Generalized homological-product code 


Stabilizer code whose properties are determined from an underlying chain complex, which often consists of some type of product of other chain complexes. The Qubit 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. Homology can be used to design codes for qubits, modular qudits, Galois qudits, as well as rotors; most codes are CSS codes. However, products can be of more than two underlying codes, in which case the output code need not be CSS (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.


  • 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.


  • Generalized homological-product CSS 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.
  • XYZ product code — XYZ product codes result from a tensor product of three classical-code chain complexes.


  • Cycle code — Cycle codes have been known in classical coding theory, and have been rediscovered in the quantum context; see Ref. [1] for a brief exposition.


G. Zémor, “On Cayley Graphs, Surface Codes, and the Limits of Homological Coding for Quantum Error Correction”, Lecture Notes in Computer Science 259 (2009) DOI
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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.
@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={} }
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Cite as:

“Generalized homological-product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.