Generalized homological product code

Description

Stabilizer code formulated using the CSS chain-complex construction, with chain complexes consisting of products of other chain complexes. The chain-complex construction of codes 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

  • Balanced product code — Balanced product codes result from a tensor product of two classical-code chain complexes, followed by a factoring out of certain symmetries.
  • Ramanujan-complex product code — Ramanujan codes result from a tensor product of a classical-code and a quantum-code chain complex.
  • XYZ product code — XYZ product codes result from a tensor product of three classical-code chain complexes.

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.

Zoo code information

Internal code ID: generalized_homological_product

Your contribution is welcome!

on github.com (edit & pull request)

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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} }
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/qldpc/generalized_homological_product.yml.