some tests

thesis/chapters/graph_simulator.tex

@@ -37,21 +37,21 @@ Where $X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$,
         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.
-    $C_1 =: C_L$ is called the local Clifford group.
+    $C_1$ is called the local Clifford group.
 \end{definition}
 
-One can show that the Clifford group $C_n$ can be generated using the elements of $C_L$  acting on all qbits and
+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 $C_L$ gates.
+generated using $CZ$ and $C_1$ gates.
 
 \begin{lemma}
-    Let $a \in C_L$ then $\forall \phi \in [0, 2\pi)$ also $\exp(i\phi)a \in C_L$.
+    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 $a' := \exp(i\phi)a$. $a' \in C_L$ iff $a'pa^{\prime\dagger} \in P \forall p \in P$.
+    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}
@@ -115,16 +115,40 @@ generated using $CZ$ and $C_L$ gates.
 \end{proof}
 
 \begin{corrolary}
-    One can disregard global phases of elements of the $C_L$ group.
+    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 $C_L$ operators such that they do not yield
+    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 
+\end{proof}
+
 
 \subsection{Graph Storage}
 

thesis/main.tex

@@ -17,6 +17,7 @@
 \newtheorem{postulate}{Postulate}
 \newtheorem{corrolary}{Corrolary}
 \newtheorem{lemma}{Lemma}
+\newtheorem{theorem}{Theorem}
 
 \numberwithin{equation}{section}
 
@@ -48,6 +49,12 @@ Simulator with a Focus on Simulation in the Graph Formalism }
 
 \include{chapters/graph_simulator}
 
+\section{Appendix}
+
+\subsection{Computing the Local Clifford Group and the Products of its Elements}
+
+%\include{chapters/C_L_elements_and_products}
+
 %\backmatter
 
 \bibliographystyle{unsrt}