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.

Parents

Child

Cousins

  • 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
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Zoo Code ID: group_cluster_state

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
BibTeX:
@incollection{eczoo_group_cluster_state, title={Group-based cluster-state code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/group_cluster_state} }
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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

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