a few more lines

thesis/chapters/stabilizer.tex

@@ -659,4 +659,18 @@ as $o_a$ is a Clifford operator:
 \end{equation}
 
 Where $\tilde{g}_a \in \{+1, -1\}\cdot\{X_a, Y_a, Z_a\}$. It is therefore enough to study the measurements
-of any Pauli operator on the vertex operator free graph states. 
+of any Pauli operator on the vertex operator free graph states. The commutators of the observable
+with the $K_G^{(i)}$ are quite easy to compute, note that Pauli matrices wither commute or anticommute 
+so it is easier to list the operators that anticommute:
+
+\begin{equation}
+    \begin{aligned}
+        A_{\pm X_a} &= \left\{j | \{j, a\} \in E\right\}\\
+        A_{\pm Y_a} &= \left\{j | \{j, a\} \in E\right\} \cup \{a\} \\
+        A_{\pm Z_a} &= \{a\}\\
+    \end{aligned}
+\end{equation}
+
+This gives one immediate result: if a qbit $a$ is isolated and the operator $\tilde{g}_a = X_a$ 
+is measured the result $+1$ is obtained with probability $1$ and $(V, E, O)$ is unchanged.
+