Abelian LP code

Abelian LP code[1,2]

Description

A lifted-product code whose lift group \(G\) is Abelian. The one-by-one (scalar) case, built from a pair of single group-algebra elements \(a,b \in \mathbb{F}_q[G]\), yields Abelian 2BGA codes. The case of \(G\) being a cyclic group is a GB code (a.k.a. a quasi-cyclic LP code) [2; Sec. III.E]. A particular family with \(G=\mathbb{Z}_{\ell}\) yields codes with parameters \([[n,k=\Theta(\log n),d=\Theta(n/\log n)]]\) [2].

The Abelian LP construction has been adapted to accommodate noise bias, yielding bias-tailored LP codes [3]. See Refs. [1,2,4,5] for other explicit examples.

Protection

Permanent bounds on the distance of quasi-cyclic LDPC codes [6,7] extend to base codes over any Abelian \(G\) [8; App. G]. For binary codes whose base matrices satisfy the full-rank condition of LP codes, they bound the LP-code distance [8; Thm. 8]. In particular, a one-row base matrix bounds the distance by the sum of the weights of any two entries that are not both zero [8; Prop. 4]. Monomial \(m\times n\) base matrices bound the distance by \((m+1)!\) [6][8; Cor. 1]. Under the same condition, codes \(LP(A,A)\) with a \(2\times 4\) base matrix and check weight at most nine have distance at most 14 [8; Thm. 13].

Rate

For cyclic groups \(G=\mathbb{Z}_{\ell}\) with \(\ell=\Theta(n/\log n)\), quasi-cyclic expander LP codes yield families with parameters \([[n,k=\Theta(\log n),d=\Theta(n/\log n)]]\) [2]. Related balanced-product reformulations and other explicit Abelian LP constructions appear in [9,10].

Gates

Transversal dimension jump, a code-switching protocol between a 3D LP code \(LP(A,B,C)\) over \(\mathbb{F}_2[G]\) and any of its 2D component codes, such as \(LP(A,B)\) [11]. The 2D code embeds as a slice of the 3D code via an inclusion chain map [11]. Physical CNOTs controlled by the 3D qubits of the slice and targeting their 2D partners form a one-way transversal logical CNOT [11]. Whenever the induced map on logical \(Z\) operators is injective, one-bit teleportation through this CNOT switches logical qubits between the two codes [11].

Decoding

Ensemble BP decoder for codes without short cycles of length 4 [12].Efficient decoder correcting order \(\Theta(n/\log n)\) errors [13].

Cousins

Quasi-cyclic LDPC (QC-LDPC) code— QC-LDPC codes can be lifted to yield various Abelian LP codes [4,12,14]. Conversely, the Abelian LP construction yields notable families of QC-LDPC codes [10].
Finite-geometry LDPC (FG-LDPC) code— FG-LDPC codes can be used to construct Abelian LP codes [12].
Expander LP code— For cyclic groups \(G=\mathbb{Z}_{\ell}\) with \(\ell=\Theta(n/\log n)\), quasi-cyclic expander LP codes yield families with parameters \([[n,k=\Theta(\log n),d=\Theta(n/\log n)]]\) [2]. Related balanced-product reformulations and other explicit Abelian LP constructions appear in [9,10].
Asymmetric quantum code (AQC)— The Abelian LP construction has been adapted to accommodate noise bias, yielding bias-tailored LP codes [3].
Concatenated GKP code— GKP codes have been concatenated with Abelian LP codes [4] that are in turn based on QC-LDPC codes [15]. Concatenating Abelian LP codes with GKP codes can surpass the CSS Hamming bound [4].

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

Balanced product (BP) codeGeneralized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum

Lifted-product (LP) code

Parents

Lifted-product (LP) code

Abelian LP code

Children

Abelian two-block group-algebra codeBB

Abelian 2BGA codes are the one-by-one (scalar) Abelian LP codes.

References

[1]: P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021) arXiv:1904.02703 DOI
[2]: P. Panteleev and G. Kalachev, “Quantum LDPC Codes With Almost Linear Minimum Distance”, IEEE Transactions on Information Theory 68, 213 (2022) arXiv:2012.04068 DOI
[3]: J. Roffe, L. Z. Cohen, A. O. Quintavalle, D. Chandra, and E. T. Campbell, “Bias-tailored quantum LDPC codes”, Quantum 7, 1005 (2023) arXiv:2202.01702 DOI
[4]: N. Raveendran, N. Rengaswamy, F. Rozpędek, A. Raina, L. Jiang, and B. Vasić, “Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound”, Quantum 6, 767 (2022) arXiv:2111.07029 DOI
[5]: Q. Xu, J. P. B. Ataides, C. A. Pattison, N. Raveendran, D. Bluvstein, J. Wurtz, B. Vasic, M. D. Lukin, L. Jiang, and H. Zhou, “Constant-Overhead Fault-Tolerant Quantum Computation with Reconfigurable Atom Arrays”, (2023) arXiv:2308.08648
[6]: R. Smarandache and P. O. Vontobel, “Quasi-Cyclic LDPC Codes: Influence of Proto- and Tanner-Graph Structure on Minimum Hamming Distance Upper Bounds”, IEEE Transactions on Information Theory 58, 585 (2012) arXiv:0901.4129 DOI
[7]: D. G. M. Mitchell, R. Smarandache, and D. J. Costello, “Quasi-Cyclic LDPC Codes based on Pre-Lifted Protographs”, (2014) arXiv:1407.5364
[8]: A. Bhardwaj, M. Ma, N. Meister, R. King, D. Bluvstein, J. Preskill, M. Cain, Q. Xu, and H.-Y. Huang, “High-rate qLDPC processors”, (2026) arXiv:2607.28795
[9]: N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021) arXiv:2012.09271 DOI
[10]: F. G. Jeronimo, T. Mittal, R. O’Donnell, P. Paredes, and M. Tulsiani, “Explicit Abelian Lifts and Quantum LDPC Codes”, (2021) arXiv:2112.01647
[11]: C. Li, J. Preskill, and Q. Xu, “Transversal dimension jump for product qLDPC codes”, (2026) arXiv:2510.07269
[12]: S. Miao, J. Mandelbaum, H. Jäkel, and L. Schmalen, “A Joint Code and Belief Propagation Decoder Design for Quantum LDPC Codes”, (2024) arXiv:2401.06874
[13]: L. Golowich and V. Guruswami, “Decoding Quasi-Cyclic Quantum LDPC Codes”, (2024) arXiv:2411.04464
[14]: T. R. Scruby, T. Hillmann, and J. Roffe, “High-Threshold, Low-Overhead and Single-Shot Decodable Fault-Tolerant Quantum Memory”, PRX Quantum 7, (2026) arXiv:2406.14445 DOI
[15]: M. P. C. Fossorier, “Quasi-Cyclic Low-Density Parity-Check Codes From Circulant Permutation Matrices”, IEEE Transactions on Information Theory 50, 1788 (2004) DOI

Page edit log

Victor V. Albert (2026-09-30) — most recent
Victor V. Albert (2026-09-26)
Victor V. Albert (2026-07-18)
Victor V. Albert (2026-06-08)
Victor V. Albert (2024-05-06)

Cite as:

“Abelian LP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/abelian_lifted_product, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/stabilizer/qldpc/balanced_product/lp/matrix/abelian_lifted_product.yml.