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Group-based cluster-state code[1]

Description

A code based on a group-based cluster state for a group \(G\) [1]. Such cluster states can be defined using a graph and conditional group multiplication operations. A group-based cluster state for \(G=\mathbb{F}_q\) 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.

Cousin

  • 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.

Member of code lists

Primary Hierarchy

Quantum Domain
Group quantum Kingdom
Group-based quantum codeQECC Quantum
Parents
Group-based quantum code
Group-based cluster states are stabilized by group-based right- and left-multiplication error operators [1,2].
Group-based cluster-state code
Children
Graph quantum code
Group-based cluster-state codes reduce to graph codes for Abelian \(G\).

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, “Noninvertible Symmetry-Protected Topological Order in a Group-Based Cluster State”, Physical Review X 15, (2025) arXiv:2312.09272 DOI
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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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https://errorcorrectionzoo.org/c/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

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