Generalized homological product code

Description

Stabilizer code formulated in terms a chain complex consisting of some type of product of other chain complexes. The CSS-to-homology correspondence yields an interpretation of codes in terms of manifolds, thus allowing for the use of various products from topology in constructing codes. The codes participating in the product can be quantum, classical, or mixed. Products can be of more than two codes, in which case the output code need not be of CSS type (e.g., for XYZ-product codes). The simplest product is a tensor product, with more general products imposing equivalence or symmetry relations on the outputs of the tensor product. A product of two codes can be interpreted as a fiber bundle, with one element of the product being the base and the other being the fiber.

Parent

  • Quantum low-density parity-check (QLDPC) code — Homological products are a primary tool for generating QLDPC codes with favorable parameters. Typically, whenever the input codes are LDPC or QLDPC, the resulting code will be QLDPC with non geometrically local stabilizer generators.

Children

Cousin

  • Calderbank-Shor-Steane (CSS) stabilizer code — The notion of homological products arose from interpreting CSS codes in terms of chain complexes over manifolds, but some generalized products no longer yield CSS codes.
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Zoo Code ID: generalized_homological_product

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

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/properties/block/qldpc/generalized_homological_product.yml.