From 6cae03db822be2afe94412d7d77819d47dff6f04 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= <daniel.knuettel@daknuett.eu>
Date: Mon, 11 Nov 2019 11:22:47 +0100
Subject: [PATCH] a little work

---
 thesis/chapters/graph_simulator.tex | 43 ++++++++++++++++++++++++-----
 1 file changed, 36 insertions(+), 7 deletions(-)

diff --git a/thesis/chapters/graph_simulator.tex b/thesis/chapters/graph_simulator.tex
index cc6ebcf..975e40c 100644
--- a/thesis/chapters/graph_simulator.tex
+++ b/thesis/chapters/graph_simulator.tex
@@ -1,13 +1,9 @@
 \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 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}.
+The following definitions and lemmata are required to understand both how the
+graph formalism works and how the simulator handles gates.
 
 \begin{definition}
     \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 
     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}
+
+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}
 
@@ -160,6 +188,7 @@ The following
 
 FIXME
 
+
 \subsection{Usage}
  FIXME