\section{The Stabilizer Formalism and VOP-Free Graph States}

\subsection{Stabilizers and Stabilizer States}

This chapter discusses the stabilizer formalism that was introduced by Gottesman\cite{gottesman1997}
for quantum error correction but soon proved to be a useful tool to describe a subset of states:
the stabilizer states which can be simulated in polynomial time \cite{gottesman2008}.



\begin{definition}
    \begin{equation}
        p \in P_n \Rightarrow p = \bigotimes\limits_{i=0}^n p_i \\
 \forall i: p_i \in P := \{\pm 1, \pm i\} \cdot \{I, X, Y, Z\}
    \end{equation}

Where $X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$,
    $Y = \left(\begin{array}{cc} 0 & i \\ -i & 0\end{array}\right)$ and
    $Z = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right)$ are the Pauli matrices and 
    $I$ is the identity. 

    $p \in P_n$ is called a multi-local Pauli operator.
\end{definition}

\begin{definition}
    For a group $G$, $g_1, ..., g_n$ are called the generators of $G$ iff $\forall g \in G: g = \prod\limits_{i \in I} g_i$ for a
    subsed $I$ of $\{1, ..., n\}$. We write $G = \langle g_i \rangle_i$ if G is generated by the $g_i$. The generators $g_i$ are chosen 
    to be the smallest set of generators of $G$.
\end{definition}

\begin{definition}
    \label{def:stabilizer}
    For a $n$ qbit state $\ket{\psi}$ $\langle S_i \rangle_i$ is called the stabilizer of $\ket{\psi}$ if

    \begin{enumerate}
        \item{$\forall i = 1, ..., n$ $S_i \in P_n$}
        \item{$\forall i,j = 1, ..., n$ $[S_i, S_j] = 0$ $S_i$ and $S_j$ commute}
        \item{$\forall i = 1, ..., n$ $S_i\ket{\psi} = +1 \ket{\psi}$}
    \end{enumerate}
\end{definition}

\begin{lemma}
    For every $\langle S_i \rangle_i$ fulfilling the first two conditions in definition \ref{def:stabilizer} there exists
    a (up to a global phase) unique state $\ket{\psi}$ fulfilling the third condition. This state is called stabilizer state.
\end{lemma}

\begin{proof}
    All multi-local Pauli operators are hermitian (observables in terms of quantum mechanics), as the $S_i$
    commute they have a common set of eigenstates. Because each $S_i$ has eigenvalues $+1, -1$, there 
    exist $2^n$ eigenstates, one state $\ket{\psi}$ with eigenvalue $+1$ for all $S_i$. As the dimension of $n$ qbits is $2^n$
    the state $\ket{psi}$ is unique up to a global phase.
\end{proof}

One can study the dynamics of stabilizer states using only the stabilizers\cite{nielsen_chuang_2010}. Two important
cases are the unitary transformation of a state and the measurement of a qbit. When applying a unitary gate to a stabilizer
state $\ket{\psi}$ the resulting state will in general be no stabilizer state anymore, however there exists a group of
transformations that map stabilizers to other stabilizers: the Clifford group.

\begin{definition}
    \begin{equation}
        C_n := \{U \in SU(2) | UpU^\dagger \in P_n \forall p \in P_n\}
    \end{equation}
    is called the Clifford group on $n$ qbits.
    $C_1$ is called the local Clifford group.
\end{definition}

The properties of this group will be discussed later, for the time being is existence is enough. 

\begin{lemma}
    Let $\ket{\psi}$ be stabilized by $\langle S_i \rangle_i$, then $U\ket{\psi}$ is stabilized
    by $\langle US_iU^\dagger \rangle_i$.
\end{lemma}

\begin{proof}
$$ U\ket{\psi} = US_i\ket{\psi} = US_iU^\dagger U\ket{\psi}$$
So $U\ket{\psi}$ is a $+1$ eigenstate of $US_iU^\dagger$.
\end{proof}

This is an important insight that is used for simulations\cite{gottesman_aaronson2008}, as
updating the $n$ stabilizers that are a tensor product of $n$ Pauli matrices scales with roughly
$\mathcal{O}(n^2)$ instead of $\mathcal{O}(2^n)$ for the state vector approach.


\subsection{The Vertex Operator-Free Graph States}

In order to understand some essential transformations of graph states it is necessary
to study the vertex operator-free graph states first, partially because the graph states as used in this paper
were derived from the vertex operator-free graph states.

\begin{definition}
    \label{def:vop_free_g_state}
    A $n$ qbit vertex operator-free graph state $\ket{\overline{G}}$ is associated with a graph $(V, E)$
    by the $n$ operators

    \begin{equation}
        K^{(i)}_G := X_i \left(\prod\limits_{\{i, j\} \in E} Z_j\right) 
    \end{equation}

    for all $i \in V$ where for some operator $O$ $O_i$ indicates that it acts on the $i$-th qbit.

    A state $\ket{\overline{G}}$ is a $+1$ eigenstate of all $n$ $K^{(i)}_G$.
\end{definition}

\begin{corrolary}
    All $K^{(i)}_G$ commute and are hermitian. Therefore they have a common set of eigenstates
    (in particular definition \ref{def:vop_free_g_state} is well defined).
    In terms of quantum mechanics $K^{(i)}_G$ are observables.

    Further as $\ket{\overline{G}}$ is a $+1$ eigenstate of all $n$ $K^{(i)}_G$ which are
    multi-local Pauli operators, $\{K^{(i)}_G | i \in \{0, ..., n-1\}\}$ is the stabilizer
    of $\ket{\overline{G}}$ and $\ket{\overline{G}}$ is a stabilizer state.
\end{corrolary}

\begin{proof}
    As $X_i$ and $Z_i$ are hermitian their product is hermitian.

    Consider the case $\{i,j\} \notin E$ first:
    \begin{equation}
    \begin{aligned}
        \left[K^{(i)}_G, K^{(j)}_G\right] = \left[X_i \prod\limits_{\{i, n\} \in E} Z_n, X_j \prod\limits_{\{j, m\} \in E} Z_m\right] = 0
    \end{aligned}
    \end{equation}

    As operators acting on different qbits commute. The case $\{i,j\} \in E$ is slightly less trivial:
    \begin{equation}
    \begin{aligned}
        \left[K^{(i)}_G, K^{(j)}_G\right] &= \left[X_i \left(\prod\limits_{\{i, n\} \in E, n \neq j} Z_n\right) Z_j, X_j \left(\prod\limits_{\{j, m\} \in E, m \neq i} Z_m\right) Z_i\right] \\
        &= \left[X_i Z_j \prod\limits_n Z_n, X_j Z_i \prod\limits_m Z_m\right]\\
        &= \left(X_i Z_j X_j Z_i - X_j Z_i X_i Z_j\right)  \prod\limits_n Z_n \prod\limits_m Z_m \\
        &= \left(Z_j X_j X_i Z_i - X_j Z_j Z_i X_i\right)  \prod\limits_n Z_n \prod\limits_m Z_m \\
        &= \left((-1)^2X_j Z_j Z_i X_i - X_j Z_j Z_i X_i\right)  \prod\limits_n Z_n \prod\limits_m Z_m \\
        &= 0
    \end{aligned}
    \end{equation}

    as $X$, $Z$ anticommute.
\end{proof}

\begin{lemma}
    \begin{equation}
        \ket{\overline{G}} = \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j} \right) \left(\prod\limits_{l \in V} H_l\right) \ket{0}
    \end{equation}

    In particular definitions \ref{def:vop_free_g_state} and \ref{def:graph_state} are consistent, when there are no 
    vertex operators on the graph state $\ket{G}$.
\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}

These insights can be used to understand how measurement works on the vop-free graph state \cite{nielsen_chuang_2010}:
Consider a state $\ket{\psi}$ that is stabilized by $g_1, ... g_n$ and a hermitian $g$ that is to be measured.