Description
A QLDPC code such that cyclic shifts of the subsystems by a fixed \(\ell\geq 1\) leave the codespace invariant. Stabilizer generator matrices of such codes can be put into block form, where each nonzero block is a circulant matrix [1,2].Cousins
- Quasi-cyclic LDPC (QC-LDPC) code— QC-QLDPC codes are quantum counterparts of QC-LDPC codes. QC-LDPC codes can be used to make qubit QLDPC codes using various non-CSS constructions [3]. There exist explicit constructions of both whose parity-check (stabilizer generator) matrices have column weight 2 and girth 12 [4].
- Lattice stabilizer code— Lattice stabilizer codes are QLDPC codes that are invariant under translations by a lattice unit cell in the bulk.
- EA QC-QLDPC code— EA QC-QLDPC codes are entanglement-assisted versions of QC-QLDPC codes.
- Group-action lift with active orthogonality (GALA) code— GALA codes with a cyclic Abelian lift action are QC-QLDPC codes, but the framework also permits more general Abelian permutation actions [5].
- Kasai code— Kasai codes with trivial multiplier \(a=1\) reduce to the QC-QLDPC codes of Hagiwara and Imai: the lifts are commuting cyclic shifts, so orthogonality holds on all rows and the distance is capped by the row weight when a removed row is independent of the retained rows [1,6].
- Two-branch coset CSS code— The optional cyclic lift produces QC-QLDPC codes whose stabilizer generator matrices are arrays of circulant permutation blocks and are invariant under a simultaneous shift of the lift coordinates [7].
- Galois-qudit BCH code— Some Galois-qudit BCH codes are QC-QLDPC [8; Ch. 16].
Member of code lists
Primary Hierarchy
Parents
Quasi-cyclic QLDPC (QC-QLDPC) code
Children
Both stabilizer generator matrices of a pair-partition CPM code are arrays of circulant permutation matrices, so the code is invariant under simultaneous cyclic shifts of the lift coordinates, an automorphism of order equal to the prime lift size [9].
References
- [1]
- M. Hagiwara and H. Imai, “Quantum Quasi-Cyclic LDPC Codes”, 2007 IEEE International Symposium on Information Theory 806 (2007) arXiv:quant-ph/0701020 DOI
- [2]
- K. Kasai, M. Hagiwara, H. Imai, and K. Sakaniwa, “Quantum Error Correction Beyond the Bounded Distance Decoding Limit”, IEEE Transactions on Information Theory 58, 1223 (2012) arXiv:1007.1778 DOI
- [3]
- P. Tan and J. Li, “Efficient Quantum Stabilizer Codes: LDPC and LDPC-Convolutional Constructions”, IEEE Transactions on Information Theory 56, 476 (2010) DOI
- [4]
- D. Komoto and K. Kasai, “Explicit Construction of Quantum Quasi-Cyclic Low-Density Parity-Check Codes with Column Weight 2 and Girth 12”, (2025) arXiv:2501.13444
- [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]
- K. Okada and K. Kasai, “A Two-Branch Finite-Field Construction for Regular CSS LDPC Bases”, (2026) arXiv:2605.23894
- [8]
- S. A. Aly, “On Quantum and Classical Error Control Codes: Constructions and Applications”, (2008) arXiv:0812.5104
- [9]
- K. Okada and K. Kasai, “Pair-Partition Constructions for CPM-Based Quantum LDPC Codes”, (2026) arXiv:2607.14091
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2024-08-01)
Cite as:
“Quasi-cyclic QLDPC (QC-QLDPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/quasi_cyclic_qldpc, arXiv:2606.11484