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Generalized homological-product CSS code

Description

CSS code whose properties are determined from an underlying chain complex, which often consists of some type of product of other chain complexes.

Cousins

  • Tanner code— Tanner codes can be generalized to sheaf codes, whose local codes satisfy a certain hierarchy. This allows for a way to formulate and understand many generalized homological-product CSS codes [1] and LTCs [2].
  • Hayden-Nezami-Salton-Sanders bosonic code— Hayden-Nezami-Salton-Sanders codes utilize chain complexes in code construction, but the complexes have trivial homology.
  • Tiger code— Tiger codes are CSS-like multi-mode bosonic non-stabilizer codes constructed from chain complexes over the integers [3]. The homology group of the logical operators has a torsion component because the chain complexes are defined over the ring of integers, which yields codes with finite logical dimension.
  • Homological number-phase code— Homological number-phase codes are non-stabilizer codes constructed from chain complexes over the integers. The homology group of the logical operators has a torsion component because the chain complexes are defined over the ring of integers, which yields codes with finite logical dimension.

Primary Hierarchy

Parents
The notion of homological products arises from interpreting CSS codes in terms of chain complexes over manifolds, but some products no longer yield CSS codes.
Generalized homological-product CSS code
Children
Homological rotor codes are rotor CSS codes constructed from chain complexes over the integers in an extension of the qubit CSS-to-homology correspondence to rotors. The homology group of the logical operators has a torsion component because the chain complexes are defined over the ring of integers, which yields codes with finite logical dimension. Products of chain complexes can also yield rotor codes.
Integer-homology bosonic CSS codes are constructed from chain complexes over the integers and realize homological rotor codes out of continuous displacement stabilizer groups. The homology group of the logical operators has a torsion component because the chain complexes are defined over the ring of integers, which yields codes with finite logical dimension.
Balanced product codes result from a tensor product of two classical-code chain complexes, followed by a factoring out of certain symmetries.

References

[1]
P. Panteleev and G. Kalachev, “Maximally Extendable Sheaf Codes”, (2024) arXiv:2403.03651
[2]
U. A. First and T. Kaufman, “Cosystolic Expansion of Sheaves on Posets with Applications to Good 2-Query Locally Testable Codes and Lifted Codes”, (2024) arXiv:2403.19388
[3]
Y. Xu, Y. Wang, C. Vuillot, and V. V. Albert, “Letting the tiger out of its cage: bosonic coding without concatenation”, (2024) arXiv:2411.09668
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Zoo Code ID: generalized_homological_product_css

Cite as:
“Generalized homological-product CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/generalized_homological_product_css
BibTeX:
@incollection{eczoo_generalized_homological_product_css, title={Generalized homological-product CSS code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/generalized_homological_product_css} }
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Cite as:

“Generalized homological-product CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/generalized_homological_product_css

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/stabilizer/qldpc/generalized_homological_product_css.yml.