Double Homological Product Code

[Fe-r-oz]

This PR creates the ECC ZOO wiki for Double Homological Product Code construction of https://arxiv.org/pdf/1805.09271.

New Codes:

• Double Homological Product Code
• By: Earl T. Campbell

Checklist:

I remembered to:

• Include relevant citations I could think of (with \cite{...})

• Create links to the other referenced codes (with
  \hyperref[code:...]{...})

• Update the relevant meta changelog fields with my user_id (see
  users/users_db.yml; add yourself in the PR if you aren't there already)

[Fe-r-oz]

Hopefully, the documentation is satisfactory.

Future work

The double HP code presented in Table 1 of arXiv:1805.09271 is a variant of multi-sector homological product codes. Another variant appears in Table III of arXiv:2407.18490, where the classical seed code is specifically quasi-cyclic -- that code construction is formulated via a tensor product of chain complexes, followed by taking the total complex of the resulting complex. A natural generalization of this approach leads to D-dimensional multi-sector HP codes, for which I would be happy to draft a dedicated wiki page after completing this one.

Given that existing literature categorizes HP codes into multi-sector and single-sector variants, I’m wondering whether the ECC Zoo wiki might formally adopt this division/convention? Or refer the single/multi-sector characterization in the documentation of HP code? It appears that the square HP code is single-sector variant because it refers to arxiv:1311.0885 which introduced to the single-sector codes.

For now, we can just consider the relation: parent id:homological_product

CC: @valbert4

[Fe-r-oz]

Maybe we can include this detail:

This specific variant of multi-sector HP code is related to the quantum LDPC codes via hypergraph product introduced by Tillich and Zemor in 0903.0566.

Reference

Furthermore, there is some freedom in the (chain complex) notation and we use a convention such that the homological (multi-sector) product in this section is manifestly equivalent to the hypergraph product of Tillich and Zemor ~ Page 12