$[[6,1,2]]$ semi-self-dual CSS code

$[[6,1,2]]$ semi-self-dual CSS code [1]

Description

A six-qubit CSS stabilizer code that is an example of a semi-self-dual CSS code, i.e., a CSS code whose $X$-type stabilizers are contained in the $Z$-type stabilizers [1].

A stabilizer tableau for the code is given by [2; ID 59] \begin{align} \begin{array}{cccccc} Z & I & Z & I & I & Z \\ I & Z & Z & I & Z & I \\ I & I & I & Z & Z & Z \\ X & I & X & X & X & I \\ I & X & X & X & I & X \end{array}~.\tag*{$(1)$} \end{align}An equivalent self-dual semi-CSS presentation has generators $XXXXII$, $IIXXXX$, $ZZZZII$, $IIZZZZ$, and $IYIYIY$ [1]. A qubit permutation $0\mapsto 0$, $1\mapsto 2$, $2\mapsto 4$, $3\mapsto 5$, $4\mapsto 1$, $5\mapsto 3$, followed by the same one-qubit Clifford on every qubit sending $X\mapsto Z$, $Z\mapsto Y$, and $Y\mapsto X$, converts the CSS generators to that semi-CSS presentation. This makes the code a concrete example of the equivalence between the semi-self-dual CSS and self-dual semi-CSS cases in the $U(\ell,R_8)$ family [1].

Protection

Detects any single-qubit error as a distance-two stabilizer code.

Transversal and Permutation-Based Gates

The physical gate $S^{\dagger}_{0}S^{\dagger}_{1}S^{\dagger}_{2}S_{3}S_{4}S_{5}$ implements the logical action $\bar{S}$ [2; ID 59].In the equivalent self-dual semi-CSS presentation obtained by fixing one logical qubit of the $[[6,2,2]]$ $C_6$ code to $|Y^{-}\rangle_L$, transversal gates support a fault-tolerant magic-state-preparation circuit [1][2; ID 59].

Cousin

$[[6,2,2]]$ $C_6$ code— Fixing one logical qubit of the $[[6,2,2]]$ $C_6$ code to $|Y^{-}\rangle_L$ yields this $[[6,1,2]]$ code [1][2; ID 59].

Member of code lists

Primary Hierarchy

References

[1]
S. Dasu and S. Burton, “A Classification of Transversal Clifford Gates for Qubit Stabilizer Codes”, (2025) arXiv:2507.10519
[2]
Qiskit Community, “Qiskit QEC framework”, URL

Page edit log

Victor V. Albert (2026-06-08) — most recent
Victor V. Albert (2026-04-24)

Cite as:

“$[[6,1,2]]$ semi-self-dual CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/css_6_1_2, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/6/css_6_1_2.yml.

\([[6,1,2]]\) semi-self-dual CSS code [1]