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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 category. The categories of interest are typically fusion categories, which subsume all finite groups and many state spaces associated with topological codes. 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.

Cousin

Member of code lists

Primary Hierarchy

Quantum Domain
Category Kingdom
Parents
Quantum error-correcting code (QECC)Quantum
Category-based quantum code
Children
Cage-net code
Two-gauge theory codeCDSC Kitaev surface
Hopf-algebra cluster-state code
Multi-fusion string-net codeCDSC Kitaev surface
\(G\)-enriched Walker-Wang model code
Group-based quantum codeFock-state bosonic Stabilizer CSS QLDPC Generalized homological-product Lattice stabilizer Bosonic stabilizer Quantum lattice Analog stabilizer Fracton stabilizer Qubit Hastings-Haah Floquet Fermion Self-complementary qubit Quantum QR code BB Homological Fermion-into-qubit Quantum RM code Color Twist-defect color CDSC Twist-defect surface Kitaev surface
Finite-group-based quantum codes, whose basis states are parameterized by a finite group, correspond to category-based codes for the fusion category \(Vec G\). Extensions of such categories to Lie groups can also be done [2–5] (see also [6]).

References

[1]
J. Berger and T. J. Osborne, “Perfect tangles”, (2018) arXiv:1804.03199
[2]
Fredenhagen, Klaus. “Superselection sectors with infinite statistical dimension.” Subfactors (Kyuzeso, 1993) (1994): 242-258.
[3]
R. Thorngren and Y. Wang, “Fusion Category Symmetry II: Categoriosities at c = 1 and Beyond”, (2021) arXiv:2106.12577
[4]
Kevin Walker. Lie group symmetries and non-unital higher categories. Subfactors and Fusion (2-)Categories, Banff International Research Station, 2023.
[5]
A. Marín-Salvador, “Continuous Tambara-Yamagami tensor categories”, (2025) arXiv:2503.14596
[6]
C. J. Isham, “A New Approach to Quantising Space-Time: I. Quantising on a General Category”, (2003) arXiv:gr-qc/0303060
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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.