\([[9,2,3]]\) automorphism-maximal code[1]
Description
A non-CSS stabilizer code on which single-qubit Clifford gates together with qubit permutations realize the largest logical group attainable on two logical qubits. It is the smallest distance-three stabilizer code with this property.
A stabilizer tableau for the code is [1][2; ID 80585] \begin{align} \begin{array}{ccccccccc} X & Z & I & I & Y & X & I & I & I \\ I & I & X & Z & I & I & Y & X & I \\ Y & X & I & I & I & X & I & I & Z \\ I & I & Y & X & I & I & I & X & Z \\ Y & Y & I & I & Z & Y & I & I & I \\ I & I & Y & Y & I & I & Z & Y & I \\ X & X & X & X & Z & I & Z & I & X \end{array}~. \tag*{(1)}\end{align} The logical basis is \(\overline{X}_1=IIZIYXIZI\), \(\overline{X}_2=IYIIIYYXI\), \(\overline{Z}_1=IXIXXIXII\), and \(\overline{Z}_2=IYIYIYIYI\) [1].
Transversal and Permutation-Based Gates
Single-qubit Clifford gates together with qubit permutations realize the logical group \(O^{+}(4,2)\) of order 72 [1]. In the displayed logical basis, this group is generated by \(H_1\), \(H_2\), and \(\mathrm{CZ}_{12}\). A different logical basis gives the generators \(H_1\), \(S_1\), and \(\mathrm{SWAP}_{12}\) [1; Tab. X]. This is the largest logical group attainable this way at \(k=2\), exceeding the maximum of 48 attainable by a CSS code [1].Cousin
- \([[6,2,2]]\) \(C_6\) code— A non-CSS code with the same parameters as the \([[6,2,2]]\) \(C_6\) code attains the logical group \(O^{+}(4,2)\) by single-qubit Clifford gates and qubit permutations [1]. That code and the \([[9,2,3]]\) automorphism-maximal code are the unique smallest \(k=2\) stabilizer codes doing so at distances two and three, up to local-Clifford and permutation equivalence [1].
Primary Hierarchy
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
- [2]
- Qiskit Community, “Qiskit QEC framework”, URL
Page edit log
- Victor V. Albert (2026-09-21) — most recent
Cite as:
“\([[9,2,3]]\) automorphism-maximal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_9_2_3, arXiv:2606.11484