Two-block CSS code[1]
Description
Galois-qudit CSS code whose stabilizer generator matrices \(H_X=(A_1,B_1)\) and \(H_Z=(B^T_2,-A^T_2)\), are constructed from four matrices satisfying \(A_1 B_2 - B_1 A_2 = 0\). In the case the two pairs are equal, we have \(H_X=(A,B)\) and \(H_Z=(B^T,-A^T)\), constructed from a pair of square commuting matrices \(A\) and \(B\).
Generalized constructions utilizing more than two blocks yield multi-block CSS codes [2,3]. Two-block CSS codes are not hypergraph-product codes in general; they reduce to hypergraph products only in special cases, such as 2BGA codes whose two support subgroups intersect trivially [4].
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 [4].Cousins
- 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.
- Three-block CSS code— Three-block (two-block) CSS codes are constructed from three (two) commuting square matrices.
Member of code lists
Primary Hierarchy
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]
- H.-K. Lin, P. K. Lim, A. A. Kovalev, and L. P. Pryadko, “Abelian multi-cycle codes for single-shot error correction”, (2026) arXiv:2506.16910
- [3]
- V. Menon, J. P. Bonilla Ataides, R. Mehta, A. Gu, D. B. Tan, and M. D. Lukin, “Magic Tricycles: Efficient Magic-State Generation with Finite Block-Length Quantum LDPC Codes”, Physical Review X 16, (2026) arXiv:2508.10714 DOI
- [4]
- H.-K. Lin and L. P. Pryadko, “Quantum two-block group algebra codes”, (2023) arXiv:2306.16400
Page edit log
- Victor V. Albert (2026-07-18) — most recent
- Victor V. Albert (2026-06-08)
- Victor V. Albert (2023-10-16)
- Leonid Pryadko (2023-10-10)
Cite as:
“Two-block CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/two_block_quantum, arXiv:2606.11484