added some more stuff to the graph states

thesis/chapters/stabilizer.tex

@@ -505,3 +505,20 @@ $$ \sqrt{-iX} = \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & i \\ i & 1 \end{ar
 $$ \sqrt{-iZ} = \exp(-i\frac{\pi}{4})\left(\begin{array}{cc} 1 & 0 \\ 0 & -i \end{array}\right)$$
 
 
+\subsubsection{Dynamics of Graph States}
+
+So far the graphical representation of stabilizer states is just another way to store
+basically a stabilizer tableaux that might require less memory than the tableaux used in
+CHP. The true power of this formalism is seen when studying its dynamics. The simplest case
+is a local Clifford operator $c_j$ acting on a qbit $j$: The stabilizers of are changed to 
+$\langle c_j S^{(i)} c_j^\dagger\rangle_i$. Using the definition of the graphical representation
+it is clear that just the vertex operators are changed and the new vertex operators are given by
+
+\begin{equation}
+    \begin{aligned}
+        o_i' &= o_i &\mbox{if } i \neq j\\
+        o_i' &= c o_i c^\dagger &\mbox{if } i = j\\
+    \end{aligned}
+\end{equation}
+
+The action of a $CZ$ gate on the state $(V, E, O)$ is less trivial