From 579f7057cd4a328cfe1399ed5e958ac5153e0e68 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= Date: Thu, 13 Feb 2020 10:37:46 +0100 Subject: [PATCH] just a paragraph --- thesis/chapters/stabilizer.tex | 18 +++++++++++++++++- 1 file changed, 17 insertions(+), 1 deletion(-) diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex index abe74a1..fafe142 100644 --- a/thesis/chapters/stabilizer.tex +++ b/thesis/chapters/stabilizer.tex @@ -521,4 +521,20 @@ it is clear that just the vertex operators are changed and the new vertex operat \end{aligned} \end{equation} -The action of a $CZ$ gate on the state $(V, E, O)$ is less trivial +The action of a $CZ$ gate on the state $(V, E, O)$ is in most cases less trivial. +Let $i \neq j$ be two qbits, now consider the action of $CZ_{i,j}$ on $(V, E, O)$. +The following discussion closely follows \cite{andersbriegel2005} and is analogous +to the implementation. + + +\textbf{Case 1:} +Both $o_i$ and $o_j$ commute with $CZ_{i,j}$. This is the case for exactly +four vertex operators: $\mathcal{Z} = \left\{I, Z, S, S^\dagger\right\}$. +In this case the CZ can be pulled past the vertex operators and just the edges +are changed to $E' = E \Delta \left\{\{i,j\}\right\}$. + +\textbf{Case 2:} +At least one vertex operator does not commute with $CZ_{i,j}$. + +\textbf{Sub-Case 2.1} +Both