# Group-based cluster-state code[1]

## Description

A code based on a group-based cluster state for a finite group \(G\) [1]. Such cluster states can be defined using a graph and conditional group multiplication operations. A group-based cluster state for \(G=GF(q)\) for prime-power \(q\) is called a Galois-qudit cluster state, while the state for \(G=\mathbb{Z}_q\) for positive \(q\) is called a modular-qudit cluster state.

## Gates

1D group-based cluster states for certain non-Abelian groups [2] are resources for universal MBQC.

## Parent

- Group GKP code — Group-based cluster states are stabilized by group-based error operators [1,2].

## Children

- Analog-cluster-state code — Analog cluster states are group-based cluster states for \(G=\mathbb{R}\).
- Modular-qudit cluster-state code — Group-based cluster-state codes reduce to modular-qudit cluster-state codes for \(G=\mathbb{Z}_q\).

## Cousins

- Hopf-algebra cluster-state code — Hopf-algebra cluster-state codes reduce to group-based cluster-state codes for finite groups when the Hopf algebra reduces to a finite group.
- Graph quantum code — A graph quantum code for Abelian \(G\) reduces to a group-based cluster state when its logical dimension is one [3].
- Galois-qudit CWS code — A single Galois-qudit cluster state is used to construct a modular-qudit CWS code.

## References

- [1]
- C. G. Brell, “Generalized cluster states based on finite groups”, New Journal of Physics 17, 023029 (2015) arXiv:1408.6237 DOI
- [2]
- C. Fechisin, N. Tantivasadakarn, and V. V. Albert, “Non-invertible symmetry-protected topological order in a group-based cluster state”, (2024) arXiv:2312.09272
- [3]
- Y. Hwang and J. Heo, “On the relation between a graph code and a graph state”, (2015) arXiv:1511.05647

## Page edit log

- Victor V. Albert (2024-04-03) — most recent

## Cite as:

“Group-based cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/group_cluster_state