Description
CSS code whose properties are determined from an underlying chain complex, which often consists of some type of product of other chain complexes.
Parents
- Calderbank-Shor-Steane (CSS) stabilizer code
- Generalized homological-product code — 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.
Children
- Homological rotor code — Homological rotor codes are formulated using 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.
- Generalized homological-product qubit CSS code
- Modular-qudit color code
- Modular-qudit surface code
- Balanced product (BP) code — Balanced product codes result from a tensor product of two classical-code chain complexes, followed by a factoring out of certain symmetries.
- Distance-balanced code
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].
- Homological bosonic code — Homological CV codes utilize chain complexes in code construction, but the complexes have trivial homology.
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
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