the lattice such that the relevant face is on the bottom left. (b) For $g = Z_{q_{1}}Z_{q_{2}}Z_{q_{3}}Z_{q_{4}}$, the flag error set is $\mathcal{E}(g) = \{ I,Z_{q_{3}}Z_{q_{4}},X_{q_{2}}Z_{q_{3}}Z_{q_{4}},Z_{q_{1}}X_{q_{2}},Z_{q_{4}},$ $X_{q_{3}}Z_{q_{4}},X_{q_{3}}Z_{q_{3}}Z_{q_{4}} \}$ which contains all errors arising from a single fault that causes the stabilizer measurement circuit $C(g)$ to flag. Since the Steane code is a CSS code, the $X$ component of an error will be corrected independently allowing us to consider the $Z$-part of the flag error set $\mathcal{E}_Z(g)=\{I,Z_{q_1},Z_{q_4},Z_{q_3}Z_{q_4} \}$. As required, the elements of $\mathcal{E}_Z(g)$ all have distinct syndromes (with satisfied stabilizers represented by a plus). []{data-label="fig:Steane"}](SteaneCodeColor.png "fig:"){width="\textwidth"}

[0.45]{} ![ (a) A representation of the Steane code where each circle is a qubit, and there is an $X$- and a $Z$-type stabilizer generator for each face. Stabilizer cicuits are specified from that in Fig. \[fig:ErrorPropSteane\](a) after rotating the lattice such that the relevant face is on the bottom left. (b) For $g = Z_{q_{1}}Z_{q_{2}}Z_{q_{3}}Z_{q_{4}}$, the flag error set is $\mathcal{E}(g) = \{ I,Z_{q_{3}}Z_{q_{4}},X_{q_{2}}Z_{q_{3}}Z_{q_{4}},Z_{q_{1}}X_{q_{2}},Z_{q_{4}},$ $X_{q_{3}}Z_{q_{4}},X_{q_{3}}Z_{q_{3}}Z_{q_{4}} \}$ which contains all errors arising from a single fault that causes the stabilizer measurement circuit $C(g)$ to flag. Since the Steane code is a CSS code, the $X$ component of an error will be corrected independently allowing 