some more work on the graph formalism

thesis/chapters/graph_simulator.tex

@@ -187,24 +187,86 @@ 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}).
 
+\subsubsection{The Vertex Operator-Free Graph States}
+
+In order to understand some essential transformations of graph states it is necessary
+to study the vertex operator-free graph states first, partially because the graph states as used in this paper
+were derived from the vertex operator-free graph states.
+
+\begin{definition}
+    \label{def:vop_free_g_state}
+    A $n$ qbit vertex operator-free graph state $\ket{\overline{G}}$ is associated with a graph $(V, E)$
+    by the $n$ operators
+
+    \begin{equation}
+        K^{(i)}_G := X_i \left(\prod\limits_{\{i, j\} \in E} Z_j\right) 
+    \end{equation}
+
+    for all $i \in V$ where for some operator $O$ $O_i$ indicates that it acts on the $i$-th qbit.
+
+    A state $\ket{\overline{G}}$ is a $+1$ eigenstate of all $n$ $K^{(i)}_G$.
+\end{definition}
+
+\begin{corrolary}
+    All $K^{(i)}_G$ commute and are hermitian. Therefore they have a common set of eigen states
+    (in particular definition \ref{def:vop_free_g_state} is well defined).
+    In terms of quantum mechanics $K^{(i)}_G$ are observables.
+
+    Further as $\ket{\overline{G}}$ is a $+1$ eigenstate of all $n$ $K^{(i)}_G$ which are
+    multi-local Pauli operators, $\{K^{(i)}_G | i \in \{0, ..., n-1\}\}$ is the stabilizer
+    of $\ket{\overline{G}}$ and $\ket{\overline{G}}$ is a stabilizer state.
+\end{corrolary}
+
+\begin{proof}
+    As $X_i$ and $Z_i$ are hermitian their product is hermitian.
+
+    Consider the case $\{i,j\} \notin E$ first:
+    \begin{equation}
+    \begin{aligned}
+        \left[K^{(i)}_G, K^{(j)}_G\right] = \left[X_i \prod\limits_{\{i, n\} \in E} Z_n, X_j \prod\limits_{\{j, m\} \in E} Z_m\right] = 0
+    \end{aligned}
+    \end{equation}
+
+    As operators acting on different qbits commute. The case $\{i,j\} \in E$ is slightly less trivial:
+    \begin{equation}
+    \begin{aligned}
+        \left[K^{(i)}_G, K^{(j)}_G\right] &= \left[X_i \left(\prod\limits_{\{i, n\} \in E, n \neq j} Z_n\right) Z_j, X_j \left(\prod\limits_{\{j, m\} \in E, m \neq i} Z_m\right) Z_i\right] \\
+        &= \left[X_i Z_j \prod\limits_n Z_n, X_j Z_i \prod\limits_m Z_m\right]\\
+        &= \left(X_i Z_j X_j Z_i - X_j Z_i X_i Z_j\right)  \prod\limits_n Z_n \prod\limits_m Z_m \\
+        &= \left(Z_j X_j X_i Z_i - X_j Z_j Z_i X_i\right)  \prod\limits_n Z_n \prod\limits_m Z_m \\
+        &= \left((-1)^2X_j Z_j Z_i X_i - X_j Z_j Z_i X_i\right)  \prod\limits_n Z_n \prod\limits_m Z_m \\
+        &= 0
+    \end{aligned}
+    \end{equation}
+
+    as $X$, $Z$ anticommute.
+
+\end{proof}
+
+
+
+
+
+
+
 \subsection{Operations on the Graph State}
 
 \subsubsection{Single Qbit Gates}
 
 Recalling \eqref{eq:g_state} 
-Makes it clear that for any single  qbit gate $o \in C_L$ with $o^{(k)}$ being the gate
+Makes it clear that for any single  qbit gate $g \in C_L$ with $g_k$ being the gate
 acting on qbit $k$ the state changes according to
 
 
 \begin{equation}
     \begin{aligned}
-        o^{(k)} \ket{G} &= o^{(k)} \left(\bigotimes\limits_{i=0}^{n-1} o_{i} \right)\left(\bigotimes\limits_{\{i, j\} \in E} CZ_{i,j} \right) \ket{+} \\
-        &= \left(\bigotimes\limits_{i=0}^{n-1} o^{\delta_{i,k}}o_{i} \right)\left(\bigotimes\limits_{\{i, j\} \in E} CZ_{i,j} \right)\ket{+}
+        g_k \ket{G} &= g_k \left(\bigotimes\limits_{i=0}^{n-1} o_{i} \right)\left(\bigotimes\limits_{\{i, j\} \in E} CZ_{i,j} \right) \ket{+} \\
+        &= \left(\bigotimes\limits_{i=0}^{n-1} g_k^{\delta_{i,k}}o_{i} \right)\left(\bigotimes\limits_{\{i, j\} \in E} CZ_{i,j} \right)\ket{+}
     \end{aligned}
 \end{equation}
 
-meaning that the graph state $(V, E, O)$ changes to $(V, E, \{o_0, ..., o_{k-1}, oo_k, o_{k+1}, ..., o_{n-1}\})$
-as $C_L$ is almost a group the element $oo_k \in C_L$ up to a global phase that is disregarded. All the results
+meaning that the graph state $(V, E, O)$ changes to $(V, E, \{o_0, ..., o_{k-1}, go_k, o_{k+1}, ..., o_{n-1}\})$
+as $C_L$ is almost a group the element $go_k \in C_L$ up to a global phase that is disregarded. All the results
 of $C_L \times C_L \rightarrow C_L, a,b \mapsto ab$ have been precomputed in a lookup table and the vertex operators
 are updated according to that lookup table.