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\((5,1,2)\)-convolutional code [1]

Description

Family of quantum convolutional codes that are 1D lattice generalizations of the five-qubit perfect code, with the former’s lattice-translation symmetry being the extension of the latter’s cyclic permutation symmetry.

Their stabilizer generators for semi-open boundary conditions are \begin{align} \begin{array}{cccccccc} X & Z & I & I & I & I & I & \cdots\\ Z & X & X & Z & I & I & I & \cdots\\ I & Z & X & X & Z & I & I & \cdots\\ I & I & Z & X & X & Z & I & \cdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array}~.\tag*{$(1)$} \end{align}

Cousin

  • Twisted XZZX toric code— \((5,1,2)\)-convolutional codes (twisted XZZX toric codes) are 1D (2D) lattice extensions of the five-qubit perfect code.

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
Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum
Quantum convolutional codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
Parents
Quantum convolutional code
\((5,1,2)\)-convolutional code
Children
\([[5,1,3]]\) Five-qubit perfect code
The \((5,1,2)\)-convolutional code is a 1D lattice extension of the five-qubit perfect code, with the former’s lattice-translation symmetry being the extension of the latter’s cyclic permutation symmetry. The \((5,1,2)\)-convolutional code reduces to the five-qubit code for a five-qubit chain and periodic boundary conditions. See Ref. [2] for the first few codes in a different extension of the five-qubit perfect code.

References

  1. [1]

    H. Ollivier and J.-P. Tillich, “Description of a Quantum Convolutional Code”, Physical Review Letters 91, (2003) arXiv:quant-ph/0304189 DOI

  2. [2]

    I. A. Simakov and I. S. Besedin, “Low-overhead quantum error-correction codes with a cyclic topology”, Physical Review A 111, (2025) arXiv:2211.03094 DOI

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Zoo Code ID: stab_5_1_2_convolutional

Cite as:
“\((5,1,2)\)-convolutional code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_5_1_2_convolutional, arXiv:2606.11484
BibTeX:
@incollection{eczoo_stab_5_1_2_convolutional,
title={\((5,1,2)\)-convolutional 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/stab_5_1_2_convolutional}
}
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Permanent link:
https://errorcorrectionzoo.org/c/stab_5_1_2_convolutional

Cite as:

“\((5,1,2)\)-convolutional code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_5_1_2_convolutional, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/convolutional/stab_5_1_2_convolutional.yml.