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\([[7,1,3]]\) XZZX cyclic code[1]

Description

A \([[7,1,3]]\) cyclic non-CSS code whose stabilizer generators are the seven cyclic shifts of the weight-four Pauli string \(XZIZXII\), any six of which are independent. The non-identity support of this generator is \(XZZX\) (at positions 0, 1, 3, 4), making this a member of the twisted XZZX code family. It is one of sixteen distinct indecomposable \([[7,1,3]]\) codes [2].

A stabilizer tableau for the code is given by [3; ID 227] \begin{align} \begin{array}{ccccccc} X & Z & I & Z & X & I & I \\ I & X & Z & I & Z & X & I \\ I & I & X & Z & I & Z & X \\ X & I & I & X & Z & I & Z \\ Z & X & I & I & X & Z & I \\ I & Z & X & I & I & X & Z \end{array}~. \tag*{(1)}\end{align}

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum
Twist-defect surface codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
XZZX surface code
Twisted XZZX toric codeCyclic quantum QECC Quantum
Parents
Twisted XZZX toric code
The \([[7,1,3]]\) XZZX cyclic code is a cyclic non-CSS code whose generators are the cyclic shifts of the weight-four Pauli string \(XZIZXII\), which has \(XZZX\) support.
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[7,1,3]]\) XZZX cyclic code

References

[1]
A. Robertson, C. Granade, S. D. Bartlett, and S. T. Flammia, “Tailored Codes for Small Quantum Memories”, Physical Review Applied 8, (2017) arXiv:1703.08179 DOI
[2]
S. Yu, Q. Chen, and C. H. Oh, “Graphical Quantum Error-Correcting Codes”, (2007) arXiv:0709.1780
[3]
Qiskit Community, “Qiskit QEC framework”, URL
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Zoo Code ID: xzzx_7_1_3

Cite as:
“\([[7,1,3]]\) XZZX cyclic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/xzzx_7_1_3
BibTeX:
@incollection{eczoo_xzzx_7_1_3, title={\([[7,1,3]]\) XZZX cyclic code}, booktitle={The Error Correction Zoo}, year={2026}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/xzzx_7_1_3} }
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Permanent link:
https://errorcorrectionzoo.org/c/xzzx_7_1_3

Cite as:

“\([[7,1,3]]\) XZZX cyclic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/xzzx_7_1_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/7/xzzx_7_1_3.yml.