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\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 := \{\bigotimes\limits_{i=0}^{n-1} p_i | p_i \in P\}
    \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 $\rangle S^{(i)} \rangle_{i=0, ..., n-1}$ 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