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
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.
Generalized homological-product qubit CSS codeQuantum Reed-Muller Color BB Homological Kitaev surface
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
Page edit log
- Victor V. Albert (2022-12-04) — most recent
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