From cb0edce102e9e771d376bd82d024e0038bd249d1 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= Date: Tue, 11 Feb 2020 10:13:06 +0100 Subject: [PATCH] added some more stuff to the graph states --- thesis/chapters/stabilizer.tex | 17 +++++++++++++++++ 1 file changed, 17 insertions(+) diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex index 2a08437..abe74a1 100644 --- a/thesis/chapters/stabilizer.tex +++ b/thesis/chapters/stabilizer.tex @@ -505,3 +505,20 @@ $$ \sqrt{-iX} = \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & i \\ i & 1 \end{ar $$ \sqrt{-iZ} = \exp(-i\frac{\pi}{4})\left(\begin{array}{cc} 1 & 0 \\ 0 & -i \end{array}\right)$$ +\subsubsection{Dynamics of Graph States} + +So far the graphical representation of stabilizer states is just another way to store +basically a stabilizer tableaux that might require less memory than the tableaux used in +CHP. The true power of this formalism is seen when studying its dynamics. The simplest case +is a local Clifford operator $c_j$ acting on a qbit $j$: The stabilizers of are changed to +$\langle c_j S^{(i)} c_j^\dagger\rangle_i$. Using the definition of the graphical representation +it is clear that just the vertex operators are changed and the new vertex operators are given by + +\begin{equation} + \begin{aligned} + o_i' &= o_i &\mbox{if } i \neq j\\ + o_i' &= c o_i c^\dagger &\mbox{if } i = j\\ + \end{aligned} +\end{equation} + +The action of a $CZ$ gate on the state $(V, E, O)$ is less trivial