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\([[8,1,3]]\) permutation-Hadamard code[1]

Description

A non-CSS cyclic stabilizer code that is the shortest code to realize a logical Hadamard gate via a single cyclic relabelling of the physical qubits.

Its stabilizer group is generated by the first six cyclic shifts of \(g=IIXXIYXZ\) together with \(h=-X^{\otimes8}\) [1].

A logical Pauli basis is \(\overline{X}=IIIXYIIZ\) and \(\overline{Z}=ZIIIXYII\) [1; Constr. 7].

Transversal and Permutation-Based Gates

The cyclic qubit permutation \(\rho=(12345678)\) implements a logical Hadamard gate [1].

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Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeQubit Stabilizer Hamiltonian-based QECC Quantum
Small-distance qubit stabilizer codeSmall-distance block quantum QECC Quantum
Parents
Small-distance qubit stabilizer code
Cyclic quantum codeQECC Quantum
\([[8,1,3]]\) permutation-Hadamard code

References

[1]
J. M. Koh, S. Majidy, A. Chakraborty, A. Gong, S. J. S. Tan, and N. Y. Yao, “Achieving the limits of automorphism gates”, (2026) arXiv:2609.19250
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Zoo Code ID: stab_8_1_3

Cite as:
“\([[8,1,3]]\) permutation-Hadamard code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_8_1_3, arXiv:2606.11484
BibTeX:
@incollection{eczoo_stab_8_1_3,
title={\([[8,1,3]]\) permutation-Hadamard 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_8_1_3}
}
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Permanent link:
https://errorcorrectionzoo.org/c/stab_8_1_3

Cite as:

“\([[8,1,3]]\) permutation-Hadamard code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_8_1_3, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/8/stab_8_1_3.yml.