\([[6,1,2]]\) 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].