$1$-flag circuits $\{ C(g_{1}),C(g_{2}), \cdots , C(g_{r}) \}$. For every generator $g_{i}$, all pairs of elements $E,E'\in \mathcal{E}(g_{i})$ satisfy $s(E)\neq s(E')$ or $E \sim E'$. \[Def:Flag1FTECcondition\]

In other words, we require that any two errors that arise when a circuit $C(g_{i})$ flags due to a single fault must be either distinguishable or logically equivalent. For the following protocol to satisfy the FTEC conditions in \[Def:FaultTolerantDef\], one can assume there is at most 1 fault. If the Flag $1$-FTEC condition is satisfied, the protocol is implemented as follows:

![Tree diagram illustrating the possible paths of the Flag $1$-FTEC Protocol. Numbers enclosed in red circles at the end of the edges indicate which step to implement in the Flag $1$-FTEC Protocol. A dashed line is followed when any of the 1-flag circuits $C(g_{i})$ flags. Solid squares indicate a syndrome measurement using 1-flag circuits whereas rings indicate a decision based on syndrome outcomes. Note that the syndrome measurement is repeated at most three times.[]{data-label="fig:TreeD3Diag"}](TreeD3.png){width="50.00000%"}

[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 