diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex
index abe74a1..fafe142 100644
--- a/thesis/chapters/stabilizer.tex
+++ b/thesis/chapters/stabilizer.tex
@@ -521,4 +521,20 @@ it is clear that just the vertex operators are changed and the new vertex operat
     \end{aligned}
 \end{equation}
 
-The action of a $CZ$ gate on the state $(V, E, O)$ is less trivial 
+The action of a $CZ$ gate on the state $(V, E, O)$ is in most cases less trivial.
+Let $i \neq j$ be two qbits, now consider the action of $CZ_{i,j}$ on $(V, E, O)$.
+The following discussion closely follows \cite{andersbriegel2005} and is analogous
+to the implementation.
+
+
+\textbf{Case 1:}
+Both $o_i$ and $o_j$ commute with $CZ_{i,j}$. This is the case for exactly
+four vertex operators: $\mathcal{Z} = \left\{I, Z, S, S^\dagger\right\}$.
+In this case the CZ can be pulled past the vertex operators and just the edges
+are changed to $E' = E \Delta \left\{\{i,j\}\right\}$.
+
+\textbf{Case 2:}
+At least one vertex operator does not commute with $CZ_{i,j}$. 
+
+\textbf{Sub-Case 2.1}
+Both