some work on the graph states

thesis/chapters/stabilizer.tex

@@ -281,6 +281,7 @@ This section will discuss the vertex operator(VOP)-free graph states. Why they a
 vertex operator-free will be clear in the following section about graph states.
 
 \begin{definition}
+    \label{def:graph}
     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 $E = \left\{\{i, j\} | i,i \in V, i \neq j\right\}$.
@@ -452,5 +453,55 @@ just constructive. As seen in theorem \ref{thm:clifford_group_approx} any $c \in
 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}
+\subsubsection{Graph States and Vertex Operators}
+
+\begin{definition}
+    \label{def:g_state}
+    A tuple $(V, E, O)$ is called the graphical representation of a stabilizer state
+    if $(V, E)$ is a graph as in Definition \ref{def:graph} and $O = \{o_1, ..., o_n\}$ where $o_i \in C_L$.
+
+    The state $\ket{G}$ is defined by the eigenvalue relation
+
+    \begin{equation}
+        +1 \ket{G} = \left(\prod\limits_{j=1}^no_j\right) K_G^{(i)} \left(\prod\limits_{j=1}^no_j\right)^\dagger \ket{G}
+    \end{equation}
+
+    $o_i$ are called the vertex operators of $\ket{G}$.
+\end{definition}
+
+Recalling the dynamics of stabilizer states the following relation follows immediately:
+
+\begin{equation}
+    \ket{G} = \left(\prod\limits_{j=1}^no_j\right) \ket{\bar{G}}
+\end{equation}
+
+The great advantage of this representation of a stabilizer state is its space requirement:
+Instead of storing $n^2$ $P_1$ matrices only some vertices (which often are implicit),
+the edges and some vertex operators ($n$ matrices) have to be stored. The following theorem
+will improve this even further: instead of $n$ matrices it is enough to store $n$ integers
+representing the vertex operators is enough:
+
+\begin{theorem}
+    $C_L$ has $24$ degrees of freedom.
+\end{theorem}
+\begin{proof}
+    It is clear that $\forall a \in C_L$ a is a group isomorphism $P_1 \circlearrowleft$: $apa^\dagger a p' a^\dagger = a pp'a^\dagger$.
+    Therefore $a$ will preserve the (anti-)commutator relations of $P$.
+    Further note that $Y = iXZ$, so one has to consider the anti-commutator relations
+    of $X,Z$ only.
+
+    As the transformations are unitary they preserve eigenvalues, so $X$ can be mapped
+    to $\pm X, \pm Y, \pm Z$ which gives $6$ degrees of freedom, the image of $Z$ has to 
+    anti-commute with the image of $X$ so $Z$ has four possible images under the transformation.
+    This gives another $4$ degrees of freedom and a total of $24$.
+\end{proof}
+
+From now on $C_L = \langle H, S \rangle$ (disregarding a global phase) will be used,
+one can show (by construction) that $H, S$ generate a possible choice of $C_L$, as is 
+$C_L = \langle \sqrt{-iX}, \sqrt{-iZ}\rangle$ which will be used in one specific operation on graph states.
+
+$$ S = \left(\begin{array}{cc} 1 & 0 \\ 0 & i \end{array}\right)$$
+$$ \sqrt{-iX} = \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & i \\ i & 1 \end{array}\right)$$
+$$ \sqrt{-iZ} = \exp(-i\frac{\pi}{4})\left(\begin{array}{cc} 1 & 0 \\ 0 & -i \end{array}\right)$$
+