some more work

thesis/Makefile

@@ -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

thesis/chapters/stabilizer.tex

@@ -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}
+

thesis/main.tex

@@ -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}