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
Parents
Group-based cluster states are stabilized by group-based right- and left-multiplication error operators [1,2].
Group-based cluster-state code
Children
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
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2024-04-03)
Cite as:
“Group-based cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/group_cluster_state, arXiv:2606.11484