Two-block CSS code

Two-block CSS code[1]

Alternative Names: Two-sublattice code, Two-square-block code.

Description

An even-length 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 \([[2n,2k,d^{\prime}]]\) two-block CSS code with \(d\leq d^{\prime}\leq 2d\) 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 \([[2n,2k,d^{\prime}]]_{\mathbb{Z}_q}\) two-block CSS code with \(d\leq d^{\prime}\leq 2d\) 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

Quantum Domain

Galois-qudit Kingdom

Galois-qudit codeQECC Quantum

Galois-qudit USt code

Galois-qudit stabilizer codeStabilizer Hamiltonian-based QECC Quantum

True Galois-qudit stabilizer code

Galois-qudit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum

Multi-block CSS code

Parents

Multi-block CSS code

Two-block CSS codes are multi-block CSS codes with \(t=2\).

Two-block CSS code

Children

Actively orthogonal CSS code

Retaining the first \(J\) block rows of the parent matrices \(\hat{H}_X=[F|G]\) and \(\hat{H}_Z=[G^{T}|F^{T}]\) gives \(H_X=(A_1,B_1)\) and \(H_Z=(B_2^{T},A_2^{T})\), where \(A_1,B_1\) are the active block rows of the two block circulants and \(A_2,B_2\) their active block columns. Active orthogonality \(H_XH_Z^{T}=0\) is exactly the two-block condition \(A_1B_2-B_1A_2=0\), and the GALA sectors are defined as two-block CSS codes outright [6][5; Defs. 4, 6, and 8].

Coset-based quantum LDPC codeBB

Coset-based codes are two-block CSS codes whose commuting blocks \(\mathbf{L}(a)\) and \(\mathbf{R}(b)\) are the matrices of a group acting on the left and right of the cosets of a subgroup [7].

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
[5]: W. Yang, C. Duckering, and A. Dua, “Designer Codes from GALA: Compact, Self-Dual, and Rate-1/2 QEC on Reconfigurable Atom Arrays”, (2026) arXiv:2608.07431
[6]: K. Kasai, “Breaking the Orthogonality Barrier in Quantum LDPC Codes”, (2026) arXiv:2601.08824
[7]: A. Aydin, I. Tamo, and A. Barg, “Breaking the bicycle frame: Coset-based quantum LDPC codes”, (2026) arXiv:2606.17268

Page edit log

Victor V. Albert (2026-08-19) — most recent
Victor V. Albert (2026-07-18)
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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/stabilizer/css/two_block_quantum.yml.