diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex
index e6f7dbc..e2807ac 100644
--- a/thesis/chapters/stabilizer.tex
+++ b/thesis/chapters/stabilizer.tex
@@ -2,15 +2,17 @@
 
 \section{The Stabilizer Formalism}
 
-The stabilizer formalism was originally introduced by Gottesman \cite{gottesman1997}
-for quantum error correction and is a useful tool to encode quantum information
-such that it is protected against noise. The prominent Shor code \cite{shor1995}
-is an example of a stabilizer code (although it was discovered before the stabilizer 
-formalism was discovered), as are the 3-qbit bit-flip and phase-flip codes.
+The stabilizer formalism was originally introduced by Gottesman
+\cite{gottesman1997} for quantum error correction and is a useful tool to
+encode quantum information such that it is protected against noise. The
+prominent Shor code \cite{shor1995} is an example of a stabilizer code
+(although it was discovered before the stabilizer formalism was discovered), as
+are the 3-qbit bit-flip and phase-flip codes.
 
-It was only later that Gottesman and Knill discovered that stabilizer states can
-be simulated in polynomial time on a classical machine \cite{gottesman2008}. This
-performance has since been improved to $n\log(n)$ time on average \cite{andersbriegel2005}.
+It was only later that Gottesman and Knill discovered that stabilizer states
+can be simulated in polynomial time on a classical machine
+\cite{gottesman2008}. This performance has since been improved to $n\log(n)$
+time on average \cite{andersbriegel2005}.
 
 \subsection{Stabilizers and Stabilizer States}
 
@@ -24,8 +26,8 @@ performance has since been improved to $n\log(n)$ time on average \cite{andersbr
     with the matrix product is called the Pauli group \cite{andersbriegel2005}.
 \end{definition}
 
-The group property of $P$ can be verified easily. Note that 
-the elements of $P$ either commute or anticommute.
+The group property of $P$ can be verified easily. Note that the elements of $P$
+either commute or anticommute.
 
 \begin{definition}
     For $n$ qbits
@@ -37,8 +39,8 @@ the elements of $P$ either commute or anticommute.
     is called the multilocal Pauli group on $n$ qbits \cite{andersbriegel2005}. 
 \end{definition}
 
-The group property of $P_n$ and the (anti-)commutator relationships follow directly from its definition
-via the tensor product.
+The group property of $P_n$ and the (anti-)commutator relationships follow
+directly from its definition via the tensor product.
 %Further are $p \in P_n$ hermitian and have the eigenvalues $\pm 1$ for
 %$p \neq \pm I$, $+1$ for $p = I$  and $-1$ for $p = -I$.
 
@@ -81,31 +83,35 @@ The discussion below follows the argumentation given in \cite{nielsen_chuang_201
 
 \end{proof}
 
-Considering all the elements of a group can be impractical for some calculations,
-the generators of a group are introduced. Often it is enough to discuss the generator's
-properties in order to understand the properties of the group.
+Considering all the elements of a group can be impractical for some
+calculations, the generators of a group are introduced. Often it is enough to
+discuss the generator's properties in order to understand the properties of the
+group.
 
 \begin{definition}
-    For a finite group $G$ and some $m \in \mathbb{N}$ one denotes the generators
-    of G
+    For a finite group $G$ and some $m \in \mathbb{N}$ one denotes the
+    generators of G
 
-    \begin{equation} \langle g_1, ..., g_m \rangle \equiv \langle g_i \rangle_{i=1,...,m}\end{equation}
+    \begin{equation} \langle g_1, ..., g_m \rangle \equiv \langle g_i
+    \rangle_{i=1,...,m}\end{equation}
 
-    where $g_i \in G$, every element in $G$ can be written as a product of the $g_i$
-    and $m$ is the smallest integer for which these statements hold.
+    where $g_i \in G$, every element in $G$ can be written as a product of the
+    $g_i$ and $m$ is the smallest integer for which these statements hold.
 \end{definition}
 
-In the following discussions $\langle S^{(i)} \rangle_{i=0, ..., n-1}$ will be used as
-the required properties of a set of stabilizers that can be studied on its
-generators.
+In the following discussions $\langle S^{(i)} \rangle_{i=0, ..., n-1}$ will be
+used as the required properties of a set of stabilizers that can be studied on
+its generators.
 
 \subsubsection{Stabilizer States}
 
-One important basic property of quantum mechanics is that hermitian operators have real eigenvalues
-and eigenspaces which are associated with these eigenvalues. Finding these eigenvalues and eigenvectors
-is what one calls solving a quantum mechanical system. One of the most fundamental insights of 
-quantum mechanics is that commuting operators have a common set of eigenvectors, i.e. they
-can be diagonalized simultaneously. This motivates and justifies the following definition.
+One important basic property of quantum mechanics is that hermitian operators
+have real eigenvalues and eigenspaces which are associated with these
+eigenvalues. Finding these eigenvalues and eigenvectors is what one calls
+solving a quantum mechanical system. One of the most fundamental insights of
+quantum mechanics is that commuting operators have a common set of
+eigenvectors, i.e. they can be diagonalized simultaneously. This motivates and
+justifies the following definition.
 
 \begin{definition}
     For a set of stabilizers $S$ the vector space
@@ -118,38 +124,35 @@ can be diagonalized simultaneously. This motivates and justifies the following d
     $\ket{\psi}$ is stabilized by $S$ \cite{nielsen_chuang_2010}.
 \end{definition}
 
-It is clear that to show the stabilization property of
-$S$ the proof for the generators is sufficient,
-as all the generators forming an element in $S$ can be absorbed into $\ket{\psi}$.
-The dimension of $V_S$ is not immediately clear. One can however show that
-for a set of stabilizers $\langle S^{(i)} \rangle_{i=1, ..., n-m}$ the dimension
-$\dim V_S = 2^m$ \cite[Chapter 10.5]{nielsen_chuang_2010}. This yields the following important
-result:
+It is clear that to show the stabilization property of $S$ the proof for the
+generators is sufficient, as all the generators forming an element in $S$ can
+be absorbed into $\ket{\psi}$.  The dimension of $V_S$ is not immediately
+clear. One can however show that for a set of stabilizers $\langle S^{(i)}
+\rangle_{i=1, ..., n-m}$ the dimension $\dim V_S = 2^m$ \cite[Chapter
+10.5]{nielsen_chuang_2010}. This yields the following important result:
 
-\begin{theorem}
-    \label{thm:unique_s_state}
-    For a $n$ qbit system and stabilizers $S = \langle S^{(i)} \rangle_{i=1, ..., n}$ the stabilizer
-    space $V_S$ has $\dim V_S = 1$, in particular there exists an up to a trivial phase unique
-    state $\ket{\psi}$ that is stabilized by $S$.
+\begin{theorem} \label{thm:unique_s_state} For a $n$ qbit system and
+    stabilizers $S = \langle S^{(i)} \rangle_{i=1, ..., n}$ the stabilizer
+    space $V_S$ has $\dim V_S = 1$, in particular there exists an up to
+    a trivial phase unique state $\ket{\psi}$ that is stabilized by $S$.
     
-    Without proof.
-\end{theorem}
+    Without proof.  \end{theorem}
 
-In the following discussions for $n$ qbits a set $S = \langle S^{(i)} \rangle_{i=1,...,n}$ 
-of $n$ independent stabilizers will be assumed.
+In the following discussions for $n$ qbits a set $S = \langle S^{(i)}
+\rangle_{i=1,...,n}$ of $n$ independent stabilizers will be assumed.
 
 
 \subsubsection{Dynamics of Stabilizer States}
 
-Consider a $n$ qbit state $\ket{\psi}$ that is the $+1$ eigenstate of $S = \langle S^{(i)} \rangle_{i=1,...,n}$
-and a unitary transformation $U$ that describes the dynamics of the system, i.e. 
+Consider a $n$ qbit state $\ket{\psi}$ that is the $+1$ eigenstate of $S
+= \langle S^{(i)} \rangle_{i=1,...,n}$ and a unitary transformation $U$ that
+describes the dynamics of the system, i.e. 
 
-\begin{equation}
-    \ket{\psi'} = U \ket{\psi}
-\end{equation}
+\begin{equation} \ket{\psi'} = U \ket{\psi} \end{equation}
 
-It is clear that in general $\ket{\psi'}$ will not be stabilized by $S$ anymore. There are
-however some statements that can still be made \cite{nielsen_chuang_2010}:
+It is clear that in general $\ket{\psi'}$ will not be stabilized by $S$
+anymore. There are however some statements that can still be made
+\cite{nielsen_chuang_2010}:
 
 \begin{equation}
     \begin{aligned}
@@ -161,9 +164,10 @@ however some statements that can still be made \cite{nielsen_chuang_2010}:
     \end{aligned}
 \end{equation}
 
-Note that in Definition \ref{def:stabilizer} it has been demanded that stabilizers are a 
-subgroup of the multilocal Pauli operators. This does not hold true for an arbitrary
-$U$ but there exists a group for which $S'$ will be a set of stabilizers.
+Note that in Definition \ref{def:stabilizer} it has been demanded that
+stabilizers are a subgroup of the multilocal Pauli operators. This does not
+hold true for an arbitrary $U$ but there exists a group for which $S'$ will be
+a set of stabilizers.
 
 \begin{definition}
     For $n$ qbits
@@ -171,7 +175,8 @@ $U$ but there exists a group for which $S'$ will be a set of stabilizers.
         C_n := \left\{U \in SU(2^n) \middle| \forall p \in P_n: UpU^\dagger \in P_n \right\}
     \end{equation}
 
-    is called the Clifford group. $C_1 =: C_L$ is called the local Clifford group \cite{andersbriegel2005}.
+    is called the Clifford group. $C_1 =: C_L$ is called the local Clifford
+    group \cite{andersbriegel2005}.
 \end{definition}
 
 \begin{theorem}
@@ -179,10 +184,11 @@ $U$ but there exists a group for which $S'$ will be a set of stabilizers.
     \begin{enumerate}
         \item{$C_L$ can be generated using only $H$ and $S$.}
         \item{$C_L$ can be generated from $\sqrt{iZ} = \exp(\frac{i\pi}{4}) S^\dagger$ 
-            and $\sqrt{-iX} = \frac{1}{\sqrt{2}} \left(\begin{array}{cc} 1 & -i \\ -i & 1 \end{array}\right)$. 
+            and $\sqrt{-iX} = \frac{1}{\sqrt{2}} \left(\begin{array}{cc} 1 & -i
+            \\ -i & 1 \end{array}\right)$. 
 
-                Also $C_L$ is generated by a product of at most $5$ matrices $\sqrt{iZ}$, $\sqrt{-iX}$.
-                }
+                Also $C_L$ is generated by a product of at most $5$ matrices
+            $\sqrt{iZ}$, $\sqrt{-iX}$.  }
             \item{$C_n$ can be generated using $C_L$ and $CZ$ or $CX$.} 
     \end{enumerate} 
 \end{theorem}
@@ -202,26 +208,26 @@ $U$ but there exists a group for which $S'$ will be a set of stabilizers.
     \end{enumerate} 
 \end{proof}
 
-This is quite an important result: As under a transformation $U \in C_n$ $S' = U^\dagger S U$ is a set of 
-$n$ independent stabilizers and $\ket{\psi'}$ is stabilized by $S'$ one can consider
-the dynamics of the stabilizers instead of the actual state. This is considerably more
-efficient as only $n$ stabilizers have to be modified, each being just the tensor
-product of $n$ Pauli matrices. This has led to the simulation using stabilizer tableaux
+This is quite an important result: As under a transformation $U \in C_n$ $S'
+= U^\dagger S U$ is a set of $n$ independent stabilizers and $\ket{\psi'}$ is
+stabilized by $S'$ one can consider the dynamics of the stabilizers instead of
+the actual state. This is considerably more efficient as only $n$ stabilizers
+have to be modified, each being just the tensor product of $n$ Pauli matrices.
+This has led to the simulation using stabilizer tableaux
 \cite{gottesman_aaronson2008}.
 
-\subsubsection{Measurements on Stabilizer States}
-\label{ref:meas_stab}
+\subsubsection{Measurements on Stabilizer States} \label{ref:meas_stab}
 
-Interestingly also measurements are dynamics covered by the stabilizers.
-When an observable $g_a \in \{\pm X_a, \pm Y_a, \pm Z_a\}$ acting on qbit $a$ is measured
-one has to consider the projector
+Interestingly also measurements are dynamics covered by the stabilizers.  When
+an observable $g_a \in \{\pm X_a, \pm Y_a, \pm Z_a\}$ acting on qbit $a$ is
+measured one has to consider the projector
 
-\begin{equation}
-    P_{g_a,s} = \frac{I + (-1)^s g_a}{2}.
+\begin{equation} 
+P_{g_a,s} = \frac{I + (-1)^s g_a}{2}.  
 \end{equation}
 
-If now $g_a$ commutes with all $S^{(i)}$ a result of $s=0$ is measured with probability $1$
-and the stabilizers are left unchanged:
+If now $g_a$ commutes with all $S^{(i)}$ a result of $s=0$ is measured with
+probability $1$ and the stabilizers are left unchanged:
 
 \begin{equation}
     \begin{aligned}
@@ -232,26 +238,26 @@ and the stabilizers are left unchanged:
     \end{aligned}
 \end{equation}
 
-As the state that is stabilized by $S$ is unique $\ket{\psi'} = \ket{\psi}$ \cite{nielsen_chuang_2010}.
+As the state that is stabilized by $S$ is unique $\ket{\psi'} = \ket{\psi}$
+\cite{nielsen_chuang_2010}.  
 
-If $g_a$ does not commute with all stabilizers the following lemma gives 
-the result of the measurement.
+If $g_a$ does not commute with all stabilizers the following lemma gives the
+result of the measurement.
 
 \begin{lemma}
-    \label{lemma:stab_measurement}
-    Let $J := \left\{ S^{(i)} \middle| [g_a, S^{(i)}] \neq 0\right\} \neq \{\}$ and
-    $J^c := \left\{S^{(i)} \middle| S^{(i)} \notin J \right\}$. When measuring 
-    $\frac{I + (-1)^s g_a}{2} $
-    $s=1$ and $s=0$ are obtained with probability $\frac{1}{2}$ and after choosing
-    a $j \in J$ the new state $\ket{\psi'}$ is stabilized by \cite{nielsen_chuang_2010} 
+    \label{lemma:stab_measurement} Let $J := \left\{ S^{(i)} \middle| [g_a,
+    S^{(i)}] \neq 0\right\} \neq \{\}$ and $J^c := \left\{S^{(i)} \middle|
+S^{(i)} \notin J \right\}$. When measuring $\frac{I + (-1)^s g_a}{2} $ $s=1$
+and $s=0$ are obtained with probability $\frac{1}{2}$ and after choosing a $j
+\in J$ the new state $\ket{\psi'}$ is stabilized by \cite{nielsen_chuang_2010} 
     \begin{equation}
         \langle \{(-1)^s g_a\} \cup \left\{S^{(i)} S^{(j)} \middle| S^{(i)} \in J \setminus \{S^{(j)}\} \right\} \cup J^c \rangle.
     \end{equation}
 \end{lemma}
 
 \begin{proof}
-    As $g_a$ is a Pauli operator and $S^{(i)} \in J$ are multi-local Pauli operators,
-    $S^{(i)}$ and $g_a$ anticommute. Choose a $S^{(j)} \in J$. Then
+    As $g_a$ is a Pauli operator and $S^{(i)} \in J$ are multi-local Pauli
+    operators, $S^{(i)}$ and $g_a$ anticommute. Choose a $S^{(j)} \in J$. Then
 
     \begin{equation}
     \begin{aligned}
@@ -278,41 +284,44 @@ the result of the measurement.
     \notag
     \end{equation}
 
-    The state after measurement is stabilized by $S^{(j)}S^{(i)}$ $i,j \in J$, and by 
-    $S^{(i)} \in J^c$. $(-1)^sg_a$ trivially stabilizes $\ket{\psi'}$ \cite{nielsen_chuang_2010}.
+    The state after measurement is stabilized by $S^{(j)}S^{(i)}$ $i,j \in J$,
+    and by $S^{(i)} \in J^c$. $(-1)^sg_a$ trivially stabilizes $\ket{\psi'}$
+    \cite{nielsen_chuang_2010}.
 \end{proof}
 
 \subsection{The VOP-free Graph States}
 \subsubsection{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 the following section about graph states.
+This section will discuss the vertex operator (VOP)-free graph states. Why they
+are called vertex operator-free will be clear in the following section about
+graph states.
 
-\begin{definition}
-    \label{def:graph}
-    The tuple $(V, E)$ is called a graph iff $V$ is a set of vertices with $|V| = n \in \mathbb{N}$ elements.
-    In the following $V = \{0, ..., n-1\}$ will be used.
-    $E$ is the set of edges $E \subset \left\{\{i, j\} \middle| i,j \in V, i \neq j\right\}$.
+\begin{definition} \label{def:graph} The tuple $(V, E)$ is called a graph iff
+    $V$ is a set of vertices with $|V| = n \in \mathbb{N}$ elements.  In the
+    following $V = \{0, ..., n-1\}$ will be used.  $E$ is the set of edges $E
+    \subset \left\{\{i, j\} \middle| i,j \in V, i \neq j\right\}$.
 
-    For a vertex $i$ $n_i := \left\{j \in V \middle| \{i, j\} \in E\right\}$ is called the neighbourhood
-    of $i$ \cite{hein_eisert_briegel2008}.
+    For a vertex $i$ $n_i := \left\{j \in V \middle| \{i, j\} \in E\right\}$ is
+called the neighbourhood of $i$ \cite{hein_eisert_briegel2008}.
 \end{definition}
 
-This definition of a graph is way less general than the definition of a graph in graph theory.
-Using this definition will however allow to avoid an extensive list of constraints on the 
-graph from graph theory that are implied in this definition.
+This definition of a graph is way less general than the definition of a graph
+in graph theory.  Using this definition will however allow to avoid an
+extensive list of constraints on the graph from graph theory that are implied
+in this definition.
 
 \begin{definition}
     For a graph $G = (V = \{0, ..., n-1\}, E)$ the associated stabilizers are 
     \begin{equation}
         K_G^{(i)} := X_i \prod\limits_{\{i,j\} \in E} Z_j
     \end{equation}
-    for all $i \in V$. The vertex operator free graph state $\ket{\bar{G}}$ is the state stabilized by 
-    $\langle K_G^{(i)} \rangle_{i = 0, ..., n-1}$ \cite{hein_eisert_briegel2008}.
+    for all $i \in V$. The vertex operator free graph state $\ket{\bar{G}}$ is
+    the state stabilized by $\langle K_G^{(i)} \rangle_{i = 0, ..., n-1}$
+    \cite{hein_eisert_briegel2008}.
 \end{definition}
 
-It is clear that the $K_G^{(i)}$ are multilocal Pauli operators. That they commute
-is clear if $\{a,b\} \notin E$. If $\{a, b\} \in E$
+It is clear that the $K_G^{(i)}$ are multilocal Pauli operators. That they
+commute is clear if $\{a,b\} \notin E$. If $\{a, b\} \in E$
 
 \begin{equation}
 \begin{aligned}
@@ -334,10 +343,10 @@ is clear if $\{a,b\} \notin E$. If $\{a, b\} \in E$
 \end{equation}
 
 
-This definition of a graph state might not seem to be straight forward
-but recalling Theorem \ref{thm:unique_s_state} it is clear that $\ket{\bar{G}}$
-is unique. The following lemma will provide a way to construct this state
-from the graph. 
+This definition of a graph state might not seem to be straight forward but
+recalling Theorem \ref{thm:unique_s_state} it is clear that $\ket{\bar{G}}$ is
+unique. The following lemma will provide a way to construct this state from the
+graph. 
 
 \begin{lemma}
     For a graph $G = (V, E)$ the associated state $\ket{\bar{G}}$ is
@@ -355,10 +364,11 @@ from the graph.
 
 
     Let $\ket{+} := \left(\prod\limits_{l \in V} H_l\right) \ket{0}$ as before.
-    Note that for any $X_i$: $X_i \ket{+} = +1 \ket{+}$.
-    In the following discussion the direction $\prod\limits_{\{l,k\} \in E} := \prod\limits_{\{l,k\} \in E, l < k}$
-    is introduced as the graph is undirected and edges must not be handled twice.
-    Set $\ket{\tilde{G}} := \left(\prod\limits_{\{l,k\} \in E} CZ_{l,k} \right)\ket{+}$.
+    Note that for any $X_i$: $X_i \ket{+} = +1 \ket{+}$.  In the following
+    discussion the direction $\prod\limits_{\{l,k\} \in E} :=
+    \prod\limits_{\{l,k\} \in E, l < k}$ is introduced as the graph is
+    undirected and edges must not be handled twice.  Set $\ket{\tilde{G}} :=
+    \left(\prod\limits_{\{l,k\} \in E} CZ_{l,k} \right)\ket{+}$.
 
     \begin{equation}
         \begin{aligned}
@@ -371,10 +381,12 @@ from the graph.
                 \left( \ket{0}\bra{0}_k \otimes I_l + (-1)^{\delta_{i,l}}\ket{1}\bra{1}_k \otimes Z_l\right) X_i \ket{+} \\
         \end{aligned}
     \end{equation}
-    As $X,Z$ anticommute. $X_i$ can now be absorbed into $\ket{+}$. The next step is a bit tricky:
-    A $Z_j$ can be absorbed into a $\ket{0}\bra{0}_j$ giving no phase or into a $\ket{1}\bra{1}_j$ yielding
-    a phase of $-1$. If there is no projector on $j$ the $Z_j$ can be commuted to the next projector. 
-    It is guaranteed that a projector on $j$ exists by the definition of $\ket{\tilde{G}}$.
+    As $X,Z$ anticommute. $X_i$ can now be absorbed into $\ket{+}$. The next
+    step is a bit tricky: A $Z_j$ can be absorbed into a $\ket{0}\bra{0}_j$
+    giving no phase or into a $\ket{1}\bra{1}_j$ yielding a phase of $-1$. If
+    there is no projector on $j$ the $Z_j$ can be commuted to the next
+    projector.  It is guaranteed that a projector on $j$ exists by the
+    definition of $\ket{\tilde{G}}$.
 
     \begin{equation}
         \begin{aligned}
@@ -385,15 +397,17 @@ from the graph.
         \end{aligned}
     \end{equation}
 
-    The $\delta_{i,l} + \delta_{j,k}$ is either $0$ or $2$ by the definitions of $K_G^{(i)}$ and $\ket{\tilde{G}}$.
+    The $\delta_{i,l} + \delta_{j,k}$ is either $0$ or $2$ by the definitions
+    of $K_G^{(i)}$ and $\ket{\tilde{G}}$.
 \end{proof}
 
 \subsubsection{Dynamics of the VOP-free Graph States}
 
-This representation gives an immediate result to how the stabilizers $\langle K_G^{(i)} \rangle_i$ change
-under the $CZ$ gate: When applying $CZ_{i,j}$ on $G = (V, E)$ the edge $\{i,j\}$ is toggled,
-resulting in a multiplication of $Z_j$ to $K_G^{(i)}$ and $Z_i$ to $K_G^{(j)}$. Toggling edges
-is done by using the symmetric set difference:
+This representation gives an immediate result to how the stabilizers $\langle
+K_G^{(i)} \rangle_i$ change under the $CZ$ gate: When applying $CZ_{i,j}$ on $G
+= (V, E)$ the edge $\{i,j\}$ is toggled, resulting in a multiplication of $Z_j$
+to $K_G^{(i)}$ and $Z_i$ to $K_G^{(j)}$. Toggling edges is done by using the
+symmetric set difference:
 
 \begin{definition}
     For two finite sets $A,B$ the symmetric set difference $\Delta$ is 
@@ -405,7 +419,8 @@ is done by using the symmetric set difference:
 \end{definition}
 
 Toggling an edge $\{i, j\}$ updates $E' = E \Delta \left\{\{i,j\}\right\}$.
-Another transformation on the VOP-free graph states for a vertex $a \in V$ is \cite{andersbriegel2005}
+Another transformation on the VOP-free graph states for a vertex $a \in V$ is
+\cite{andersbriegel2005}
 
 \begin{equation}
     M_a := \sqrt{-iX_a} \prod\limits_{j\in n_a} \sqrt{iZ_j}.
@@ -428,10 +443,10 @@ that will be used later\cite{andersbriegel2005}.
     \end{equation}
 \end{lemma}
 \begin{proof}
-    $\ket{\bar{G}'}$ is stabilized by $\langle M_a K_G^{(i)} M_a^\dagger \rangle_i$, so it is sufficient
-    to study how the $ K_G^{(i)}$ change under $M_a$.
-    At first note that $M_a^2 \alpha K_G^{(a)} \Rightarrow [K_G^{(a)}, M_a] = 0$.
-    Further $\forall i\in V\setminus n_a$ $[K_G^{(i)}, M_a] = 0$,
+    $\ket{\bar{G}'}$ is stabilized by $\langle M_a K_G^{(i)} M_a^\dagger
+    \rangle_i$, so it is sufficient to study how the $ K_G^{(i)}$ change under
+    $M_a$.  At first note that $M_a^2 \alpha K_G^{(a)} \Rightarrow [K_G^{(a)},
+    M_a] = 0$.  Further $\forall i\in V\setminus n_a$ $[K_G^{(i)}, M_a] = 0$,
     so the first two equations follow trivially. For $j \in n_a$ set 
 
     \begin{equation}
@@ -453,11 +468,12 @@ that will be used later\cite{andersbriegel2005}.
         \end{aligned}
     \end{equation}
 
-    One can now construct a new set of $K_{G'}^{(i)}$ such that $M_a \ket{\bar{G}}$ is the $+1$ eigenstate
-    of the new $K_{G'}^{(i)}$. Because $\forall i\in V\setminus n_a$ $[K_G^{(i)}, M_a] = 0$ 
-    it is clear that $\forall j \notin n_a$  $K_{G'}^{(j)} = K_G^{(j)}$.
-    To construct the $K_{G'}^{(i)}$ let for some $j \in n_a$ $n_a =: \{j\} \cup F$ and $n_j =: \{a\} \cup D$.
-    Then follows:
+    One can now construct a new set of $K_{G'}^{(i)}$ such that $M_a
+    \ket{\bar{G}}$ is the $+1$ eigenstate of the new $K_{G'}^{(i)}$. Because
+    $\forall i\in V\setminus n_a$ $[K_G^{(i)}, M_a] = 0$ it is clear that
+    $\forall j \notin n_a$  $K_{G'}^{(j)} = K_G^{(j)}$.  To construct the
+    $K_{G'}^{(i)}$ let for some $j \in n_a$ $n_a =: \{j\} \cup F$ and $n_j =:
+    \{a\} \cup D$.  Then follows:
 
     \begin{equation}
         \begin{aligned}
@@ -479,34 +495,36 @@ that will be used later\cite{andersbriegel2005}.
                         = K_{G'}^{(j)}K_{G}^{(a)}\ket{\bar{G}'} = K_{G'}^{(j)}\ket{\bar{G}'}
     \end{equation}
 
-    Because $\left\{K_G^{(i)} \middle| i \notin n_a\right\} \cup \left\{S^{(i)} \middle| i\in n_a\right\}$ and 
-    $\left\{K_G^{(i)} \middle| i \notin n_a\right\} \cup \left\{K_{G'}^{(i)} \middle| i\in n_a \right\}$ are both $n$ commuting
-    multi-local Pauli operators where the $S^{(i)}$ can be generated from the $K_{G'}^{(i)}$
-    and $\ket{\bar{G}'}$ is a $+1$ eigenstate of $K_{G'}^{(j)}$ 
-    $\langle\left\{K_G^{(i)} \middle| i \notin n_a\right\} \cup \left\{K_{G'}^{(i)} \middle| i\in n_a \right\}\rangle$
-    are the stabilizers of $\ket{\bar{G}'}$. Therefore the associated graph is changed as given
-    in the third equation.
+    Because $\left\{K_G^{(i)} \middle| i \notin n_a\right\} \cup \left\{S^{(i)}
+    \middle| i\in n_a\right\}$ and $\left\{K_G^{(i)} \middle| i \notin
+n_a\right\} \cup \left\{K_{G'}^{(i)} \middle| i\in n_a \right\}$ are both $n$
+commuting multi-local Pauli operators where the $S^{(i)}$ can be generated from
+the $K_{G'}^{(i)}$ and $\ket{\bar{G}'}$ is a $+1$ eigenstate of $K_{G'}^{(j)}$
+$\langle\left\{K_G^{(i)} \middle| i \notin n_a\right\} \cup \left\{K_{G'}^{(i)}
+\middle| i\in n_a \right\}\rangle$ are the stabilizers of $\ket{\bar{G}'}$.
+Therefore the associated graph is changed as given in the third equation.
 \end{proof}
 
 \subsection{Graph States}
 
-The definition of a VOP-free graph state above raises an obvious question: 
-Can any stabilizer state be described using just a graph?
-The answer is straight forward: No. The most simple cases are the single qbit
-stated $\ket{0},\ket{1}$ and $\ket{+_Y}, \ket{-_Y}$. But there is an extension
-to the VOP-free graph states that allows the representation of an arbitrary
-stabilizer state. The proof that indeed any state can be represented is
-purely constructive. As seen in Theorem \ref{thm:clifford_group_approx} any $c \in C_n$
-can be constructed from $CZ$ and $C_L$. In the following discussion it will become
-clear that both $C_L$ and $CZ$ can be applied to a general graph state.
+The definition of a VOP-free graph state above raises an obvious question: Can
+any stabilizer state be described using just a graph?  The answer is straight
+forward: No. The most simple cases are the single qbit stated $\ket{0},\ket{1}$
+and $\ket{+_Y}, \ket{-_Y}$. But there is an extension to the VOP-free graph
+states that allows the representation of an arbitrary stabilizer state. The
+proof that indeed any state can be represented is purely constructive. As seen
+in Theorem \ref{thm:clifford_group_approx} any $c \in C_n$ can be constructed
+from $CZ$ and $C_L$. In the following discussion it will become clear that both
+$C_L$ and $CZ$ can be applied to a general graph state.
 
 \subsubsection{Graph States and Vertex Operators}
 \label{ref:g_states_vops}
 
 \begin{definition}
     \label{def:g_state}
-    A tuple $(V, E, O)$ is called the graphical representation of a stabilizer state
-    if $(V, E)$ is a graph as in Definition \ref{def:graph} and $O = \{o_1, ..., o_n\}$ where $o_i \in C_L$.
+    A tuple $(V, E, O)$ is called the graphical representation of a stabilizer
+    state if $(V, E)$ is a graph as in Definition \ref{def:graph} and $O
+    = \{o_1, ..., o_n\}$ where $o_i \in C_L$.
 
     The state $\ket{G}$ is defined by the eigenvalue relation:
 
@@ -517,36 +535,39 @@ clear that both $C_L$ and $CZ$ can be applied to a general graph state.
     $o_i$ are called the vertex operators of $\ket{G}$ \cite{andersbriegel2005}. 
 \end{definition}
 
-Recalling the dynamics of stabilizer states the following relation follows immediately:
+Recalling the dynamics of stabilizer states the following relation follows
+immediately:
 
 \begin{equation}
     \ket{G} = \left(\prod\limits_{j=1}^no_j\right) \ket{\bar{G}}
 \end{equation}
 
-The great advantage of this representation of a stabilizer state is its space requirement:
-Instead of storing $n^2$ $P$ matrices only some vertices (which often are implicit),
-the edges and some vertex operators ($n$ matrices) have to be stored. The following theorem
-will improve this even further: instead of $n$ matrices it is sufficient to store $n$ integers
-representing the vertex operators:
+The great advantage of this representation of a stabilizer state is its space
+requirement: Instead of storing $n^2$ $P$ matrices only some vertices (which
+often are implicit), the edges and some vertex operators ($n$ matrices) have to
+be stored. The following theorem will improve this even further: instead of $n$
+matrices it is sufficient to store $n$ integers representing the vertex
+operators:
 
 \begin{theorem}
     $C_L$ has $24$ degrees of freedom \cite{andersbriegel2005}.
 \end{theorem}
-\begin{proof}
-    It is clear that $\forall a \in C_L$ a is a group isomorphism $P \rightarrow P$: $apa^\dagger a p' a^\dagger = a pp'a^\dagger$.
-    Therefore $a$ will preserve the (anti-)commutator relations of $P$.
-    Further note that $Y = iXZ$, so one has to consider the anti-commutator relations
-    of $X,Z$ only.
+\begin{proof} It is clear that $\forall a \in C_L$ a is a group isomorphism $P
+    \rightarrow P$: $apa^\dagger a p' a^\dagger = a pp'a^\dagger$.  Therefore
+    $a$ will preserve the (anti-)commutator relations of $P$.  Further note
+    that $Y = iXZ$, so one has to consider the anti-commutator relations of
+    $X,Z$ only.
 
-    As the transformations are unitary they preserve eigenvalues, so $X$ can be mapped
-    to $\pm X, \pm Y, \pm Z$ which gives $6$ degrees of freedom. Furthermore the image of $Z$ has to 
-    anti-commute with the image of $X$ so $Z$ has four possible images under the transformation.
-    This gives another $4$ degrees of freedom and a total of $24$.
-\end{proof}
+    As the transformations are unitary they preserve eigenvalues, so $X$ can be
+    mapped to $\pm X, \pm Y, \pm Z$ which gives $6$ degrees of freedom.
+    Furthermore the image of $Z$ has to anti-commute with the image of $X$ so
+    $Z$ has four possible images under the transformation.  This gives another
+$4$ degrees of freedom and a total of $24$.  \end{proof}
 
-From now on $C_L = \langle H, S \rangle$ (disregarding a global phase) will be used.
-One can show (by construction) that $H, S$ generate a possible choice of $C_L$, as is 
-$C_L = \langle \sqrt{-iX}, \sqrt{-iZ}\rangle$ which is required in one specific operation on graph states \cite{andersbriegel2005}.
+From now on $C_L = \langle H, S \rangle$ (disregarding a global phase) will be
+used.  One can show (by construction) that $H, S$ generate a possible choice of
+$C_L$, as is $C_L = \langle \sqrt{-iX}, \sqrt{-iZ}\rangle$ which is required in
+one specific operation on graph states \cite{andersbriegel2005}.
 
 \begin{equation}
     S = \left(\begin{array}{cc} 1 & 0 \\ 0 & i \end{array}\right)
@@ -561,12 +582,14 @@ $C_L = \langle \sqrt{-iX}, \sqrt{-iZ}\rangle$ which is required in one specific
 
 \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\cite{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 are changed 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
+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\cite{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 are changed 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
 
 \begin{equation}
     \begin{aligned}
@@ -575,30 +598,33 @@ one sees that just the vertex operators are changed and the new vertex operators
     \end{aligned}
 \end{equation}
 
-The action of a $CZ$ gate on the state $(V, E, O)$ is in most cases less trivial.
-Let $i \neq j$ be two qbits, now consider the action of $CZ_{a,b}$ on $(V, E, O)$.
-The cases given here follow the implementation of a $CZ$ application in \cite{pyqcs},
-the respective paragraphs from \cite{andersbriegel2005} are given in italic.
-Most of the discussion follows the one given in \cite{andersbriegel2005} closely.
+The action of a $CZ$ gate on the state $(V, E, O)$ is in most cases less
+trivial.  Let $i \neq j$ be two qbits, now consider the action of $CZ_{a,b}$ on
+$(V, E, O)$.  The cases given here follow the implementation of a $CZ$
+application in \cite{pyqcs}, the respective paragraphs from
+\cite{andersbriegel2005} are given in italic.  Most of the discussion follows
+the one given in \cite{andersbriegel2005} closely.
 
 
 \textbf{Case 1}(\textit{Case 1})\textbf{:}\\
-Both $o_a$ and $o_b$ commute with $CZ_{a,b}$. This is the case for exactly
-four vertex operators: $\mathcal{Z} = \left\{I, Z, S, S^\dagger\right\}$.
-In this case the CZ can be pulled past the vertex operators and just the edges
-are changed to $E' = E \Delta \left\{\{a,b\}\right\}$.
+Both $o_a$ and $o_b$ commute with $CZ_{a,b}$. This is the case for exactly four
+vertex operators: $\mathcal{Z} = \left\{I, Z, S, S^\dagger\right\}$.  In this
+case the CZ can be pulled past the vertex operators and just the edges are
+changed to $E' = E \Delta \left\{\{a,b\}\right\}$.
 
 \textbf{Case 2}(\textit{Sub-Sub-Case 2.2.1})\textbf{:}\\
-The two qbits are isolated: From the definition of the graph state one can derive that
-any isolated clique of the graph can be treated independently. Therefore the two isolated qbits
-can be treated as an independent state and the set of two qbit stabilizer states is finite. An
-upper bound to the number of two qbit stabilizer states is given by $2\cdot24^2$:
-A factor of two for the options with and withoit an edge between the 
-qbits and $24$ Clifford operators on each vertex.
+The two qbits are isolated: From the definition of the graph state one can
+derive that any isolated clique of the graph can be treated independently.
+Therefore the two isolated qbits can be treated as an independent state and the
+set of two qbit stabilizer states is finite. An upper bound to the number of
+two qbit stabilizer states is given by $2\cdot24^2$: A factor of two for the
+options with and withoit an edge between the qbits and $24$ Clifford operators
+on each vertex.
 
-All those states and the resulting state after a $CZ$ application can be computed which leads to
-another interesting result that will be useful later: If one vertex has the vertex operator $I$ the
-resulting state can be chosen such that at least one of the vertex operators is $I$ again and in particular
+All those states and the resulting state after a $CZ$ application can be
+computed which leads to another interesting result that will be useful later:
+If one vertex has the vertex operator $I$ the resulting state can be chosen
+such that at least one of the vertex operators is $I$ again and in particular
 the identity on the vertex can be preserved under the application of a $CZ$.
 
 \textbf{Case 3}(\textit{Case 2, but one sub-case has been handled})\textbf{:}\\
@@ -607,10 +633,9 @@ has non-operand (i.e. neighbours that are neither $a$ nor $b$) neighbours. In
 this case one can try to clear the vertex operators which will succeed for at
 least one qbit.
 
-The transformation given in 
-Lemma \ref{lemma:M_a} is used to "clear" the vertex operators. Recalling that
-the transformation $M_j$ toggles the neighbourhood of vertex $j$ gives substance
-to the following theorem:
+The transformation given in Lemma \ref{lemma:M_a} is used to "clear" the vertex
+operators. Recalling that the transformation $M_j$ toggles the neighbourhood of
+vertex $j$ gives substance to the following theorem:
 
 \begin{theorem}
     A graph state $\ket{G}$ associated with $(V, E, O)$ is invariant when
@@ -623,11 +648,13 @@ to the following theorem:
     Without proof.
 \end{theorem}
 
-As stated in \ref{ref:g_states_vops} $C_L$ is also generated by $\sqrt{-iX}$ and $\sqrt{iZ}$.
-This yields an algorithm to reduce the vertex operator of a non-isolated qbit $j$ to the identity.
-The combined operation of toggling the neighbourhood of $j$ and right-multiplying
-$M_j^\dagger$ is called $L_j$ transformation, which transforms $(V, E, O)$ into a so-called
-local Clifford equivalent graphical representation. The algorithm is given by the following steps:
+As stated in \ref{ref:g_states_vops} $C_L$ is also generated by $\sqrt{-iX}$
+and $\sqrt{iZ}$.  This yields an algorithm to reduce the vertex operator of
+a non-isolated qbit $j$ to the identity.  The combined operation of toggling
+the neighbourhood of $j$ and right-multiplying $M_j^\dagger$ is called $L_j$
+transformation, which transforms $(V, E, O)$ into a so-called local Clifford
+equivalent graphical representation. The algorithm is given by the following
+steps:
 
 \begin{enumerate}
     \item{Check whether $n_a \setminus \{b\} \neq \{\}$. If this condition does not hold
@@ -668,7 +695,8 @@ non-operand neighbours and $b$ does not. Now the state $\ket{G}$ has the form
     \end{aligned}
 \end{equation}
 
-As $o_b$ commutes with all operators but $CZ_{a,b}$ and $s \in \{0, 1\}$ indicates whether there is an edge between $a$ and $b$.
+As $o_b$ commutes with all operators but $CZ_{a,b}$ and $s \in \{0, 1\}$
+indicates whether there is an edge between $a$ and $b$.
 
 \begin{equation}
     \begin{aligned}
@@ -707,8 +735,8 @@ operators. For this consider the projector $P_{a,s} = \frac{I + (-1)^sZ_a}{2}$
     \end{aligned}
 \end{equation}
 
-This transformed projector has the important property that it still is a Pauli projector
-as $o_a$ is a Clifford operator 
+This transformed projector has the important property that it still is a Pauli
+projector as $o_a$ is a Clifford operator 
 
 \begin{equation}
     \begin{aligned}
@@ -719,9 +747,10 @@ as $o_a$ is a Clifford operator
     \end{aligned}
 \end{equation}
 
-Where $\tilde{g}_a \in \{+1, -1\}\cdot\{X_a, Y_a, Z_a\}$. Therefore, it is enough to study the measurements
-of any Pauli operator on the vertex operator free graph states. The commutators of the observable
-with the $K_G^{(i)}$ are quite easy to compute. Note that Pauli matrices either commute or anticommute 
+Where $\tilde{g}_a \in \{+1, -1\}\cdot\{X_a, Y_a, Z_a\}$. Therefore, it is
+enough to study the measurements of any Pauli operator on the vertex operator
+free graph states. The commutators of the observable with the $K_G^{(i)}$ are
+quite easy to compute. Note that Pauli matrices either commute or anticommute
 and it is easier to list the operators that anticommute
 
 \begin{equation}
@@ -732,15 +761,17 @@ and it is easier to list the operators that anticommute
     \end{aligned}
 \end{equation}
 
-This gives one immediate result: if a qbit $a$ is isolated and the operator $\tilde{g}_a = X_a (-X_a)$ 
-is measured the result $s=0(1)$ is obtained with probability $1$ and $(V, E, O)$ is unchanged.
-In any other case the results $s=1$ and $s=0$ have probability $\frac{1}{2}$ and both 
-graph and vertex operators have to be updated. Further it is clear that measurements of $-\tilde{g}_a$
-and $\tilde{g}_a$ are related by just inverting the result $s$.
+This gives one immediate result: if a qbit $a$ is isolated and the operator
+$\tilde{g}_a = X_a (-X_a)$ is measured the result $s=0(1)$ is obtained with
+probability $1$ and $(V, E, O)$ is unchanged.  In any other case the results
+$s=1$ and $s=0$ have probability $\frac{1}{2}$ and both graph and vertex
+operators have to be updated. Further it is clear that measurements of
+$-\tilde{g}_a$ and $\tilde{g}_a$ are related by just inverting the result $s$.
 
-The calculations to obtain the transformation on graph and vertex operators are lengthy and follow
-the scheme of Lemma \ref{lemma:M_a}. \cite[Section IV]{hein_eisert_briegel2008} also contains 
-the steps required to obtain the following results
+The calculations to obtain the transformation on graph and vertex operators are
+lengthy and follow the scheme of Lemma \ref{lemma:M_a}. \cite[Section
+IV]{hein_eisert_briegel2008} also contains the steps required to obtain the
+following results
 
 \begin{equation}
     \begin{aligned}
@@ -749,10 +780,11 @@ the steps required to obtain the following results
     \end{aligned}
 \end{equation}
 
-These transformations split it two parts: the first is a result of Lemma \ref{lemma:stab_measurement}.
-The second part makes sure that the qbit $a$ is diagonal in measured observable and has the correct eigenvalue.
-When comparing with Lemma \ref{lemma:stab_measurement} in both cases $Y,Z$ the anticommuting stabilizer 
-$K_G^{(a)}$ is chosen. The graph is changed according to
+These transformations split it two parts: the first is a result of Lemma
+\ref{lemma:stab_measurement}.  The second part makes sure that the qbit $a$ is
+diagonal in measured observable and has the correct eigenvalue.  When comparing
+with Lemma \ref{lemma:stab_measurement} in both cases $Y,Z$ the anticommuting
+stabilizer $K_G^{(a)}$ is chosen. The graph is changed according to
 
 \begin{equation}
     \begin{aligned}