From d243e31253a025b002f96950aa87ef56b0970a74 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= <daniel.knuettel@daknuett.eu>
Date: Wed, 20 Nov 2019 19:28:42 +0100
Subject: [PATCH] some work on the thesis

---
 thesis/chapters/graph_simulator.tex | 41 +++++++++++++++++++++++++++++
 thesis/main.tex                     |  1 +
 2 files changed, 42 insertions(+)

diff --git a/thesis/chapters/graph_simulator.tex b/thesis/chapters/graph_simulator.tex
index 975e40c..4ac9723 100644
--- a/thesis/chapters/graph_simulator.tex
+++ b/thesis/chapters/graph_simulator.tex
@@ -170,13 +170,54 @@ A graph state now represents the state by the gates that have been applied to it
     \ket{+} := \bigotimes\limits_{i=0}^{n-1} H_i \ket{0}
 \end{equation}
 
+\begin{definition}
+\label{def:graph_state}
+
 A graph state $\ket{G}$ is a 3-tuple $(V, E, O)$ where $(V = \{0, ..., n-1\}, E)$ is a graph with the vertices $V$, edges $E$ 
 and vertex operators $O = \{o_i | i = 0, ..., n-1; o_i \in C_L \forall i\}$. The vertex operators and edges are defined
 by the following relation:
 
 \begin{equation}
+    \label{eq:g_state}
     \ket{G} = \left(\bigotimes\limits_{i=0}^{n-1} o_i \right)\left(\bigotimes\limits_{\{i, j\} \in E} CZ_{i,j} \right) \ket{+}
 \end{equation}
+\end{definition}
+
+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}
+
+\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
+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{+}
+    \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
+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.
+
+\subsubsection{Controlled Phase Gate}
+
+Recalling \eqref{eq:g_state} 
+it is clear that some $CZ$ application is less trivial 
+than a single qbit gate.
+
+%\begin{struktogramm}(100, 50)
+%    \ifthenelse[10]{1, 4}
+%    {Both Vertex operators Commute with CZ}{\sTrue}{\sFalse}
+%    \change
+%    \ifend
+%\end{struktogramm}
 
 \subsection{Graph Storage}
 
diff --git a/thesis/main.tex b/thesis/main.tex
index 74aea51..beda6df 100644
--- a/thesis/main.tex
+++ b/thesis/main.tex
@@ -10,6 +10,7 @@
 \usepackage{enumerate}
 \usepackage{physics}
 \usepackage{listings}
+\usepackage{struktex}
 
 \geometry{left=2.5cm,right=2.5cm,top=2.5cm,bottom=2.5cm}