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Category-based quantum code

Root code for the Category Kingdom

Description

Encodes a finite-dimensional logical Hilbert space into a physical Hilbert space associated with a finite category. Codes on modular fusion categories are often associated with a particular topological quantum field theory (TQFT), as the data of such theories is described by such categories.

Cousins

Member of code lists

Primary Hierarchy

Quantum Domain
Category Kingdom
Parents
Finite-dimensional quantum error-correcting codeQECC Quantum
Category-based quantum code
Children
Two-gauge theory codeCDSC Kitaev surface
Hopf-algebra cluster-state code
Multi-fusion string-net codeCDSC Kitaev surface
\(G\)-enriched Walker-Wang model code
Modular-qudit codeQubit Fermion Self-complementary qubit Hastings-Haah Floquet Fermion-into-qubit Twist-defect color Twist-defect surface Fracton stabilizer Color BB Homological Quantum Reed-Muller CDSC Kitaev surface
Category quantum codes whose physical spaces are constructed using the group \(\mathbb{Z}_q\) as the category are modular-qudit codes.
Galois-qudit codeQubit Hastings-Haah Floquet Fermion Self-complementary qubit Fermion-into-qubit Twist-defect color CDSC Twist-defect surface Color Homological BB Kitaev surface Quantum QR code Quantum Reed-Muller
Category quantum codes whose physical spaces are constructed using the group \(GF(q)\) as the category are Galois-qudit codes. More general versions of such qudits can be valued in a Galois ring [2], over which there also exists a Fourier transform [3].

References

[1]
J. Berger and T. J. Osborne, “Perfect tangles”, (2018) arXiv:1804.03199
[2]
R. Zucchini, “Calibrated hypergraph states: II calibrated hypergraph state construction and applications”, (2025) arXiv:2501.18968
[3]
Y. Zhang, “Quantum Fourier Transform Over Galois Rings”, (2009) arXiv:0904.2560
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Zoo Code ID: category_quantum

Cite as:
“Category-based quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/category_quantum
BibTeX:
@incollection{eczoo_category_quantum, title={Category-based quantum code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/category_quantum} }
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Permanent link:
https://errorcorrectionzoo.org/c/category_quantum

Cite as:

“Category-based quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/category_quantum

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/categories/category_quantum.yml.