did some work on the paper

thesis/chapters/graph_simulator.tex

@@ -183,6 +183,8 @@ by the following relation:
 \end{equation}
 \end{definition}
 
+Where every $o_i$ acts on the $i$-th qbit.
+
 One can show that any stabilizer state can be realized as a graph state (for instance in \cite{schlingenmann2001}).
 
 \subsection{Operations on the Graph State}
@@ -219,6 +221,34 @@ than a single qbit gate.
 %    \ifend
 %\end{struktogramm}
 
+
+\subsubsection{Measurement of a Qbit}
+
+Note that the $Z$ gate is diagonal in the $\{\ket{1}_s,\ket{0}_s\}$ basis with eigen values $1, -1$.
+This gives a simple expression for the projector on the $\ket{0}_s$ and $\ket{1}_s$ state of qbit $k$:
+
+\begin{equation}
+    \tilde{P^{v}_{k}} := \frac{I_k + (-1)^vZ_k}{2}, v \in {0, 1}
+\end{equation}
+
+This projector trivially commutes with any non-operand vertex operator yielding for a state
+
+
+\begin{equation}
+    \begin{aligned}
+        \ket{\psi} := \left(\bigotimes\limits_{\{i,j\} \in E} CZ_{i,j}\right) \ket{+} \\
+        \tilde{P^{v}_{k}} \left(\bigotimes\limits_{i=0}{n-1} o_i\right)\ket{\psi} &=  
+        \left(\bigotimes\limits_{i=0,i \ne k}{n-1} o_i\right) \tilde{P^{v}_{k}} o_k \ket{\psi} \\
+        &= \left(\bigotimes\limits_{i=0,i \ne k}{n-1} o_i\right) o_ko_k^\dagger \tilde{P^{v}_{k}} o_k \ket{\psi} \\
+        &= \left(\bigotimes\limits_{i=0,i \ne k}{n-1} o_i\right) o_k = \frac{I_k + (-1)^v o_k^\dagger Z_ko_k}{2} \ket{\psi}
+    \end{aligned}
+\end{equation}
+
+Now $o_k$ stabilizes $P$, as it is an element of $C_L$, meaning that $o_k^\dagger Z_k o_k \in \{-1,1\}\{X_k, Y_k, Z_k\}$
+Which gives a new projector on a Pauli matrix: $P_k := o_k^\dagger Z_ko_k$
+
+
+
 \subsection{Graph Storage}
 
 One of the gread advantages of simulating in the graph formalism is a great increase