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Bicycle code [1]

Description

An even-length CSS code whose stabilizer generator matrix blocks are \(H_{X}=H_{Z}=(A|A^T)\), where \(A\) is a sparse circulant matrix. The fact that \(A\) commutes with its transpose ensures that the CSS condition is satisfied. Bicycle codes were among the first QLDPC code families [2; Sec. III].

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 \(\mathbb{Z}_2^n\) [3].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeQubit Stabilizer Hamiltonian-based QECC Quantum
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum
Permutationally self-dual (PSD) CSS code
Self-dual CSS codeQubit Stabilizer Hamiltonian-based QECC Quantum
Parents
Self-dual CSS code
Bicycle codes use the same check matrix \(H_X=H_Z=(A|A^T)\) for both check types [1]. Their \(X\)- and \(Z\)-type stabilizer spaces therefore coincide.
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.
Mirror codeQLDPC Qubit Stabilizer Hamiltonian-based QECC Quantum
Bicycle codes are mirror codes up to qubit permutations and Hadamard gates [4; Prop. 3.21].
Bicycle code

References

  1. [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. [2]

    Z. Babar, P. Botsinis, D. Alanis, Soon Xin Ng, and L. Hanzo, “The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure”, IEEE Access 3, 146 (2015) arXiv:1503.02372 DOI

  3. [3]

    A. Couvreur, N. Delfosse, and G. Zémor, “A Construction of Quantum LDPC Codes from Cayley Graphs”, (2013) arXiv:1206.2656

  4. [4]

    A. B. Khesin and J. Z. Lu, “Mirror codes: High-threshold quantum LDPC codes beyond the CSS regime”, (2026) arXiv:2603.05496

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Zoo Code ID: bicycle

Cite as:
“Bicycle code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/bicycle, arXiv:2606.11484
BibTeX:
@incollection{eczoo_bicycle,
title={Bicycle code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/bicycle}
}
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Permanent link:
https://errorcorrectionzoo.org/c/bicycle

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/generalized_bicycle/bicycle.yml.