% vim: ft=tex

\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.

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}

\subsubsection{Local Pauli Group and Multilocal Pauli Group}

\begin{definition}
    \begin{equation}
        P := \{\pm 1, \pm i\} \cdot \{I, X, Y, Z\}
    \end{equation}

    Is called the Pauli group.
\end{definition}

The group property of $P$ can be verified easily. Note that 
the elements of $P$ either commute or anticommute.

\begin{definition}
    For $n$ qbits

    \begin{equation}
        P_n := \left\{\bigotimes\limits_{i=0}^{n-1} p_i | p_i \in P\right\}
    \end{equation}

    is called the multilocal Pauli group on $n$ qbits. 
\end{definition}

The group property of $P_n$ follows directly from its definition
via the tensor product as do the (anti-)commutator relationships.
%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 $p = -I$.

\subsubsection{Stabilizers}

\begin{definition}
    \label{def:stabilizer}
    An abelian subgroup $S = \{S^{(0)}, ..., S^{(N)}\}$ of $P_n$ is called a set of stabilizers iff
    \begin{enumerate}
        \item{$\forall i,j = 1, ..., N$ $[S^{(i)}, S^{(j)}] = 0$ $S^{(i)}$ and $S^{(j)}$ commute}
        \item{$-I \notin S$}
    \end{enumerate}
\end{definition}

\begin{lemma}
    If $S$ is a set of stabilizers, the following statements are follow
    directly

    \begin{enumerate}
        \item{$\pm iI \notin S$}
        \item{$(S^{(i)})^2 = I$ for all $i$}
        \item{$S^{(i)}$ are hermitian for all $i$}
    \end{enumerate}
\end{lemma}
\begin{proof}

    \begin{enumerate}
        \item{$(iI)^2 = (-iI)^2 = -I$. Which contradicts the definition of $S$.}
        \item{From the definition of $S$ ($G_n$ respectively) follows that any
            $S^{(i)} \in S$ has the form $\pm i^l (\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j)$ where
            $\tilde{p}_j \in \{X, Y, Z, I\}$ and $l \in \{0, 1\}$. As $(\bigotimes\limits_{j=0}^{n-1} \tilde{p}_j)$
            is hermitian $(S^{(i)})^2$ is either $+I$ or $-I$. As $-I \notin S$ $(S^{(i)})^2 = I$ follows directly. 
            }
        \item{Following the argumentation above $(S^{(i)})^2 = -I \Leftrightarrow l=1$ 
            therefore $(S^{(i)})^2 = -I \Leftrightarrow (S^{(i)})^\dagger \neq (S^{(i)})$.}
    \end{enumerate}

\end{proof}

As considering all elements of a group can be unpractical for some calculations
the generators of a group are introduced. It is usually enough to discuss the generator's
properties 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

    $$ \langle g_1, ..., g_m \rangle \equiv \langle g_i \rangle_{i=1,...,m}$$

    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 properties of a set of stabilizers that are used in the discussions can be studied using only its
generators.

\subsubsection{Stabilizer States}

One important basic property of quantum mechanics is that hermitian operators have real eigenvalues
and eigenspaces 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 operators that commute 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

    \begin{equation}
        V_S := \{\ket{\psi} | S^{(i)}\ket{\psi} = +1\ket{\psi} \forall S^{(i)} \in S\}
    \end{equation}

    is called the space of stabilizer states associated with $S$ and one says
    $\ket{\psi}$ is stabilized by $S$.
\end{definition}

It is clear that it is sufficient to show the stabilization property for the generators of
$S$, 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 a set $S = \langle S^{(i)} \rangle_{i=1, ..., n}$ 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}

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. 

\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:

\begin{equation}
    \begin{aligned}
    \ket{\psi'} &= U \ket{\psi} \\
                &= U S^{(i)} \ket{\psi} \\
                &= U S^{(i)} U^\dagger U\ket{\psi} \\
                &= U S^{(i)} U^\dagger \ket{\psi'} \\
                &= S^{\prime(i)} \ket{\psi'} \\
    \end{aligned}
\end{equation}

Note that in \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
    \begin{equation}
        C_n := \left\{U \in SU(n) | UpU^\dagger \in P_n \forall p \in P_n\right\}
    \end{equation}

    is called the Clifford group. $C_1 =: C_L$ is called the local Clifford group.
\end{definition}

\begin{theorem}
    \label{thm:clifford_group_approx}
    \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 $\sqrt{-iX} = \frac{1}{\sqrt{2}} \left(\begin{array}{cc} 1 & -i \\ -i & 1 \end{array}\right)$. 

                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}

\begin{proof}
    \begin{enumerate}
        \item{See \cite[Theorem 10.6]{nielsen_chuang_2010}}
        \item{
            One can easily verify that $\sqrt{iZ} \in C_L$ and $\sqrt{-iX} \in C_L$.
            Further one can show easily that (up to a global phase)
            $H = \sqrt{iZ} \sqrt{-iX}^3 \sqrt{iZ}$ and $S = \sqrt{iZ}^3$.

            The length of the product can be seen when explicitly calculating 
            $C_L$.
            }
        \item{See \cite[Theorem 10.6]{nielsen_chuang_2010}}
    \end{enumerate} 
\end{proof}

This is quite an important result: As under a transformation $U \in C_n$ $S'$ 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}

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} 
\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:

\begin{equation}
    \begin{aligned}
        \ket{\psi'} &= \frac{I + g_a}{2}\ket{\psi} \\
                    &= \frac{I + g_a}{2}S^{(i)} \ket{\psi} \\
                    &= S^{(i)} \frac{I + g_a}{2}\ket{\psi} \\
                    &= S^{(i)}\ket{\psi'} \\
    \end{aligned}
\end{equation}

As the state that is stabilized by $S$ is unique $\ket{\psi'} = \ket{\psi}$.

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 $J := \{ S^{(i)} | [g_a, S^{(i)}] \neq 0\} \neq \{\}$. When measuring 
    $\frac{I + (-1)^s g_a}{2} $
    $1$ and $0$ are obtained with probability $\frac{1}{2}$ and after choosing
    a $j \in J$ the new state $\ket{\psi'}$ is stabilized by 
    \begin{equation}
        \langle \{(-1)^s g_a\} \cup \{S^{(i)} S^{(j)} | S^{(i)} \in J \setminus \{S^{(j)}\} \} \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

    \begin{equation}
    \begin{aligned}
        P(s=+1) &= \hbox{Tr}\left(\frac{I + g_a}{2}\ket{\psi}\bra{\psi}\right) \\
                &= \hbox{Tr}\left(\frac{I + g_a}{2}S^{(j)} \ket{\psi}\bra{\psi}\right)\\
                &= \hbox{Tr}\left(S^{(j)}\frac{I - g_a}{2}\ket{\psi}\bra{\psi}\right)\\
                &= \hbox{Tr}\left(\frac{I - g_a}{2}\ket{\psi}\bra{\psi}S^{(j)}\right)\\
                &= \hbox{Tr}\left(\frac{I - g_a}{2}\ket{\psi}\bra{\psi}\right)\\
                &= P(s=-1)
    \end{aligned}
    \notag
    \end{equation}

    With $P(s=+1) + P(s=-1) = 1$ follows $P(s=+1) = \frac{1}{2} = P(s=-1)$.
    Further for $S^{(i)},S^{(j)} \in J$

    \begin{equation}
    \begin{aligned}
        \frac{I + (-1)^sg_a}{2}\ket{\psi} &= \frac{I + (-1)^sg_a}{2}S^{(j)}S^{(i)} \ket{\psi} \\
            &= S^{(j)}\frac{I + (-1)^{s + 1}g_a}{2}S^{(i)} \ket{\psi} \\
            &=  S^{(j)}S^{(i)}\frac{I + (-1)^{s + 2}g_a}{2}\ket{\psi} \\
            &=  S^{(j)}S^{(i)}\frac{I + (-1)^sg_a}{2}\ket{\psi}
    \end{aligned}
    \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 $\ket{\psi'}$.
\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.

\begin{definition}
    \label{def:graph}
    The tuple $(V, E)$ is called a graph iff $V$ is a set of vertices with $|V| = n \in \mathbb{N}$.
    In the following $V = \{0, ..., n-1\}$ will be used.
    $E$ is the set of edges $E = \left\{\{i, j\} | i,i \in V, i \neq j\right\}$.

    For a vertex $i$ $n_i := \left\{j \in V | \{i, j\} \in E\right\}$ is called the neighbourhood
    of $i$.
\end{definition}

This definition of a graph is way less general than the definition of a mathematical graph.
Using this definition will however allow to avoid an extensive list of constraints on the 
mathematical graph 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_i
    \end{equation}
    for all $i \in V$. The vertex operator free graph state $\ket{\bar{G}}$ is the state stabilized by 
    $\langle K_G^{(i)} \rangle_{i = 0, ..., n-1}$.
\end{definition}

It is clear that the $K_G^{(i)}$ multilocal Pauli operators. That they commute 
follows from the fact that $\{i,j\} \in E \Leftrightarrow \{j,i\} \in E$ so for two
operators $K_G^{(i)}$ and $K_G^{(j)}$ either $\{i, j\} \notin E$ so they commute trivially
if $\{i,j\} \in E$  $X_i$, $Z_j$ and $X_j$, $Z_i$ anticommute which yields that the 
operators commute.

This definition of a graph state might not seem to be quite 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
    constructed using

    \begin{equation}
        \begin{aligned}
            \ket{\bar{G}} &= \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j}\right)\left(\prod\limits_{i \in V} H_i\right) \ket{0} \\
                        &= \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j}\right) \ket{+} \\
        \end{aligned}
    \end{equation}
\end{lemma}
\begin{proof}
    Let $\ket{+} := \left(\prod\limits_{l \in V} H_l\right) \ket{0}$ as before. Note that for any $X_i$ $X_i \ket{+} = +1 \ket{+}$.
    Set $\ket{\tilde{G}} := \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j} \right)\ket{+}$.

    \begin{equation}
        \begin{aligned}
            K_G^{(i)} \ket{\tilde{G}} & = X_i \left(\prod\limits_{\{i,j\} \in E} Z_j\right)\left(\prod\limits_{\{l,j\} \in E} CZ_{l,j} \right) \ket{+} \\
            & = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)X_i\prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + \ket{1}\bra{1}_j \otimes Z_l\right) \ket{+} \\
            & = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)\prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + (-1)^{\delta_{i,l}}\ket{1}\bra{1}_j \otimes Z_l\right) X_i \ket{+} \\
            & = \prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + (-1)^{2\delta_{i,l}}\ket{1}\bra{1}_j \otimes Z_l\right) \ket{+} \\
			& = \prod\limits_{\{l,j\} \in E}\left( \ket{0}\bra{0}_j \otimes I_l + \ket{1}\bra{1}_j \otimes Z_l\right) \ket{+} \\	
		    & = +1 \ket{\tilde{G}}	
        \end{aligned}
    \end{equation}

    as $X, Z$ anticommute and $Z\ket{1} = -1\ket{1}$.

\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:

\begin{definition}
    For to finite sets $A,B$ the symmetric set difference $\Delta$ is 
    defined as

    \begin{equation}
        A \Delta B = (A \cup B) \setminus (A \cap B)
    \end{equation}
\end{definition}

Toggling an edge $\{i, j\}$ updates $E' = E \Delta \left\{\{i,j\}\right\}$.
Another transformation on the VOP-free graph states is for a vertex $a \in V$

\begin{equation}
    M_a := \sqrt{-iX_a} \prod\limits_{j\in n_a} \sqrt{iZ_j}
\end{equation}

This transformation toggles the neighbourhood of $a$ which is an operation
that will be used later.

\begin{lemma}
    \label{lemma:M_a}
    When applying $M_a$ to a state $\ket{\bar{G}}$ the new state
    $\ket{\bar{G}'}$ is again a VOP-free graph state and the 
    graph is updated according to 
    \begin{equation}
        \begin{aligned}
            n_a' &= n_a \\
            n_j' &= n_j, \hbox{ if } j \notin n_a\\
            n_j' &= n_j \Delta n_a, \hbox{ if } j \in n_a
        \end{aligned}
    \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 $[K_G^{(a)}, M_a] = 0$ and $\forall i\in V\setminus n_a$ $[K_G^{(i)}, M_a] = 0$,
    so the first two equations follow trivially. For $j \in n_a$ set 

    \begin{equation}
        \begin{aligned}
            S^{(j)} &= M_a K_G^{(j)} M_a^\dagger \\
                &= \sqrt{-iX_a} \left(\prod\limits_{l \in n_a \setminus \{j\}} \sqrt{iZ_l}\right)
                                 \sqrt{iZ_j}  K_G^{(j)} \sqrt{iZ_j}^\dagger 
                                 \left(\prod\limits_{l \in n_a \setminus \{j\}} \sqrt{iZ_l}^\dagger\right)
                                 \sqrt{-iX_a}^\dagger \\
                &= \sqrt{-iX_a} \left(\prod\limits_{l \in n_a \setminus \{j\}} \sqrt{iZ_l}\right)\sqrt{iZ_j}
                                X_j \left(\prod\limits_{m \in n_j \setminus \{a\}} Z_m\right) Z_a
                                \sqrt{iZ_j}^\dagger
                                \left(\prod\limits_{l \in n_a \setminus \{j\}} \sqrt{iZ_l}^\dagger\right)
                                \sqrt{-iX_a}^\dagger \\
                &= \sqrt{iZ_j} X_j\sqrt{iZ_j}^\dagger\left(\prod\limits_{m \in n_j \setminus \{a\}} Z_m\right) 
                                \sqrt{-iX_a} Z_a \sqrt{-iX_a}^\dagger \\
                &= (-1)^2 Y_j Y_a \left(\prod\limits_{m \in n_j \setminus \{a\}} Z_m\right)\\
                &= (-1)i^2 Z_j X_a X_j Z_a \left(\prod\limits_{m \in n_j \setminus \{a\}} Z_m\right)
        \end{aligned}
    \end{equation}

    One can now construct a new set of $K_{G'}^{(i)}$ s.t. $M_a \ket{\bar{G}}$ is the $+1$ eigenvalue
    of the new $K_{G'}^{(i)}$. It is clear that $\forall j \notin n_a$  $K_{G'}^{(j)} = K_G^{(j)}$.
    To construct the $K_{G'}^{(i)}$ let for some $j \in n_a$ $n_a = \{j\} \cup I$ and $n_j = \{a\} \cup J$.
    Then

    \begin{equation}
        \begin{aligned}
            S^{(j)} &= Z_j X_a X_j Z_a \prod\limits_{l \in J} Z_l\\
                    &= Z_j X_a X_j Z_a \left(\prod\limits_{l \in J} Z_l\right)
                                        \left(\prod\limits_{l \in I}Z_l\right)
                                        \left(\prod\limits_{l \in I}Z_l\right) \\
                    &= Z_j X_a X_j Z_a \left(\prod\limits_{l \in ((I\cup J) \setminus (I\cap J))} Z_L\right)
                                        \left(\prod\limits_{l \in I}Z_l\right) \\
                    &= K_{G'}^{(a)} K_{G'}^{(j)} \\
                    &= K_{G}^{(a)} K_{G'}^{(j)}
        \end{aligned}
    \end{equation}

    Using this ine can show that $\ket{\bar{G}'}$ is a $+1$ eigenstate of $K_{G'}^{(j)}$:

    \begin{equation}
        \ket{\bar{G}'} = S^{(j)}\ket{\bar{G}'} = K_{G}^{(a)} K_{G'}^{(j)}\ket{\bar{G}'} 
                        = K_{G'}^{(j)}K_{G}^{(a)}\ket{\bar{G}'} = K_{G'}^{(j)}\ket{\bar{G}'}
    \end{equation}

    Because $\{K_G^{(i)} | i \notin n_a\} \cup \{S^{(i)} | i\in n_a\}$ and 
    $\{K_G^{(i)} | i \notin n_a\} \cup \{K_{G'}^{(i)} | i\in n_a \}$ 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 $K_{G'}^{(j)}$ 
    $\langle\{K_G^{(i)} | i \notin n_a\} \cup \{K_{G'}^{(i)} | i\in n_a \}\rangle$
    are the stabilizers of $\ket{\bar{G}'}$ and 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 quite simple: No. The most simple cases are the single qbit
stated $\ket{0},\ket{1}$ and $\ket{+_Y}, \ket{-_Y}$. But there is a simple 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
just constructive. As seen in theorem \ref{thm:clifford_group_approx} any $c \in C_n$
can be constructed from $CZ$ and $C_L$ and 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 $o_i \in C_L$.

    The state $\ket{G}$ is defined by the eigenvalue relation

    \begin{equation}
        +1 \ket{G} = \left(\prod\limits_{j=1}^no_j\right) K_G^{(i)} \left(\prod\limits_{j=1}^no_j\right)^\dagger \ket{G}
    \end{equation}

    $o_i$ are called the vertex operators of $\ket{G}$.
\end{definition}

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 $n^2$ $P_1$ 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 enough to store $n$ integers
representing the vertex operators is enough:

\begin{theorem}
    $C_L$ has $24$ degrees of freedom.
\end{theorem}
\begin{proof}
    It is clear that $\forall a \in C_L$ a is a group isomorphism $P_1 \circlearrowleft$: $apa^\dagger a p' a^\dagger = a pp'a^\dagger$.
    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, 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 will be used in one specific operation on graph states.

$$ S = \left(\begin{array}{cc} 1 & 0 \\ 0 & i \end{array}\right)$$
$$ \sqrt{-iX} = \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & i \\ i & 1 \end{array}\right)$$
$$ \sqrt{-iZ} = \exp(-i\frac{\pi}{4})\left(\begin{array}{cc} 1 & 0 \\ 0 & -i \end{array}\right)$$


\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. 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 of are changed to 
$\langle c_j S^{(i)} c_j^\dagger\rangle_i$. Using the definition of the graphical representation
it is clear that just the vertex operators are changed and the new vertex operators are given by

\begin{equation}
    \begin{aligned}
        o_i' &= o_i &\mbox{if } i \neq j\\
        o_i' &= c o_i c^\dagger &\mbox{if } i = j\\
    \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.


\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 $E' = E \Delta \left\{\{a,b\}\right\}$.

\textbf{Case 2}(\textit{Sub-Sub-Case 2.2.1})\textbf{:}\\
The two qbits are isolated: From the definition of the graph state it is clear 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 $2\cdot24^2$: with or without 
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 and while doing so one
gets 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{:}\\
At least one vertex operator does not commute with $CZ$ and at least one vertex 
has non-operand 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:

\begin{theorem}
    A graph state $\ket{G}$ associated with $(V, E, O)$ is invariant when
    simultaneously toggling the neighbourhood of a non-isolated qbit $j$
    and right-multiplying $M_j^\dagger$ to the vertex operators in the sense
    that $\sqrt{-iX}^\dagger = \sqrt{iX}$ is right-multiplied to $o_j$ and 
    $\sqrt{iZ}^\dagger = \sqrt{-iZ}$ is right-multiplied to $o_l$ for all 
    neighbours $l$ of $j$.

    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 and it transforms $(V, E, O)$ into a so-called
local Clifford equivalent graphical representation.

\begin{enumerate}
    \item{Check whether $n_a \setminus \{b\} \neq \{\}$. If this condition does not hold
            the vertex operator on $a$ cannot be cleared }
    \item{Express the vertex operator $o_a$ as a product of $\sqrt{-iX}$ and $\sqrt{iZ}$.}
    \item{Move from right to left through the product
        \begin{enumerate}
            \item{If the current matrix is $\sqrt{-iX}$ apply $L_a$}
            \item{If the current matrix is $\sqrt{iZ}$ select a vertex
                $j \in n_a \setminus \{b\}$ (which is possible as it has been checked before)
                and apply $L_j$.}
        \end{enumerate}
        }
\end{enumerate}

This algorithm has the important properties that if the algorithm succeeds 
$o_a = I$ and $o_b$ has picked up powers of $\sqrt{iZ}^\dagger$.
If $b$ has non-operand neighbours after clearing the vertex operators on $a$ the vertex operators on $b$
can be cleared using the same algorithm which gives $o_b = I$ and $o_a = (\sqrt{iZ}^\dagger)^l = S^l \in \mathcal{Z}$
for some $l \in \{1, ..., 5\}$. Therefore Case 1 can now be applied.

If $o_a$ could not be cleared $o_b$ can be cleared using the same procedure and after
clearing $o_b$ one can retry to clear $o_a$. 

In any case at least one vertex operator has been cleared. If both vertex operators have been
cleared Case 1 will be applied, if there is just one cleared vertex operator it is sure that it 
is the vertex with non-operand neighbours. Using this one can still apply a $CZ$: Without loss of generality
assume that $a$ has non-operand neighbours and $b$ does not. Now the state $\ket{G}$ has the form\cite{andersbriegel2005}

\begin{equation}
    \begin{aligned}
        \ket{G} &= \left(\prod\limits_{o_i \in O} o_i\right) \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j}\right) \ket{+}_n \\
                &= \left(\prod\limits_{o_i \in O \setminus \{o_b\}} o_i \right) \left(\prod\limits_{\{i,j\} \in E \setminus \{a,b\}} CZ_{i,j}\right) o_b (CZ_{a,b})^s \ket{+}_n \\
                &= \left(\prod\limits_{o_i \in O \setminus \{o_b\}} o_i \right) \left(\prod\limits_{\{i,j\} \in E \setminus \{a,b\}} CZ_{i,j}\right) \ket{+}_{n-2} \otimes \left(o_b (CZ_{a,b})^s \ket{+}_2\right) \\
    \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 $b$.

\begin{equation}
    \begin{aligned}
        CZ_{a,b}\ket{G} &= CZ_{a,b}\left(\prod\limits_{o_i \in O \setminus \{o_b\}} o_i \right) \left(\prod\limits_{\{i,j\} \in E \setminus \{a,b\}} CZ_{i,j}\right) o_b (CZ_{a,b})^s \ket{+} \\
                        &= \left(\prod\limits_{o_i \in O \setminus \{o_b\}} o_i \right) \left(\prod\limits_{\{i,j\} \in E \setminus \{a,b\}} CZ_{i,j}\right) \ket{+}_{n-2} \otimes \left(CZ_{a,b} o_b (CZ_{a,b})^s \ket{+}_2\right) \\
    \end{aligned}
\end{equation}

Using this one can re-use the method used in Case 2 to apply the $CZ$ while keeping the $o_a = I$.

As both $CZ$ and $C_L$ can be applied to a graph state $\ket{G}$ this proofs constructively that
the graphical representation of a stabilizer state is indeed able to represent any stabilizer state.
If one wants to do computations using this formalism it is however also necessary to perform measurements.

\subsubsection{Measurements on Graph States}

Recalling \ref{ref:meas_stab} it is clear that one has to compute the commutator of
the observable $g_a = Z_a$ with the stabilizers to get the probability amplitudes.
This is a quite expensive computation in theory however it is possible to simplify
the problem by pulling the observable behind the vertex operators. For this consider
the projector $P_a = \frac{I + (-1)^sZ_a}{2}$:

\begin{equation}
    \begin{aligned}
        P_a \ket{\psi} &= P_a \left(\prod\limits_{o_i \in O} o_i \right) \ket{\bar{G}} \\
                        &= \left(\prod\limits_{o_i \in O \setminus o_a}o_i \right)P_a  o_a \ket{\bar{G}} \\
                        &= \left(\prod\limits_{o_i \in O \setminus o_a}o_i \right) o_a o_a^\dagger P_a o_a \ket{\bar{G}} \\
                        &= \left(\prod\limits_{o_i \in O} o_i \right) o_a^\dagger P_a o_a \ket{\bar{G}} \\
                        &= \left(\prod\limits_{o_i \in O} o_i \right) \tilde{P}_a \ket{\bar{G}} \\
    \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: 

\begin{equation}
    \begin{aligned}
        \tilde{P}_a &= o_a^\dagger P_a o_a \\
                    &= o_a^\dagger \frac{I + (-1)^sZ_a}{2} o_a \\
                    &= \frac{I + (-1)^s o_a^\dagger Z_a o_a}{2} \\
                    &= \frac{I + (-1)^s \tilde{g}_a}{2} \\
    \end{aligned}
\end{equation}

Where $\tilde{g}_a \in \{+1, -1\}\cdot\{X_a, Y_a, Z_a\}$. It is therefore 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 wither commute or anticommute 
so it is easier to list the operators that anticommute:

\begin{equation}
    \begin{aligned}
        A_{\pm X_a} &= \left\{j | \{j, a\} \in E\right\}\\
        A_{\pm Y_a} &= \left\{j | \{j, a\} \in E\right\} \cup \{a\} \\
        A_{\pm Z_a} &= \{a\}\\
    \end{aligned}
\end{equation}

This gives one immediate result: if a qbit $a$ is isolated and the operator $\tilde{g}_a = X_a$ 
is measured the result $+1$ is obtained with probability $1$ and $(V, E, O)$ is unchanged.