Bicycle code[1]
Description
A CSS code whose stabilizer generator matrix blocks are \(H_{X}=H_{Z}=(A|A^T)\), where \(A\) is a circulant matrix. The fact that \(A\) commutes with its transpose ensures that the CSS condition is satisfied. Bicycle codes are the first QLDPC codes.
A notable example is an \([[2^n,2^{(n+1)/2},2^{(n-1)/2}]]\) code constructed from the repetition code and the Cayley graph of \(\mathbb{Z}_2^n\) [2].
Cousin
- Qubit QLDPC code— Bicycle codes are the first QLDPC codes [1].
Primary Hierarchy
Generalized homological-product qubit CSS codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Generalized bicycle (GB) codeCSS Generalized homological-product Lattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
A GB code whose circulants satisfy \(B = A^T\) reduces to a bicycle code.
Bicycle code
References
- [1]
- D. J. C. MacKay, G. Mitchison, and P. L. McFadden, “Sparse-Graph Codes for Quantum Error Correction”, IEEE Transactions on Information Theory 50, 2315 (2004) arXiv:quant-ph/0304161 DOI
- [2]
- A. Couvreur, N. Delfosse, and G. Zémor, “A Construction of Quantum LDPC Codes from Cayley Graphs”, (2013) arXiv:1206.2656
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2024-05-09)
Cite as:
“Bicycle code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/bicycle, arXiv:2606.11484