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Galois-qudit color code[1]

Alternative Names: \(\mathbb{F}_q\)-qudit color code.

Description

Extension of the color code to 2D lattices of Galois qudits.

Cousin

  • Abelian quantum-double stabilizer code— A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [2,4–7][3; Sec. 5.3]. Galois-qudit color codes yield Abelian quantum-double codes with Abelian-group topological order via this decomposition.

Member of code lists

Primary Hierarchy

Quantum Domain
Galois-qudit Kingdom
Galois-qudit codeQECC Quantum
Galois-qudit USt code
Galois-qudit stabilizer codeStabilizer Hamiltonian-based QECC Quantum
True Galois-qudit stabilizer code
Galois-qudit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Galois-qudit CSS code
2D lattice stabilizer codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
Generalized homological-product CSS codeGeneralized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Abelian topological codeTopological Hamiltonian-based QECC Quantum
A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [2,4–7][3; Sec. 5.3]. Galois-qudit color codes yield Abelian quantum-double codes with Abelian-group topological order via this decomposition.
Quantum-double codeTopological Hamiltonian-based QECC Quantum
A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [2,4–7][3; Sec. 5.3]. Galois-qudit color codes yield Abelian quantum-double codes with Abelian-group topological order via this decomposition.
Galois-qudit color code
Children
2D color code
Galois-qudit 2D color codes reduce to 2D color codes for \(q=2\).

References

[1]
P. Sarvepalli, “Topological color codes over higher alphabet”, 2010 IEEE Information Theory Workshop 1 (2010) 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]
D. Gottesman, Surviving as a Quantum Computer in a Classical World (2024) URL
[5]
Q. T. Nguyen, “Good binary quantum codes with transversal CCZ gate”, (2024) arXiv:2408.10140
[6]
L. Golowich and V. Guruswami, “Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates”, (2024) arXiv:2408.09254
[7]
M. Heinrich, “On stabiliser techniques and their application to simulation and certification of quantum devices”, PhD thesis, Universität zu Köln, 2021.
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Zoo Code ID: galois_color

Cite as:
“Galois-qudit color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/galois_color, arXiv:2606.11484
BibTeX:
@incollection{eczoo_galois_color,
title={Galois-qudit color code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/galois_color}
}
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Permanent link:
https://errorcorrectionzoo.org/c/galois_color

Cite as:

“Galois-qudit color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/galois_color, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/stabilizer/topological/galois_color.yml.