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thesis/Makefile
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@ -6,9 +6,11 @@ chapters=chapters/introduction.tex \
chapters/quantum_computing.tex \
chapters/stabilizer.tex
cover=cover.png
all: main.pdf
main.pdf: main.tex main.bib $(chapters)
main.pdf: main.tex main.bib $(cover) $(chapters)
$(latex) main
$(bibtex) main
$(latex) main

159
thesis/chapters/stabilizer.tex
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@ -123,6 +123,7 @@ $dim V_S = 2^m$ \cite[Chapter 10.5]{nielsen_chuang_2010}. This yields the follow
result:
\begin{theorem}
\label{thm:unique_s_state}
For a $n$ qbit system and a set $S = \langle S^{(i)} \rangle_{i=1, ..., n}$ the stabilizer
space $V_S$ has $dim V_S = 1$, in particular there exists an up to a trivial phase unique
state $\ket{\psi}$ that is stabilized by $S$.
@ -170,6 +171,7 @@ $U$ but there exists a group for which $S'$ will be a set of stabilizers.
\end{definition}
\begin{theorem}
\label{thm:clifford_group_approx}
\begin{enumerate}
\item{$C_L$ can be generated using only $H$ and $S$.}
\item{$C_L$ can be generated from $\sqrt{iZ} = \exp(\frac{i\pi}{4}) S^\dagger$
@ -273,6 +275,7 @@ the result of the measurement.
\end{proof}
\subsection{The VOP-free Graph States}
\subsubsection{VOP-free Graph States}
This section will discuss the vertex operator(VOP)-free graph states. Why they are called
vertex operator-free will be clear in the following section about graph states.
@ -281,6 +284,9 @@ vertex operator-free will be clear in the following section about graph states.
The tuple $(V, E)$ is called a graph iff $V$ is a set of vertices with $|V| = n \in \mathbb{N}$.
In the following $V = \{0, ..., n-1\}$ will be used.
$E$ is the set of edges $E = \left\{\{i, j\} | i,i \in V, i \neq j\right\}$.
For a vertex $i$ $n_i := \left\{j \in V | \{i, j\} \in E\right\}$ is called the neighbourhood
of $i$.
\end{definition}
This definition of a graph is way less general than the definition of a mathematical graph.
@ -295,3 +301,156 @@ mathematical graph that are implied in this definition.
for all $i \in V$. The vertex operator free graph state $\ket{\bar{G}}$ is the state stabilized by
$\langle K_G^{(i)} \rangle_{i = 0, ..., n-1}$.
\end{definition}
It is clear that the $K_G^{(i)}$ multilocal Pauli operators. That they commute
follows from the fact that $\{i,j\} \in E \Leftrightarrow \{j,i\} \in E$ so for two
operators $K_G^{(i)}$ and $K_G^{(j)}$ either $\{i, j\} \notin E$ so they commute trivially
if $\{i,j\} \in E$ $X_i$, $Z_j$ and $X_j$, $Z_i$ anticommute which yields that the
operators commute.
This definition of a graph state might not seem to be quite straight forward
but recalling theorem \ref{thm:unique_s_state} it is clear that $\ket{\bar{G}}$
is unique. The following lemma will provide a way to construct this state
from the graph.
\begin{lemma}
For a graph $G = (V, E)$ the associated state $\ket{\bar{G}}$ is
constructed using
\begin{equation}
\begin{aligned}
\ket{\bar{G}} &= \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j}\right)\left(\prod\limits_{i \in V} H_i\right) \ket{0} \\
&= \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j}\right) \ket{+} \\
\end{aligned}
\end{equation}
\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}
\subsubsection{Dynamics of the VOP-free Graph States}
This representation gives an immediate result to how the stabilizers $\langle K_G^{(i)} \rangle_i$ change
under the $CZ$ gate: When applying $CZ_{i,j}$ on $G = (V, E)$ the edge $\{i,j\}$ is toggled,
resulting in a multiplication of $Z_j$ to $K_G^{(i)}$ and $Z_i$ to $K_G^{(j)}$. Toggling edges
is done by using the symmetric set difference:
\begin{definition}
For to finite sets $A,B$ the symmetric set difference $\Delta$ is
defined as
\begin{equation}
A \Delta B = (A \cup B) \setminus (A \cap B)
\end{equation}
\end{definition}
Toggling an edge $\{i, j\}$ updates $E' = E \Delta \left\{\{i,j\}\right\}$.
Another transformation on the VOP-free graph states is for a vertex $a \in V$
\begin{equation}
M_a := \sqrt{-iX_a} \prod\limits_{j\in n_a} \sqrt{iZ_j}
\end{equation}
This transformation toggles the neighbourhood of $a$ which is an operation
that will be used later.
\begin{lemma}
When applying $M_a$ to a state $\ket{\bar{G}}$ the new state
$\ket{\bar{G}'}$ is again a VOP-free graph state and the
graph is updated according to
\begin{equation}
\begin{aligned}
n_a' &= n_a \\
n_j' &= n_j, \hbox{ if } j \notin n_a\\
n_j' &= n_j \Delta n_a, \hbox{ if } j \in n_a
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
$\ket{\bar{G}'}$ is stabilized by $\langle M_a K_G^{(i)} M_a^\dagger \rangle_i$, so it is sufficient
to study how the $ K_G^{(i)}$ change under $M_a$.
At first note that $[K_G^{(a)}, M_a] = 0$ and $\forall i\in V\setminus n_a$ $[K_G^{(i)}, M_a] = 0$,
so the first two equations follow trivially. For $j \in n_a$ set
\begin{equation}
\begin{aligned}
S^{(j)} &= M_a K_G^{(j)} M_a^\dagger \\
&= \sqrt{-iX_a} \left(\prod\limits_{l \in n_a \setminus \{j\}} \sqrt{iZ_l}\right)
\sqrt{iZ_j} K_G^{(j)} \sqrt{iZ_j}^\dagger
\left(\prod\limits_{l \in n_a \setminus \{j\}} \sqrt{iZ_l}^\dagger\right)
\sqrt{-iX_a}^\dagger \\
&= \sqrt{-iX_a} \left(\prod\limits_{l \in n_a \setminus \{j\}} \sqrt{iZ_l}\right)\sqrt{iZ_j}
X_j \left(\prod\limits_{m \in n_j \setminus \{a\}} Z_m\right) Z_a
\sqrt{iZ_j}^\dagger
\left(\prod\limits_{l \in n_a \setminus \{j\}} \sqrt{iZ_l}^\dagger\right)
\sqrt{-iX_a}^\dagger \\
&= \sqrt{iZ_j} X_j\sqrt{iZ_j}^\dagger\left(\prod\limits_{m \in n_j \setminus \{a\}} Z_m\right)
\sqrt{-iX_a} Z_a \sqrt{-iX_a}^\dagger \\
&= (-1)^2 Y_j Y_a \left(\prod\limits_{m \in n_j \setminus \{a\}} Z_m\right)\\
&= (-1)i^2 Z_j X_a X_j Z_a \left(\prod\limits_{m \in n_j \setminus \{a\}} Z_m\right)
\end{aligned}
\end{equation}
One can now construct a new set of $K_{G'}^{(i)}$ s.t. $M_a \ket{\bar{G}}$ is the $+1$ eigenvalue
of the new $K_{G'}^{(i)}$. It is clear that $\forall j \notin n_a$ $K_{G'}^{(j)} = K_G^{(j)}$.
To construct the $K_{G'}^{(i)}$ let for some $j \in n_a$ $n_a = \{j\} \cup I$ and $n_j = \{a\} \cup J$.
Then
\begin{equation}
\begin{aligned}
S^{(j)} &= Z_j X_a X_j Z_a \prod\limits_{l \in J} Z_l\\
&= Z_j X_a X_j Z_a \left(\prod\limits_{l \in J} Z_l\right)
\left(\prod\limits_{l \in I}Z_l\right)
\left(\prod\limits_{l \in I}Z_l\right) \\
&= Z_j X_a X_j Z_a \left(\prod\limits_{l \in ((I\cup J) \setminus (I\cap J))} Z_L\right)
\left(\prod\limits_{l \in I}Z_l\right) \\
&= K_{G'}^{(a)} K_{G'}^{(j)} \\
&= K_{G}^{(a)} K_{G'}^{(j)}
\end{aligned}
\end{equation}
Using this ine can show that $\ket{\bar{G}'}$ is a $+1$ eigenstate of $K_{G'}^{(j)}$:
\begin{equation}
\ket{\bar{G}'} = S^{(j)}\ket{\bar{G}'} = K_{G}^{(a)} K_{G'}^{(j)}\ket{\bar{G}'}
= K_{G'}^{(j)}K_{G}^{(a)}\ket{\bar{G}'} = K_{G'}^{(j)}\ket{\bar{G}'}
\end{equation}
Because $\{K_G^{(i)} | i \notin n_a\} \cup \{S^{(i)} | i\in n_a\}$ and
$\{K_G^{(i)} | i \notin n_a\} \cup \{K_{G'}^{(i)} | i\in n_a \}$ are both $n$ commuting
multi-local Pauli operators where the $S^{(i)}$ can be generated from the $K_{G'}^{(i)}$
and $\ket{\bar{G}'}$ is a $+1$ eigenstate of $K_{G'}^{(j)}$
$\langle\{K_G^{(i)} | i \notin n_a\} \cup \{K_{G'}^{(i)} | i\in n_a \}\rangle$
are the stabilizers of $\ket{\bar{G}'}$ and the associated graph is changed as given
in the third equation.
\end{proof}
\subsection{Graph States}
The definition of a VOP-free graph state above raises an obvious question:
Can any stabilizer state be described using just a graph?
The answer is quite simple: No. The most simple cases are the single qbit
stated $\ket{0},\ket{1}$ and $\ket{+_Y}, \ket{-_Y}$. But there is a simple extension
to the VOP-free graph states that allows the representation of an arbitrary
stabilizer state. The proof that indeed any state can be represented is
just constructive. As seen in theorem \ref{thm:clifford_group_approx} any $c \in C_n$
can be constructed from $CZ$ and $C_L$ and in the following discussion it will become
clear that both $C_L$ and $CZ$ can be applied to a general graph state.
\subsubsection{Graph States and Vertex operators}

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thesis/main.tex
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@ -1,4 +1,5 @@
\documentclass[a4paper,12pt]{scrartcl}
\usepackage{titlepic}
\usepackage[utf8]{inputenc}
\usepackage{graphicx}
\usepackage{amssymb, amsthm}
@ -22,13 +23,16 @@
\numberwithin{equation}{section}
\title{An Efficient Quantum Computing Simulator using a Graphical Description for Many-Qbit Systems}
\author{Daniel Knüttel}
\date{10.04.2020}
\titlepic{\includegraphics[width=\textwidth]{cover.png}}
\begin{document}
\maketitle
%\frontmatter
\newpage
\tableofcontents
\include{chapters/introduction}