\section{The Graph Simulator}

\subsection{Mathematical Prerequisites}

The following definitions and lemmata are required to understand both how the
graph formalism works and how the simulator handles gates.

%A $n$ qbit graph or stabilizer state is a $+1$ eigenstate of some $ p \in P_n$ where $P_n$ is the Pauli group\cite{andersbriegel2005}.

\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}

One can show that the Clifford group $C_n$ can be generated using the elements of $C_1$  acting on all qbits and
the controlled phase gate $CZ$ between all qbits\cite{andersbriegel2005}. It is worth noting that the $CX$ gate can be
generated using $CZ$ and $C_1$ gates.

\begin{lemma}
    Let $a \in C_1$ then $\forall \phi \in [0, 2\pi)$ also $\exp(i\phi)a \in C_1$.

    \textbf{Note}: This is also true for $C_n \forall n >= 1$.
\end{lemma}
\begin{proof}

    Let $a' := \exp(i\phi)a$. $a' \in C_1$ iff $a'pa^{\prime\dagger} \in P \forall p \in P$.

    \begin{equation}
    \begin{aligned}
        a'pa^{\prime\dagger} &= (\exp(i\phi)a)p(\exp(i\phi)a)^\dagger \\
        & = \exp(i\phi)ap\exp(-i\phi)a^\dagger \\
        & = \exp(i\phi)\exp(-i\phi) apa^\dagger \\
        & = apa^\dagger \in P
    \end{aligned}
    \end{equation}
\end{proof}

\begin{lemma}
    One cannot measure phases by projecting states.
\end{lemma}
\begin{proof}
    Let $\ket{\psi}$ be a state, $\ket{\varphi}\bra{\varphi}$ a projector. $\ket{\psi'} := \exp(i\phi)\ket{\psi}$ for some $\phi \in [0, 2\pi)$.

    \begin{equation}
    \begin{aligned}
        \bra{\psi'}\ket{\varphi}\bra{\varphi}\ket{\psi'} &= \exp(-i\phi)\bra{\psi}\ket{\varphi}\bra{\varphi}\exp(i\phi)\ket{\psi} \\
        & = \exp(-i\phi)\exp(i\phi)\bra{\psi}\ket{\varphi}\bra{\varphi}\ket{\psi} \\
        & = \bra{\psi}\ket{\varphi}\bra{\varphi}\ket{\psi}
    \end{aligned}
    \end{equation}
\end{proof}

\begin{definition}
    A phase $\phi \in [0, 2\pi)$ is called qbit-global, if for some qbit states $\ket{\psi}, \ket{\varphi}$ $\ket{\psi} = \exp(i\phi)\ket{\varphi}$.
\end{definition}

\begin{lemma}
    When entangling qbits via projections one can disregard qbit-global phases.
    
    Two qbits are entangled via projection, if for some single qbit gates $M,N$
    and two orthonormal states $\ket{a}, \ket{b}$

    \begin{equation}
        C^{M,N}(i,j) = \ket{a}\bra{a}_j \otimes M_i + \ket{b}\bra{b}_j \otimes N_i
    \end{equation}

    \textbf{Remark.}
    In particular when entangling states using $CX$ and $CZ$ one can disregard qbit-global phases.
    This is immideatly clear when recalling \eqref{eq:CX_pr} and \eqref{eq:CZ_pr}.
\end{lemma}

\begin{proof}
    Let $\alpha, \beta \in [0, 2\pi)$ be some phases, $\ket{\psi}, \ket{\varphi}, \ket{\psi'} := \exp(i\alpha)\ket{\psi}, \ket{\varphi'} := \exp(i\beta)\ket{\varphi}$ some single qbit states,
    $M, N, \ket{a}, \ket{b}, C^{M,N}(i,j)$ as defined above.

\begin{equation}
\begin{aligned}
     C^{M,N}(1, 0) (\ket{\psi'}\otimes\ket{\varphi'}) & = \ket{a}\braket{a}{\varphi'}\otimes M\ket{\psi'} + \ket{b}\braket{b}{\varphi'} \otimes N\ket{\psi'} \\
     & = \exp(i\beta)\ket{a}\braket{a}{\varphi}\otimes\exp(i\alpha)M\ket{\phi} + \exp(i\beta)\ket{b}\braket{b}{\varphi}\otimes\exp(i\alpha)N\ket{\phi}\\
     & = \exp(i(\beta + \alpha))(\ket{a}\braket{a}{\varphi}\otimes M\ket{\psi} + \ket{b}\braket{b}{\varphi} \otimes N\ket{\psi})\\
     & = \exp(i(\beta + \alpha))C^{M,N}(1, 0) (\ket{\psi}\otimes\ket{\varphi})
\end{aligned}
\end{equation}

    Where $\exp(i(\beta + \alpha))$ is a multi-qbit-global phase which can be (following the above argumentation) disregarded.
    
\end{proof}

\begin{corrolary}
    One can disregard global phases of elements of the $C_1$ group.
\end{corrolary}

\begin{proof}
    As it has been shown above a quantum computer cannot measure global phases. Also
    the entanglement gates $CX, CZ$ map qbit-global phases to multi-qbit-global phases which cannot 
    be measured. It has been shown above that one can choose the $C_1$ operators such that they do not yield
    a phase.
\end{proof}

\begin{definition}
    \begin{equation}
        C_L := \{a \in C_1 | \nexists \phi \in [0, 2\pi), b \in C_L : a = \exp(i\phi)b\}
    \end{equation}
    Is called the non-trivial local Clifford group.
\end{definition}

\textbf{Remark.} When computing the elements of $C_L$ and their products one will realize that $C_L$ is not a group.
If one however disregards a global phase the product of two $C_L$ elements will be in $C_L$ again. Because the global phases
can be disregarded as discussed above $C_L$ will be used from now on instead of $C_1$. 

\begin{theorem}
    \begin{equation}
        | C_L | = 24
    \end{equation}
\end{theorem}

\begin{proof}
    It is clear that $\forall a \in C_L$ a is a group isomorphism $P \circlearrowleft$: $apa^\dagger a p' a^\dagger = a pp'a^\dagger$.
    Therefore $a$ will preserve the (anti-)commutator relations of $P$. Also $P$ is generated by $X,Z$ when disregarding a phase wich 
    does not matter for anticommutator relations. 
    This means that $X$ can be mapped to any $p \in P$ which are six elements disregarding 

    FIXME
\end{proof}

\subsection{Introduction to the Graph Formalism}

The first step towards the simulation in the graph formalism has been
the discovery of the stabilizer states and stabilizer circuits \cite{gottesman2009}\cite{gottesman1997}.
They led to the faster simulation using stabilizer tableaux\cite{gottesman_aaronson2008} and later
to the graph formalism\cite{schlingenmann2001}\cite{andersbriegel2005}\cite{vandennest_ea2004}.
The following discussion eludicates the graph formalism and explains how the graph simulator works.
Some parts will be kept short as they can be looked up in \cite{andersbriegel2005}.

A naive state is just a vector containing the coefficients $c_i$ as defined in \ref{ref:nqbitsystems}. 
It is a quite straight forward approach and gates are applied by updating the coefficients according
to the gate's matrix representation. A naive state has the time and space complexity $\mathcal{O}(2^n)$ which limits the number 
of qbits drastically. \\
The stabilizer tableaux represent the state by its stabilizers i.e. by those Pauli operators of which the
state is a $+1$ eigenstate. This has a space complexity of $\mathcal{O}(n^2)$ while updating the tableaux
has a time complexity of $\mathcal{O}(n)$ for unitary gates and $mathcal{O}(n^2)$ for measurements.

A graph state now represents the state by the gates that have been applied to it starting from the $\ket{+}$ state:

\begin{equation}
    \ket{+} := \bigotimes\limits_{i=0}^{n-1} H_i \ket{0}
\end{equation}

\begin{definition}
\label{def:graph_state}

A graph state $\ket{G}$ is a 3-tuple $(V, E, O)$ where $(V = \{0, ..., n-1\}, E)$ is a graph with the vertices $V$, edges $E$ 
and vertex operators $O = \{o_i | i = 0, ..., n-1; o_i \in C_L \forall i\}$. The vertex operators and edges are defined
by the following relation:

\begin{equation}
    \label{eq:g_state}
    \ket{G} = \left(\bigotimes\limits_{i=0}^{n-1} o_i \right)\left(\bigotimes\limits_{\{i, j\} \in E} CZ_{i,j} \right) \ket{+}
\end{equation}
\end{definition}

Where every $o_i$ acts on the $i$-th qbit.

One can show that any stabilizer state can be realized as a graph state (for instance in \cite{schlingenmann2001}).




\subsection{Operations on the Graph State}

\subsubsection{Single Qbit Gates}

Recalling \eqref{eq:g_state} 
Makes it clear that for any single  qbit gate $g \in C_L$ with $g_k$ being the gate
acting on qbit $k$ the state changes according to


\begin{equation}
    \begin{aligned}
        g_k \ket{G} &= g_k \left(\bigotimes\limits_{i=0}^{n-1} o_{i} \right)\left(\bigotimes\limits_{\{i, j\} \in E} CZ_{i,j} \right) \ket{+} \\
        &= \left(\bigotimes\limits_{i=0}^{n-1} g_k^{\delta_{i,k}}o_{i} \right)\left(\bigotimes\limits_{\{i, j\} \in E} CZ_{i,j} \right)\ket{+}
    \end{aligned}
\end{equation}

meaning that the graph state $(V, E, O)$ changes to $(V, E, \{o_0, ..., o_{k-1}, go_k, o_{k+1}, ..., o_{n-1}\})$
as $C_L$ is almost a group the element $go_k \in C_L$ up to a global phase that is disregarded. All the results
of $C_L \times C_L \rightarrow C_L, a,b \mapsto ab$ have been precomputed in a lookup table and the vertex operators
are updated according to that lookup table.

\subsubsection{Controlled Phase Gate}

Recalling \eqref{eq:g_state} 
it is clear that some $CZ$ application is less trivial 
than a single qbit gate.

%\begin{struktogramm}(100, 50)
%    \ifthenelse[10]{1, 4}
%    {Both Vertex operators Commute with CZ}{\sTrue}{\sFalse}
%    \change
%    \ifend
%\end{struktogramm}


\subsubsection{Measurement of a Qbit}

Note that the $Z$ gate is diagonal in the $\{\ket{1}_s,\ket{0}_s\}$ basis with eigen values $1, -1$.
This gives a simple expression for the projector on the $\ket{0}_s$ and $\ket{1}_s$ state of qbit $k$:

\begin{equation}
    \tilde{P^{v}_{k}} := \frac{I_k + (-1)^vZ_k}{2}, v \in {0, 1}
\end{equation}

This projector trivially commutes with any non-operand vertex operator yielding for a state


\begin{equation}
    \begin{aligned}
        \ket{\psi} := \left(\bigotimes\limits_{\{i,j\} \in E} CZ_{i,j}\right) \ket{+} \\
        \tilde{P^{v}_{k}} \left(\bigotimes\limits_{i=0}{n-1} o_i\right)\ket{\psi} &=  
        \left(\bigotimes\limits_{i=0,i \ne k}{n-1} o_i\right) \tilde{P^{v}_{k}} o_k \ket{\psi} \\
        &= \left(\bigotimes\limits_{i=0,i \ne k}{n-1} o_i\right) o_ko_k^\dagger \tilde{P^{v}_{k}} o_k \ket{\psi} \\
        &= \left(\bigotimes\limits_{i=0,i \ne k}{n-1} o_i\right) o_k = \frac{I_k + (-1)^v o_k^\dagger Z_ko_k}{2} \ket{\psi}
    \end{aligned}
\end{equation}

Now $o_k$ stabilizes $P$, as it is an element of $C_L$, meaning that $o_k^\dagger Z_k o_k \in \{-1,1\}\{X_k, Y_k, Z_k\}$
Which gives a new projector on a Pauli matrix: $P_k := o_k^\dagger Z_ko_k$



\subsection{Graph Storage}

One of the gread advantages of simulating in the graph formalism is a great increase 
in simulation performance and a lower memory requirement. The simulation of
at least $10^6$ qbits on a common desktop computer should be possible\cite{andersbriegel2005}.
Therefore  one has to take care when choosing a representation of the graph state. 
The following 

FIXME


\subsection{Usage}
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 \subsection{Performance}
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