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thesis/chapters/graph_simulator.tex
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\section{The Graph Simulator} \section{The Graph Simulator}
\subsection{Introduction to the Graph Formalism} \subsection{Mathematical Prerequisites}
The first step towards the simulation in the graph formalism has been The following definitions and lemmata are required to understand both how the
the discovery of the stabilizer states and stabilizer circuits \cite{gottesman2009}\cite{gottesman1997}. graph formalism works and how the simulator handles gates.
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}.
\begin{definition} \begin{definition}
\begin{equation} \begin{equation}
@ -147,8 +143,40 @@ can be disregarded as discussed above $C_L$ will be used from now on instead of
Therefore $a$ will preserve the (anti-)commutator relations of $P$. Also $P$ is generated by $X,Z$ when disregarding a phase wich 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. does not matter for anticommutator relations.
This means that $X$ can be mapped to any $p \in P$ which are six elements disregarding This means that $X$ can be mapped to any $p \in P$ which are six elements disregarding
FIXME
\end{proof} \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}
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}
\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}
\subsection{Graph Storage} \subsection{Graph Storage}
@ -160,6 +188,7 @@ The following
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\subsection{Usage} \subsection{Usage}
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