\([[12,2,2]]\) CSS code[1]
Description
CSS code that admits a logical \(CS\) gate via application of physical \(T\) and \(T^{\dagger}\) gates.
A stabilizer tableau for the code is given by [1; Fig. 6] \begin{align} \begin{array}{cccccccccccc} X & X & X & X & I & I & I & I & X & X & X & X \\ I & I & I & I & X & X & X & X & X & X & X & X \\ Z & Z & Z & Z & I & I & I & I & I & I & I & I \\ Z & Z & I & I & Z & Z & I & I & I & I & I & I \\ Z & I & Z & I & Z & I & Z & I & I & I & I & I \\ I & Z & Z & I & Z & I & I & Z & I & I & I & I \\ Z & I & I & I & Z & I & I & I & Z & I & I & I \\ I & Z & I & I & Z & I & I & I & I & Z & I & I \\ I & I & Z & I & Z & I & I & I & I & I & Z & I \\ Z & Z & Z & I & Z & I & I & I & I & I & I & Z \end{array}~. \tag*{(1)}\end{align}
Transversal Gates
Logical \(CS_{01}\) gate via transversal application of \(T\) on qubits \(0,3,4,7,8,11\), \(T^3\) on qubits \(1,2,5,6\), and \(T^7 = T^{\dagger}\) on qubits \(9,10\) [1; Fig. 6].Cousin
- \([[8,3,2]]\) Smallest interesting color code— The \([[12,2,2]]\) CSS code can be obtained by joining two copies of the \([[8,3,2]]\) code at a common face [1].
Primary Hierarchy
References
- [1]
- M. A. Webster, A. O. Quintavalle, and S. D. Bartlett, “Transversal diagonal logical operators for stabiliser codes”, New Journal of Physics 25, 103018 (2023) arXiv:2303.15615 DOI
Page edit log
- Victor V. Albert (2025-02-07) — most recent
Cite as:
“\([[12,2,2]]\) CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/stab_12_2_2