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\([[13,1,5]]\) twisted toric code[1]

Description

Thirteen-qubit twisted toric code whose stabilizer tableau consists of cyclic permutations of the \(XZZX\)-type Pauli string \(XIZZIXIIIIIII\). The code can be thought of as a small twisted XZZX code [1; Exam. 11 and Fig. 3].

Transversal and Permutation-Based Gates

No non-Pauli transversal gates.

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 \([[13,1,5]]\) twisted toric code is a small twisted XZZX toric code [1; Exam. 11 and Fig. 3].
Frobenius codeStabilizer Hamiltonian-based Cyclic quantum QECC Quantum
Hermitian qubit codeStabilizer Hamiltonian-based Qubit QECC Quantum
The \([[13,1,5]]\) twisted toric code is Hermitian [2; ID 67067199e766ad364e845262].
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[13,1,5]]\) twisted toric code

References

[1]
A. A. Kovalev, I. Dumer, and L. P. Pryadko, “Design of additive quantum codes via the code-word-stabilized framework”, Physical Review A 84, (2011) arXiv:1108.5490 DOI
[2]
S. Burton, “qecdb.org: Quantum Error Correction Database”, URL
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Zoo Code ID: stab_13_1_5

Cite as:
“\([[13,1,5]]\) twisted toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_13_1_5, arXiv:2606.11484
BibTeX:
@incollection{eczoo_stab_13_1_5,
title={\([[13,1,5]]\) twisted toric 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_13_1_5}
}
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Permanent link:
https://errorcorrectionzoo.org/c/stab_13_1_5

Cite as:

“\([[13,1,5]]\) twisted toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_13_1_5, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/13/stab_13_1_5.yml.