Presentation should work like that
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@ -306,6 +306,7 @@
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\begin{itemize}
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\item{
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Choose a finite Abelian subgroup $S$ of $P_n$ with $-I \notin S$.
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One can show that all elements of $S$ are hermitian.
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}
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\item{
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One says $S = \langle S^{(i)} \rangle_{i=1,...,n}$ is generated by the $S^{(i)}$
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@ -406,8 +407,9 @@
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\item{Applying a local Clifford gate $U_i$ is trivial:
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Just the vertex operator is updated to $U o_i$.}
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\item{If $o_a, o_b \in \{I, Z, R_\frac{\pi}{2}, R_\frac{\pi}{2}^\dagger\}$ applying a $CZ_{a,b}$
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just toggles the edge $\{a,b\}$ in $E$: $E' = E \Delta \{\{a,b\}\}$.
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With the symmetric set difference $\Delta$.}
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just toggles the edge $\{a,b\}$ in $E$.%: $E' = E \Delta \{\{a,b\}\}$.
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%With the symmetric set difference $\Delta$.
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}
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\item{If the vertices $a, b$ are isolated the resulting state after
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applying $CZ_{a,b}$ can be precomputed. }
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\item{When none of the above is possible one can clear at least one
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@ -453,14 +455,16 @@
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$
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$19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
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$19 = \sqrt{iZ}^2\sqrt{-iX}^3$
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\end{minipage}
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\noindent\begin{minipage}{0.5\textwidth}
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\includegraphics[width=\textwidth]{graphs/clear_vop_02.png}
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$ 21 = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]$
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$ 21 = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]$\\
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$ 21 = \sqrt{iZ}^2\sqrt{-iX}^2$
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\end{minipage}
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\end{frame}
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}
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@ -471,14 +475,16 @@
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$ 21 = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]$
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$ 21 = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]$\\
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$ 21 = \sqrt{iZ}^2\sqrt{-iX}^2$
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\end{minipage}
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\noindent\begin{minipage}{0.5\textwidth}
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\includegraphics[width=\textwidth]{graphs/clear_vop_03.png}
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$
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$11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
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$11 = \sqrt{iZ}^2\sqrt{-iX}$
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\end{minipage}
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\end{frame}
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}
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@ -489,14 +495,16 @@
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$
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$11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
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$11 = \sqrt{iZ}^2\sqrt{-iX}$
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\end{minipage}
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\noindent\begin{minipage}{0.5\textwidth}
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\includegraphics[width=\textwidth]{graphs/clear_vop_04.png}
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$
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$5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$\\
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$5 = \sqrt{iZ}^2$
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\end{minipage}
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\end{frame}
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}
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@ -507,14 +515,16 @@
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$
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$5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$\\
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$5 = \sqrt{iZ}^2$
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\end{minipage}
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\noindent\begin{minipage}{0.5\textwidth}
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\includegraphics[width=\textwidth]{graphs/clear_vop_05.png}
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$ 7 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{matrix}\right]$
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$ 7 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
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$7 = \sqrt{iZ}$
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\end{minipage}
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\end{frame}
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}
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@ -525,14 +535,15 @@
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$ 7 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{matrix}\right]$
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$ 7 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
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$7 = \sqrt{iZ}$
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\end{minipage}
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\noindent\begin{minipage}{0.5\textwidth}
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\includegraphics[width=\textwidth]{graphs/clear_vop_06.png}
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$2 = \left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]$
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$2 = \left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]$\\
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\end{minipage}
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\end{frame}
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}
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@ -543,7 +554,8 @@
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$
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$19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
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$19 = \sqrt{iZ}^2\sqrt{-iX}^3$
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\end{minipage}
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\noindent\begin{minipage}{0.5\textwidth}
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\includegraphics[width=\textwidth]{graphs/clear_vop_06.png}
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@ -556,6 +568,13 @@
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}
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\section{Implementation and Performance}
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{
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\begin{frame}{Implementation}
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\center
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\includegraphics[width=\textwidth,height=0.8\textheight,keepaspectratio=true]{screenshot_github.png}
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\end{frame}
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}
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{
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\begin{frame}{Implementation}
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\begin{itemize}
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@ -576,11 +595,11 @@
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\includegraphics[width=\textwidth]{../performance/scaling_qbits_linear.png}
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\end{frame}
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}
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{
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\begin{frame}{Performance: Dense Vector vs. Graphical Representation}
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\includegraphics[width=\textwidth]{../performance/scaling_qbits_log.png}
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\end{frame}
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}
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%{
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%\begin{frame}{Performance: Dense Vector vs. Graphical Representation}
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% \includegraphics[width=\textwidth]{../performance/scaling_qbits_log.png}
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%\end{frame}
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%}
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{
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\begin{frame}{Performance: Circuit Length on Graphical Representation}
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@ -588,48 +607,48 @@
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\end{frame}
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}
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{
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\begin{frame}{Performance: Circuit Length on Graphical Representation}
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\begin{itemize}
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\item{There seem to be three regimes: Low-linear regime, intermediate regime and high-linear regime.}
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\item{In the low-linear regime only few VOPs have to be cleared.
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The length of this regime increases with the number of qbits.
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}
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\item{In the high-linear regime the neighbourhoods are big; the probability that VOPs
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must be cleared is high. Clearing VOPs involves many vertices.}
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\item{The intermediate is dominated by growing neighbourhoods $\Rightarrow$
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no linear behaviour.}
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\end{itemize}
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\end{frame}
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}
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%{
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%\begin{frame}{Performance: Circuit Length on Graphical Representation}
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% \begin{itemize}
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% \item{There seem to be three regimes: Low-linear regime, intermediate regime and high-linear regime.}
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% \item{In the low-linear regime only few VOPs have to be cleared.
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% The length of this regime increases with the number of qbits.
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% }
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% \item{In the high-linear regime the neighbourhoods are big; the probability that VOPs
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% must be cleared is high. Clearing VOPs involves many vertices.}
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% \item{The intermediate is dominated by growing neighbourhoods $\Rightarrow$
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% no linear behaviour.}
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% \end{itemize}
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%
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%\end{frame}
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%}
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{
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\begin{frame}{Graph in the Low-Linear Regime}
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\includegraphics[width=\textwidth]{../thesis/graphics/graph_low_linear_regime.png}
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\end{frame}
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}
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{
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\begin{frame}{Window in a Graph in the Intermediate Regime}
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\includegraphics[width=\textwidth]{../thesis/graphics/graph_intermediate_regime_cut.png}
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\end{frame}
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}
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{
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\begin{frame}{Window in a Graph in the High-Linear Regime}
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\includegraphics[width=\textwidth]{../thesis/graphics/graph_high_linear_regime_cut.png}
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\end{frame}
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}
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%{
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%\begin{frame}{Graph in the Low-Linear Regime}
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% \includegraphics[width=\textwidth]{../thesis/graphics/graph_low_linear_regime.png}
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%\end{frame}
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%}
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%{
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%\begin{frame}{Window in a Graph in the Intermediate Regime}
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% \includegraphics[width=\textwidth]{../thesis/graphics/graph_intermediate_regime_cut.png}
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%\end{frame}
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%}
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%{
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%\begin{frame}{Window in a Graph in the High-Linear Regime}
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% \includegraphics[width=\textwidth]{../thesis/graphics/graph_high_linear_regime_cut.png}
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%\end{frame}
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%}
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{
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\begin{frame}{Performance: Circuit Length on Graphical Representation}
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\begin{itemize}
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\item{Pauli measurements reduce entanglement entropy.}
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\item{Pauli measurements reduce the amount of edges in the graph.}
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\item{When adding measurement to the random circuits the regimes
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do not show up.}
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\end{itemize}
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\end{frame}
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}
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%{
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%\begin{frame}{Performance: Circuit Length on Graphical Representation}
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% \begin{itemize}
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% \item{Pauli measurements reduce entanglement entropy.}
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% \item{Pauli measurements reduce the amount of edges in the graph.}
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% \item{When adding measurement to the random circuits the regimes
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% do not show up.}
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% \end{itemize}
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%\end{frame}
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%}
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{
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\begin{frame}{Performance: Circuit Length on Graphical Representation}
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@ -638,4 +657,27 @@
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}
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\section{Conclusion and Outlook}
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{
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\begin{frame}{Performance}
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\begin{itemize}
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\item{Simulation using stabilizers (in particular using the graphical representation)
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is polynomial hard.}
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\item{Simulation using dense state vectors is exponentially hard.}
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\item{The performance of the graphical simulator depends on the state/circuit.}
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\end{itemize}
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\end{frame}
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}
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{
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\begin{frame}{Physical Applications}
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\begin{itemize}
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\item{The graphical simulator cannot be used for physical applications because
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interesting problems require real (or at least rational) probability amplitudes.}
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\item{Stabilizer states do not span a vector space.}
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\item{The idea of finding a more efficient representation for a subset of
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states can be used in physics.}
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\end{itemize}
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\end{frame}
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}
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\end{document}
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@ -795,38 +795,120 @@
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{
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\begin{frame}{Clearing VOPs: Example}
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\includegraphics[width=0.5\textwidth]{graphs/clear_vop_01.png}
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\includegraphics[width=0.5\textwidth]{graphs/clear_vop_02.png}
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\noindent\begin{minipage}{0.5\textwidth}
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\includegraphics[width=\textwidth]{graphs/clear_vop_01.png}
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
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$19 = \sqrt{iZ}^2\sqrt{-iX}^3$
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\end{minipage}
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\noindent\begin{minipage}{0.5\textwidth}
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\includegraphics[width=\textwidth]{graphs/clear_vop_02.png}
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$ 21 = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]$\\
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$ 21 = \sqrt{iZ}^2\sqrt{-iX}^2$
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\end{minipage}
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\end{frame}
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}
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{
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\begin{frame}{Clearing VOPs: Example}
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\includegraphics[width=0.5\textwidth]{graphs/clear_vop_02.png}
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\includegraphics[width=0.5\textwidth]{graphs/clear_vop_03.png}
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\noindent\begin{minipage}{0.5\textwidth}
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\includegraphics[width=\textwidth]{graphs/clear_vop_02.png}
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$ 21 = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]$\\
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$ 21 = \sqrt{iZ}^2\sqrt{-iX}^2$
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\end{minipage}
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\noindent\begin{minipage}{0.5\textwidth}
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\includegraphics[width=\textwidth]{graphs/clear_vop_03.png}
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
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$11 = \sqrt{iZ}^2\sqrt{-iX}$
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\end{minipage}
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\end{frame}
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}
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{
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\begin{frame}{Clearing VOPs: Example}
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\includegraphics[width=0.5\textwidth]{graphs/clear_vop_03.png}
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\includegraphics[width=0.5\textwidth]{graphs/clear_vop_04.png}
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\noindent\begin{minipage}{0.5\textwidth}
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\includegraphics[width=\textwidth]{graphs/clear_vop_03.png}
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
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$11 = \sqrt{iZ}^2\sqrt{-iX}$
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\end{minipage}
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\noindent\begin{minipage}{0.5\textwidth}
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\includegraphics[width=\textwidth]{graphs/clear_vop_04.png}
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$\\
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$5 = \sqrt{iZ}^2$
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\end{minipage}
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\end{frame}
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}
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{
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\begin{frame}{Clearing VOPs: Example}
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\includegraphics[width=0.5\textwidth]{graphs/clear_vop_04.png}
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\includegraphics[width=0.5\textwidth]{graphs/clear_vop_05.png}
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\noindent\begin{minipage}{0.5\textwidth}
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\includegraphics[width=\textwidth]{graphs/clear_vop_04.png}
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$\\
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$5 = \sqrt{iZ}^2$
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\end{minipage}
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\noindent\begin{minipage}{0.5\textwidth}
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\includegraphics[width=\textwidth]{graphs/clear_vop_05.png}
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\end{minipage} \hfill
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\begin{minipage}{0.4\textwidth}
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Vertex operator \\
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$ 7 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
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$7 = \sqrt{iZ}$
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\end{minipage}
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\end{frame}
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}
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{
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\begin{frame}{Clearing VOPs: Example}
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\includegraphics[width=0.5\textwidth]{graphs/clear_vop_05.png}
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\includegraphics[width=0.5\textwidth]{graphs/clear_vop_06.png}
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\noindent\begin{minipage}{0.5\textwidth}
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\includegraphics[width=\textwidth]{graphs/clear_vop_05.png}
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\end{minipage} \hfill
|
||||
\begin{minipage}{0.4\textwidth}
|
||||
Vertex operator \\
|
||||
$ 7 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
|
||||
$7 = \sqrt{iZ}$
|
||||
\end{minipage}
|
||||
\noindent\begin{minipage}{0.5\textwidth}
|
||||
\includegraphics[width=\textwidth]{graphs/clear_vop_06.png}
|
||||
\end{minipage} \hfill
|
||||
\begin{minipage}{0.4\textwidth}
|
||||
Vertex operator \\
|
||||
$2 = \left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]$\\
|
||||
\end{minipage}
|
||||
\end{frame}
|
||||
}
|
||||
{
|
||||
\begin{frame}{Clearing VOPs: Example}
|
||||
\includegraphics[width=0.5\textwidth]{graphs/clear_vop_01.png}
|
||||
\includegraphics[width=0.5\textwidth]{graphs/clear_vop_06.png}
|
||||
\noindent\begin{minipage}{0.5\textwidth}
|
||||
\includegraphics[width=\textwidth]{graphs/clear_vop_01.png}
|
||||
\end{minipage} \hfill
|
||||
\begin{minipage}{0.4\textwidth}
|
||||
Vertex operator \\
|
||||
$19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
|
||||
$19 = \sqrt{iZ}^2\sqrt{-iX}^3$
|
||||
\end{minipage}
|
||||
\noindent\begin{minipage}{0.5\textwidth}
|
||||
\includegraphics[width=\textwidth]{graphs/clear_vop_06.png}
|
||||
\end{minipage} \hfill
|
||||
\begin{minipage}{0.4\textwidth}
|
||||
Vertex operator \\
|
||||
$2 = \left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]$
|
||||
\end{minipage}
|
||||
\end{frame}
|
||||
}
|
||||
|
||||
|
|
@ -867,6 +949,12 @@
|
|||
}
|
||||
|
||||
\section{Implementation and Performance}
|
||||
{
|
||||
\begin{frame}{Implementation}
|
||||
\center
|
||||
\includegraphics[width=\textwidth,height=0.8\textheight,keepaspectratio=true]{screenshot_github.png}
|
||||
\end{frame}
|
||||
}
|
||||
|
||||
{
|
||||
\begin{frame}{Implementation}
|
||||
|
|
|
|||
Loading…
Reference in New Issue