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Subsystem group-based quantum code

Root code for the Group quantum Kingdom

Description

Group-based quantum code whose codespace admits a tensor-product decomposition into logical and gauge factors.

Cousins

  • Group-based quantum code— Subsystem group-based quantum codes reduce to (subspace) group-based quantum codes when there is no gauge subsystem.
  • Quantum-double code— Subsystem versions of quantum-double codes have been formulated [1].

Member of code lists

Primary Hierarchy

Quantum Domain
Group quantum Kingdom
Parents
Subsystem QECCQuantum
Subsystem group-based quantum code
Children
Subsystem stabilizer codeLattice subsystem
Subsystem modular-qudit codeSubsystem qubit
Subsystem group quantum codes whose physical spaces are constructed using modular-integer groups \(\mathbb{Z}_q\) are subsystem modular-qudit codes.
Subsystem Galois-qudit codeSubsystem qubit
A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [2]; see Sec. 5.3 of Ref. [3]. Interpreted this way, subsystem Galois-qudit codes are subsystem group quantum codes whose physical spaces are constructed using Galois fields \(\mathbb{F}_q\) as groups. More general versions of such qudits can be valued in a Galois ring [4], over which there also exists a Fourier transform [5].

References

[1]
P. Kumar, “A Class of Quantum Double Subsystem Codes”, (2011) DOI
[2]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
[3]
A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
[4]
R. Zucchini, “Calibrated hypergraph states: II calibrated hypergraph state construction and applications”, (2025) arXiv:2501.18968
[5]
Y. Zhang, “Quantum Fourier Transform Over Galois Rings”, (2009) arXiv:0904.2560
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Zoo Code ID: subsystem_group_quantum

Cite as:
“Subsystem group-based quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/subsystem_group_quantum
BibTeX:
@incollection{eczoo_subsystem_group_quantum, title={Subsystem group-based quantum code}, booktitle={The Error Correction Zoo}, year={2026}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/subsystem_group_quantum} }
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Permanent link:
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Cite as:

“Subsystem group-based quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/subsystem_group_quantum

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