# Two-block CSS code[1]

Also known as Two-sublattice code, Two-square-block code.

## Description

Galois-qudit CSS code whose stabilizer generator matrices \(H_X=(A,B)\) and \(H_Z=(B^T,-A^T)\), are constructed from a pair of square commuting matrices \(A\) and \(B\).

Generalized constructions utilizing more than two blocks have also been considered [2].

## Protection

Code parameters are generally unknown, although they can be formally expressed in terms of ranks of some matrices related to \(A\) and \(B\). The corresponding expressions, as well as some upper and lower bounds on parameters are given in [3].

## Parent

## Child

- Two-block group-algebra (2BGA) codes — 2BGA codes are two-block quantum codes whose commuting matrices are constructed with the help of a group algebra.

## Cousins

- Quantum low-density parity-check (QLDPC) code — When matrices \(A\) and \(B\) have row and column weights bounded by \(W\), a two-block CSS code is a quantum LDPC code with stabilizer generators bounded by \(2W\).
- Lifted-product (LP) code — LP codes can be constructed using non-square matrices and taking a hypergraph product over a group algebra, while two-block CSS codes are constructed directly using square matrices.
- Qubit CSS code — Any \([[n,k,d]]\) stabilizer code can be mapped onto a \([[2n,2k,\geq d]]\) two-block CSS code via symplectic doubling, which preserves geometric locality of a code up to a constant factor.
- Modular-qudit CSS code — Any \([[n,k,d]]_{\mathbb{Z}_q}\) stabilizer code can be mapped onto a \([[2n,2k,\geq d]]_{\mathbb{Z}_q}\) two-block CSS code code via symplectic doubling, which preserves geometric locality of a code up to a constant factor.

## References

- [1]
- A. A. Kovalev and L. P. Pryadko, “Quantum Kronecker sum-product low-density parity-check codes with finite rate”, Physical Review A 88, (2013) arXiv:1212.6703 DOI
- [2]
- N. Willenborg et al., “Dihedral Quantum Codes”, (2024) arXiv:2310.15092
- [3]
- H.-K. Lin and L. P. Pryadko, “Quantum two-block group algebra codes”, (2023) arXiv:2306.16400

## Page edit log

- Victor V. Albert (2023-10-16) — most recent
- Leonid Pryadko (2023-10-10)

## Cite as:

“Two-block CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/two_block_quantum