\) tenfold code

The \([[10,1,4]]_{G}\) group code is defined using a graph [1; Prop. V.1].

Analog cluster-state code

Graph quantum codes for \(G=\mathbb{R}\) reduce to analog cluster-state codes.

Modular-qudit cluster-state code

Graph quantum codes for \(G=\mathbb{Z}_q\) reduce to modular-qudit cluster-state codes.

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, 45 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]: 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/stabilizer/graph_quantum.yml.