Quasi-cyclic QLDPC (QC-QLDPC) code

Quasi-cyclic QLDPC (QC-QLDPC) code[1,2]

Description

A QLDPC code such that cyclic shifts of the subsystems by \(\ell\geq 1\) leave the codespace invariant. Such codes have circulant stabilizer generator matrices [1,2].

Rate

There exist QC-QLDPC codes that achieve the Hashing bound [3].

Cousins

Quasi-cyclic LDPC (QC-LDPC) code— QC-QLDPC codes are quantum versions of QC-LDPC codes. QC-LDPC codes can be used to make qubit QLDPC codes using various non-CSS constructions [4]. There exist explicit constructions of both whose parity-check (stabilizer generator) matrices have column weight 2 and girth 12 [5].
Galois-qudit BCH code— Some Galois-qudit BCH codes are QC-QLDPC [6; Ch. 16].

Member of code lists

Primary Hierarchy

Quantum Domain

Quantum code

Operator-algebra QECC (OAQECC)

Quantum error-correcting code (QECC)

Hamiltonian-based code

Commuting-projector Hamiltonian code

Stabilizer codeHamiltonian-based QECC Quantum

Quantum low-weight check (QLWC) codeQECC Quantum

Quantum LDPC (QLDPC) codeQECC Quantum

Parents

Quantum LDPC (QLDPC) code

Quasi-cyclic quantum codeQECC Quantum

Quasi-cyclic QLDPC (QC-QLDPC) code

Children

Lattice stabilizer codeBB Twist-defect color Twist-defect surface CDSC Kitaev surface Fracton stabilizer

Lattice stabilizer codes are QLDPC codes that are invariant under translations by a lattice unit cell.

Abelian LP codeBB

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]: D. Komoto and K. Kasai, “Quantum Error Correction near the Coding Theoretical Bound”, (2025) arXiv:2412.21171
[4]: P. Tan and J. Li, “Efficient Quantum Stabilizer Codes: LDPC and LDPC-Convolutional Constructions”, IEEE Transactions on Information Theory 56, 476 (2010) DOI
[5]: D. Komoto and K. Kasai, “Explicit Construction of Classical and Quantum Quasi-Cyclic Low-Density Parity-Check Codes with Column Weight 2 and Girth 12”, (2025) arXiv:2501.13444
[6]: S. A. Aly, “On Quantum and Classical Error Control Codes: Constructions and Applications”, (2008) arXiv:0812.5104

Page edit log

Victor V. Albert (2024-08-01) — most recent

Cite as:

“Quasi-cyclic QLDPC (QC-QLDPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quasi_cyclic_qldpc

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/stabilizer/qldpc/quasi_cyclic_qldpc.yml.