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diff --git a/performance/regimes/circuit_scaling_graph0.csv b/performance/regimes/circuit_scaling_graph0.csv
index e247678..b0062ef 100644
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diff --git a/performance/regimes/circuit_scaling_meta.json b/performance/regimes/circuit_scaling_meta.json
index afbb888..6f22558 100644
--- a/performance/regimes/circuit_scaling_meta.json
+++ b/performance/regimes/circuit_scaling_meta.json
@@ -1 +1 @@
-{"nstart": 400, "nstop": 2800, "step": 50, "ncircuits": 100, "nqbits0": 100, "nqbits1": 50, "seed": 3735928559}
\ No newline at end of file
+{"nstart": 400, "nstop": 2600, "step": 50, "ncircuits": 50, "nqbits0": 100, "nqbits1": 50, "seed": 3735928559}
\ No newline at end of file
diff --git a/performance/regimes/generate_data_scaling_circuits.py b/performance/regimes/generate_data_scaling_circuits.py
index cb2546d..6c20e78 100644
--- a/performance/regimes/generate_data_scaling_circuits.py
+++ b/performance/regimes/generate_data_scaling_circuits.py
@@ -51,9 +51,9 @@ def test_scaling_circuits(state_factory
 
 if __name__ == "__main__":
     nstart = 400
-    nstop = 2800
+    nstop = 2600
     step = 50
-    ncircuits = 100
+    ncircuits = 50
     nqbits0 = 100
     nqbits1 = 50
     seed = 0xdeadbeef
@@ -65,7 +65,7 @@ if __name__ == "__main__":
                                     , step
                                     , nqbits0
                                     , ncircuits
-                                    , repeat=10)
+                                    , repeat=5)
     np.random.seed(seed)
     results_graph1 = test_scaling_circuits(GraphState.new_zero_state
                                     , nstart
@@ -73,7 +73,7 @@ if __name__ == "__main__":
                                     , step
                                     , nqbits1
                                     , ncircuits
-                                    , repeat=10)
+                                    , repeat=4)
 
     np.savetxt("circuit_scaling_graph0.csv", results_graph0)
     print("saved results0 to circuit_scaling_graph0.csv")
diff --git a/performance/regimes/scaling_circuits_linear.png b/performance/regimes/scaling_circuits_linear.png
index 87eec75..dbec6ba 100644
Binary files a/performance/regimes/scaling_circuits_linear.png and b/performance/regimes/scaling_circuits_linear.png differ
diff --git a/performance/scaling_qbits_linear.png b/performance/scaling_qbits_linear.png
index 91c189f..6ce4339 100644
Binary files a/performance/scaling_qbits_linear.png and b/performance/scaling_qbits_linear.png differ
diff --git a/performance/scaling_qbits_log.png b/performance/scaling_qbits_log.png
index 8024bc7..a739756 100644
Binary files a/performance/scaling_qbits_log.png and b/performance/scaling_qbits_log.png differ
diff --git a/presentation/main.tex b/presentation/main.tex
index d268007..76802c5 100644
--- a/presentation/main.tex
+++ b/presentation/main.tex
@@ -6,6 +6,7 @@
 \usepackage{setspace} 
 \usepackage{amsmath} 
 \usepackage{hyperref}
+\usepackage{url}
 \usepackage{geometry} 
 \usepackage{enumerate}
 \usepackage{physics}
@@ -13,7 +14,7 @@
 %\usepackage{struktex}
 \usepackage{qcircuit}
 \usepackage{adjustbox}
-\usepackage{tikz}
+%\usepackage{tikz}
 
 
 \usetheme{metropolis}
@@ -71,7 +72,7 @@
         \item{A universal quantum computer is a $2^n$ dimensional complex vector
             to which any unitary can be applied with a quantum mechanical measurement process.}
         \item{Algorithms using the exponentially large Hilbert space can 
-            solve classically exponentially hard problems in polynomial time.}
+            solve some classically exponentially hard problems in polynomial time.}
     \end{itemize}
 
 \end{frame}
@@ -275,9 +276,24 @@
             starting from the $\ket{0}^{\otimes n}$ state can be
             simulated and sampled efficiently, i.e. in $\mbox{poly}(n, m)$ time
             where $m$ is the amount of gates/measurements.}
+    \end{itemize}
+
+\end{frame}
+}
+{
+\begin{frame}{The Gottesman-Knill Theorem}
+    \begin{itemize}
         \item{Note that a general state has $2^n$ complex coefficients. 
             Computing operations on this state is therefore exponentially hard in
             $n$.}
+        \item{Note that using the $R_\phi$ is universal and $R_\frac{\pi}{4}$
+            allows rational approximations of universal gates.}
+        \item{Restricting $\phi$ to $\frac{\pi}{2}$ allows the simulation of
+            large numbers of qbits on a classical computer.\\
+            {\bf Goal of this project is to create a simulator that can
+                simulate several thousand up to millions of qbits.
+            }}
+
     \end{itemize}
 
 \end{frame}
@@ -349,13 +365,9 @@
     \begin{itemize}
         \item{In general a Clifford gate $U \in C_n$ will map a stabilizer state to another stabilizer state.
             The new state is stabilized by $\langle U S^{(i)} U^\dagger \rangle_i$.}
-        \item{A Pauli observable $g_a \in \{X_a, Y_a, Z_a\}$ will either commute with all stabilizers
-            (in this case $g_a$ is a stabilizer, the measurement is deterministic and the stabilizers are unchanged) or
-            or anticommute with at least one stabilizer.}
-        \item{If $g_a$ anticommutes with a subset $A := \{S^{(i)} | \{S^{(i)}, g_a\} = 0 \}$
-            the probability to measure $+1$ or $-1$ is $\frac{1}{2}$ and the stabilizers $A$
-            are changed.}
-        \item{When going from $+g_a$ to $-g_a$ the results are changed from $+1$ to $-1$ and vice versa.}
+        \item{One can show that measurements of Pauli observables are covered by the stabilizer formalism.}
+        \item{When measuring Pauli observable probability amplitudes of $0, 1$ or $\frac{1}{2}$ are
+            possible.}
     \end{itemize}
 \end{frame}
 }
@@ -363,14 +375,13 @@
 {
 \begin{frame}{Graphical States}
     \begin{itemize}
-        \item{The graphical representation of stabilizer states is an optimized way to write 
+        \item{The graphical representation is an optimized way to write 
             the stabilizers.}
         \item{
-            $(V, E, O)$ is called the graphical representation of
-                a stabilizer state if $V = \{0, ..., n-1\}$,
-                $E \subset \{\{i,j\} | i,j \in V, i \neq j \}$ and
-                $O = \{o_0, ..., o_{n-1}\}$ where $o_i \in C_1$.
-                $G = (V, E)$ is a graph, $O$ are called vertex operators.
+            For a set of vertices $V = \{0, ..., n-1\}$, some edges
+            $E \subset V \otimes V$, and local Clifford operators 
+            $O = \{o_0, ..., o_{n-1}\}$ the tuple $(V, E, O)$
+            is the graphical representation of a stabilizer state.
         }
         \item{The state associated with $(V, E, O)$ is given by
             \begin{equation}
@@ -570,8 +581,20 @@
 
 {
 \begin{frame}{Implementation}
+    \noindent\begin{minipage}{0.5\textwidth}
     \center
     \includegraphics[width=\textwidth,height=0.8\textheight,keepaspectratio=true]{screenshot_github.png}
+    \end{minipage} \hfill
+    \begin{minipage}{0.4\textwidth}
+    {\small
+    \begin{itemize}
+        \item{The code is GPLv3.0 licensed.}
+        \item{GitHub repository: \url{https://github.com/daknuett/pyqcs}}
+        \item{Install it via PyPI: \lstinline{pip3 install pyqcs}}
+        \item{The code is tested automatically on TravisCI.}
+    \end{itemize}
+    }
+    \end{minipage}
 \end{frame}
 }
 
@@ -680,4 +703,10 @@
     \end{itemize}
 \end{frame}
 }
+
+{
+\begin{frame}{References}
+    See the bibliography of my bachelor thesis.
+\end{frame}
+}
 \end{document}
diff --git a/thesis/chapters/appendix.tex b/thesis/chapters/appendix.tex
index 1d54c73..9613ba4 100644
--- a/thesis/chapters/appendix.tex
+++ b/thesis/chapters/appendix.tex
@@ -66,9 +66,9 @@ code is converted to an image using
 It has been mentioned several times that the implementation with
 \lstinline{ufuncs} as gates is faster than a pure \lstinline{python}
 implementation.  To support this statement a simple benchmark is written. The
-relatively simple Pauli $X$ gate us implemented, more complicated gates like $CX$ or $H$
+relatively simple Pauli $X$ gate is implemented, more complicated gates like $CX$ or $H$
 have worse performance when written in \lstinline{python}. The performance
-improvement when in this example is a factor around $6.4$.
+improvement in this example is a factor around $6.4$.
 One must note that the tested \lstinline{python} code is not
 realistic and in a possible application there would be a significant overhead.
 
diff --git a/thesis/chapters/conclusion.tex b/thesis/chapters/conclusion.tex
index 4f6135d..b298dc3 100644
--- a/thesis/chapters/conclusion.tex
+++ b/thesis/chapters/conclusion.tex
@@ -3,9 +3,9 @@
 
 As seen in \ref{ref:performance} simulation using stabilizers is exponentially
 faster than simulating with dense state vectors. The graphical representation
-for stabilizer states is on average more efficiently than the stabilizer
-tableaux. In particular one can simulate more qbits while only applying
-Clifford gates.
+for stabilizer states is in realistic cases more efficiently than the
+stabilizer tableaux \cite{andersbriegel2005}. In particular one can simulate
+more qbits while only applying Clifford gates.
 
 This is considerably useful when working on quantum error correcting strategies
 as they often include many qbits; the smallest quantum error correcting
@@ -119,4 +119,5 @@ The exponential speedup of quantum computing is often attributed to
 entanglement, superposition and interference effects
 \cite{uwaterloo}\cite{21732}\cite{vandennest2018}. Stabilizer states show
 entanglement, superposition and interference effects; as do computations using
-general normalizers \cite{vandennest2018}.
+general normalizers \cite{vandennest2018}. The question why quantum computing
+can speed up computations exponentially is non-trivial.
diff --git a/thesis/chapters/implementation.tex b/thesis/chapters/implementation.tex
index 0b726c6..5435e4c 100644
--- a/thesis/chapters/implementation.tex
+++ b/thesis/chapters/implementation.tex
@@ -172,7 +172,13 @@ For the graphical state $(V, E, O)$ the list of vertices $V$ can be stored impli
 by demanding $V = \{0, ..., n - 1\}$. This leaves two components that have to be stored:
 The edges $E$ and the vertex operators $O$. Storing the vertex operators is
 done using a \lstinline{uint8_t} array. Every local Clifford operator is
-associated with an integer ranging from $0$ to $23$. Their order is
+associated with an integer ranging from $0$ to $23$
+\footnote{
+    \cite{andersbriegel2005} also uses integers from $0$ to $23$ to represent
+    the Clifford operators. The authors do not describe the mapping from integer
+    to operator and the order might be different.
+}
+. Their order is
 
 \begin{equation}
 \begin{aligned}
@@ -224,7 +230,8 @@ a vertex.  Also one must take care to keep the memory required to store a state
 low.  To meet both requirements a linked list-array hybrid is used.  For every
 vertex the neighbourhood is stored in a sorted linked list (which is a sparse
 representation of a column vector) and these adjacency lists are stored in
-a length $n$ array.
+a length $n$ array. \cite{andersbriegel2005} uses the same representation for
+the graph.
 
 Using this storage method all operations - including searching and toggling edges -
 inherit their time complexity from the sorted linked list.
@@ -240,7 +247,7 @@ three classes of operations.
 \begin{description} 
     \item[\hspace{-1em}]{\lstinline{RawGraphState.apply_C_L}\\
         This method implements local Clifford gates. It takes the qbit index
-        and the index of the local Clifford operator (ranging form $0$ to $23$).}
+        and the index of the local Clifford operator (ranging from $0$ to $23$).}
     \item[\hspace{-1em}]{\lstinline{RawGraphState.apply_CZ}\\
             Applies the $CZ$ gate to the state. The first argument is the
             act-qbit, the second the control qbit (note that this is just for
@@ -270,7 +277,7 @@ of the user.
 This implementation reads byte code from a file and executes it. The execution is 
 always done in three steps:
 
-\begin{enumerate}[1]
+\begin{enumerate}[1.]
     \item{Initializing the state according the header of the byte code file.}
     \item{Applying operations given by the byte code to the state. This includes local
             Clifford gates, $CZ$ gates and measurements (the measurement outcome is ignored).}
@@ -363,7 +370,7 @@ circuits are used. Length $m$ circuits are generated from the probability space
 with the uniform distribution. The continuous part $[0, 2\pi)$ is ignored when
 generating random circuits for the graphical simulator; in order to generate
 random circuits for dense vector simulations this is used as the argument $\phi$ of the
-$R_\phi$ gate.  For $m=1$ an outcome is mapped to a gate where 
+$R_\phi$ gate.  For $m=1$ an outcome is mapped to a gate with 
 
 \begin{equation}
 \begin{aligned}
@@ -382,7 +389,12 @@ dense vector simulator $S$ can be replaced by $R_\phi$ with the parameter $x$.
 
 Using this method circuits are generated and applied both to graphical and
 dense vector states and the time required to execute the operations
-\cite{timeit} is measured. The resulting graph can be seen in
+\cite{timeit} is measured\footnote{
+    The used computer had an \lstinline{Intel(R) Core(TM) i5-4590 CPU @ 3.30GHz} processor, 
+    \lstinline{7.7 GiB} RAM, and \lstinline{27 GiB} SSD swap space (which was unused).
+    The operating system was \lstinline{Linux 4.19.0-8-amd64 #1 SMP Debian 4.19.98-1 (2020-01-26) x86_64 GNU/Linux}.
+
+}. The resulting graph can be seen in
 Figure \ref{fig:scaling_qbits_linear} and Figure \ref{fig:scaling_qbits_log}.
 Note that in both cases the length of the circuits have been scaled linearly
 with the amount of qbits and the measured time was divided by the number of
@@ -421,11 +433,11 @@ the stabilizer tableaux like CHP \cite{CHP} with an average runtime behaviour
 of $\mathcal{O}\left(n\log(n)\right)$ instead of $\mathcal{O}\left(n^2\right)$.
 
 One should be aware that the gate execution time (the time required to apply
-a gate to the state) highly depends on the  state it is applied to. For the
+a gate to the state) highly depends on the  state it is applied to \cite{andersbriegel2005}. For the
 dense vector simulator and CHP this is not true: Gate execution time is
 constant for all gates and states. Because the graphical simulator has to
-toggle neighbourhoods the gate execution time of the $CZ$ gate varies
-significantly. The plot Figure \ref{fig:scaling_circuits_linear}
+toggle neighbourhoods the gate execution time of the $CZ$ and $M$ gates vary
+significantly \cite{andersbriegel2005}. The plot Figure \ref{fig:scaling_circuits_linear}
 shows the circuit execution time for two different numbers of qbits. One can observe three
 regimes:
 
@@ -455,7 +467,7 @@ regimes:
 
 These three regimes can be explained when considering the graphical states that
 typical live in these regimes. With increased circuit length the amount of
-edges increases which makes toggling neighbourhoods harder. Graphs from the
+edges increases which makes toggling neighbourhoods harder \cite{andersbriegel2005}. Graphs from the
 low-linear, intermediate and high-linear regime can be seen in Figure
 \ref{fig:graph_low_linear_regime}, Figure \ref{fig:graph_intermediate_regime}
 and Figure \ref{fig:graph_high_linear_regime}. Due to the great amount of edges
diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex
index 96a9d55..f082683 100644
--- a/thesis/chapters/stabilizer.tex
+++ b/thesis/chapters/stabilizer.tex
@@ -16,6 +16,8 @@ This performance has since been improved to $n\log(n)$ time on average
 
 \section{Stabilizers and Stabilizer States}
 
+The discussion below follows the argumentation given in \cite{nielsen_chuang_2010}.
+
 \subsection{Local Pauli Group and Multilocal Pauli Group}
 
 \begin{definition}
@@ -46,7 +48,6 @@ deduced from its definition via the tensor product.
 
 \subsection{Stabilizers}
 
-The discussion below follows the argumentation given in \cite{nielsen_chuang_2010}.
 
 \begin{definition}
     \label{def:stabilizer}
@@ -292,6 +293,8 @@ and $s=0$ are obtained with probability $\frac{1}{2}$ and after choosing a $S^{(
 \section{The VOP-free Graph States}
 This section will discuss the vertex operator (VOP)-free graph states. Why they
 are called vertex operator-free will be clear in \ref{ref:sec_g_states}.
+Most of the discussion here is adapted from \cite{hein_eisert_briegel2008}
+whith some parts from \cite{andersbriegel2005}.
 
 \subsection{VOP-free Graph States}
 
@@ -650,7 +653,7 @@ studying its dynamics. The simplest case is a local Clifford operator $c_j$
 acting on a qbit $j$ changing the stabilizers to $\langle c_j S^{(i)}
 c_j^\dagger\rangle_i$. Using the definition of the graphical representation one
 sees that just the vertex operators are changed and the new vertex operators
-are given by
+are given by \cite{andersbriegel2005}
 
 \begin{equation}
     \begin{aligned}
diff --git a/thesis/main.tex b/thesis/main.tex
index 2d678ad..2dcbd82 100644
--- a/thesis/main.tex
+++ b/thesis/main.tex
@@ -88,6 +88,20 @@ Supervised by Prof. Dr. Christoph Lehner}
 \bibliographystyle{unsrt} 
 \bibliography{main}{}
 
+\newpage
+{\Large\textbf{Acknowledgements}}
+\\
+\\
+I want to thank Prof. Dr. Christoph Lehner for offering me the possibility to
+write this thesis in his workgroup. He also found the time to answer my
+questions and help me when I got stuck while keeping an inspiring atmosphere.\\
+Further I own a big thank you to Simon Feyrer for proof-reading my thesis. \\
+Also I want to thank Andreas Hackl for proof-reading my thesis and helping out
+with several technical problems.\\
+Thanks to Shirley Galbaw for helping to fix some of my terrible ingris grammar.
+
+
+
 \newpage
 \textbf{Erklärung zur Anfertigung:}\\
 Ich habe die Arbeit selbständig verfasst, keine anderen als die angegebenen