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

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

Hi! THanks for bringing this to attention. We should indeed have some multi-sector entry. Should it be a multi-sector hypergraph entry = a hypergraph product of two or more classical codes (1-chain complexes)? What do you think of also having a multi-sector homological product = tensor product of two quantum codes (2-chain complexes) entry? Or should these be merged together?

Should it be a multi-sector hypergraph entry = a hypergraph product of two or more classical codes (1-chain complexes)? What do you think of also having a multi-sector homological product = tensor product of two quantum codes (2-chain complexes) entry? Or should these be merged together?

There is some freedom in the notation as follows:

Furthermore, there are a few different notions of the homological product. For instance, Bravyi and Hastings use a simplified variant that they call the single sector homological product, whereas we will use a more standard textbook variant that Bravyi and Hastings would call a multi sector homological product [20]. Furthermore, there is some freedom in the notation and we use a convention such that the homological product in this section is manifestly equivalent to the hypergraph product of Tillich and Zemor.

This means that in some papers, the

multi-sector hypergraph entry = a hypergraph product of two or more classical codes (1-chain complexes)

is viewed as multi-sector homological product as per the above reference.

We should consider adding a separate multi-sector homological product entry which corresponds to tensor product of two classical codes (1-chain complexes). Currently, the [Homological product code entry appears to imply primarily on the single-sector formulation. It might be helpful to clarify that the existing content discusses the single-sector case, while introducing the multi-sector entry as generalization. Since the homological product naturally generalizes the hypergraph product construction, creating a dedicated multi-sector homological product entry would provide better coverage of this more general framework.

I am happy to set them up and request your review and there I will be more precise about conventions this my general message here.

P.S.: Pressed enter the message while typing the message so edited it.

Thanks. I left some comments on your current submissions.

I'll create multisector versions of both, and you can then edit them as you see fit.

We already have https://errorcorrectionzoo.org/c/square_homological_product btw.

We already have https://errorcorrectionzoo.org/c/square_homological_product btw.

Thank you for your feedback. This entry is great!

Added multisector_hypergraph: bb7a7a5

If you know, please comment or edit this code's relation to xyz_product, higher_dimensional_surface, and iterated_ramanujan. In particular, are any of these its children? I take it this code is CSS, but then xyz_product cannot be its child.

I checked xyz_product and iterated_ramanujan. Relation to higher_dimensional_surface and _toric still open.

Relation to higher_dimensional_surface and _toric still open.

The higher dimensional surface and toric codes (3D/4D) versions are children of multisector_hypergraph entry. Reference is https://arxiv.org/pdf/2408.08865 where they build the 3D/4D surface codes using multi-sector hypergraph product.

Thanks. It is indeed basically as you, but I was side-tracked by an error in higher_dimensional_surface. Now fixed: 45b496f

Because all the surface codes are only related when on a hypercubic lattice, multisector_hypergraph doesn't actually have a lot of children. One more child would be Earl's double_homological, but I'm thinking of upgrading this directly to multisector_homological instead of having a separate entry for the latter.

• I added this and upgraded multisector_hypergraph to homological, consolidating some cousins. Only thing i was confused about was "d_{ss} = \infty"?

• I acknowledged you both in this and the multisector entries.

• Would you be willing to do an augmented surface code entry formatted along the lines of this one?