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Presentation should work like that

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Daniel Knüttel 2020-04-01 17:03:55 +02:00
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158
presentation/main.tex
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@ -306,6 +306,7 @@
\begin{itemize} \begin{itemize}
\item{ \item{
Choose a finite Abelian subgroup $S$ of $P_n$ with $-I \notin S$. Choose a finite Abelian subgroup $S$ of $P_n$ with $-I \notin S$.
One can show that all elements of $S$ are hermitian.
} }
\item{ \item{
One says $S = \langle S^{(i)} \rangle_{i=1,...,n}$ is generated by the $S^{(i)}$ One says $S = \langle S^{(i)} \rangle_{i=1,...,n}$ is generated by the $S^{(i)}$
@ -406,8 +407,9 @@
\item{Applying a local Clifford gate $U_i$ is trivial: \item{Applying a local Clifford gate $U_i$ is trivial:
Just the vertex operator is updated to $U o_i$.} Just the vertex operator is updated to $U o_i$.}
\item{If $o_a, o_b \in \{I, Z, R_\frac{\pi}{2}, R_\frac{\pi}{2}^\dagger\}$ applying a $CZ_{a,b}$ \item{If $o_a, o_b \in \{I, Z, R_\frac{\pi}{2}, R_\frac{\pi}{2}^\dagger\}$ applying a $CZ_{a,b}$
just toggles the edge $\{a,b\}$ in $E$: $E' = E \Delta \{\{a,b\}\}$. just toggles the edge $\{a,b\}$ in $E$.%: $E' = E \Delta \{\{a,b\}\}$.
With the symmetric set difference $\Delta$.} %With the symmetric set difference $\Delta$.
}
\item{If the vertices $a, b$ are isolated the resulting state after \item{If the vertices $a, b$ are isolated the resulting state after
applying $CZ_{a,b}$ can be precomputed. } applying $CZ_{a,b}$ can be precomputed. }
\item{When none of the above is possible one can clear at least one \item{When none of the above is possible one can clear at least one
@ -453,14 +455,16 @@
\end{minipage} \hfill \end{minipage} \hfill
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
Vertex operator \\ Vertex operator \\
$19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$ $19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
$19 = \sqrt{iZ}^2\sqrt{-iX}^3$
\end{minipage} \end{minipage}
\noindent\begin{minipage}{0.5\textwidth} \noindent\begin{minipage}{0.5\textwidth}
\includegraphics[width=\textwidth]{graphs/clear_vop_02.png} \includegraphics[width=\textwidth]{graphs/clear_vop_02.png}
\end{minipage} \hfill \end{minipage} \hfill
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
Vertex operator \\ Vertex operator \\
$ 21 = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]$ $ 21 = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]$\\
$ 21 = \sqrt{iZ}^2\sqrt{-iX}^2$
\end{minipage} \end{minipage}
\end{frame} \end{frame}
} }
@ -471,14 +475,16 @@
\end{minipage} \hfill \end{minipage} \hfill
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
Vertex operator \\ Vertex operator \\
$ 21 = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]$ $ 21 = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]$\\
$ 21 = \sqrt{iZ}^2\sqrt{-iX}^2$
\end{minipage} \end{minipage}
\noindent\begin{minipage}{0.5\textwidth} \noindent\begin{minipage}{0.5\textwidth}
\includegraphics[width=\textwidth]{graphs/clear_vop_03.png} \includegraphics[width=\textwidth]{graphs/clear_vop_03.png}
\end{minipage} \hfill \end{minipage} \hfill
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
Vertex operator \\ Vertex operator \\
$11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$ $11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
$11 = \sqrt{iZ}^2\sqrt{-iX}$
\end{minipage} \end{minipage}
\end{frame} \end{frame}
} }
@ -489,14 +495,16 @@
\end{minipage} \hfill \end{minipage} \hfill
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
Vertex operator \\ Vertex operator \\
$11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$ $11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
$11 = \sqrt{iZ}^2\sqrt{-iX}$
\end{minipage} \end{minipage}
\noindent\begin{minipage}{0.5\textwidth} \noindent\begin{minipage}{0.5\textwidth}
\includegraphics[width=\textwidth]{graphs/clear_vop_04.png} \includegraphics[width=\textwidth]{graphs/clear_vop_04.png}
\end{minipage} \hfill \end{minipage} \hfill
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
Vertex operator \\ Vertex operator \\
$5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$ $5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$\\
$5 = \sqrt{iZ}^2$
\end{minipage} \end{minipage}
\end{frame} \end{frame}
} }
@ -507,14 +515,16 @@
\end{minipage} \hfill \end{minipage} \hfill
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
Vertex operator \\ Vertex operator \\
$5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$ $5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$\\
$5 = \sqrt{iZ}^2$
\end{minipage} \end{minipage}
\noindent\begin{minipage}{0.5\textwidth} \noindent\begin{minipage}{0.5\textwidth}
\includegraphics[width=\textwidth]{graphs/clear_vop_05.png} \includegraphics[width=\textwidth]{graphs/clear_vop_05.png}
\end{minipage} \hfill \end{minipage} \hfill
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
Vertex operator \\ 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 = \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} \end{minipage}
\end{frame} \end{frame}
} }
@ -525,14 +535,15 @@
\end{minipage} \hfill \end{minipage} \hfill
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
Vertex operator \\ 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 = \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} \end{minipage}
\noindent\begin{minipage}{0.5\textwidth} \noindent\begin{minipage}{0.5\textwidth}
\includegraphics[width=\textwidth]{graphs/clear_vop_06.png} \includegraphics[width=\textwidth]{graphs/clear_vop_06.png}
\end{minipage} \hfill \end{minipage} \hfill
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
Vertex operator \\ Vertex operator \\
$2 = \left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]$ $2 = \left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]$\\
\end{minipage} \end{minipage}
\end{frame} \end{frame}
} }
@ -543,7 +554,8 @@
\end{minipage} \hfill \end{minipage} \hfill
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
Vertex operator \\ Vertex operator \\
$19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$ $19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
$19 = \sqrt{iZ}^2\sqrt{-iX}^3$
\end{minipage} \end{minipage}
\noindent\begin{minipage}{0.5\textwidth} \noindent\begin{minipage}{0.5\textwidth}
\includegraphics[width=\textwidth]{graphs/clear_vop_06.png} \includegraphics[width=\textwidth]{graphs/clear_vop_06.png}
@ -556,6 +568,13 @@
} }
\section{Implementation and Performance} \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} \begin{frame}{Implementation}
\begin{itemize} \begin{itemize}
@ -576,11 +595,11 @@
\includegraphics[width=\textwidth]{../performance/scaling_qbits_linear.png} \includegraphics[width=\textwidth]{../performance/scaling_qbits_linear.png}
\end{frame} \end{frame}
} }
{ %{
\begin{frame}{Performance: Dense Vector vs. Graphical Representation} %\begin{frame}{Performance: Dense Vector vs. Graphical Representation}
\includegraphics[width=\textwidth]{../performance/scaling_qbits_log.png} % \includegraphics[width=\textwidth]{../performance/scaling_qbits_log.png}
\end{frame} %\end{frame}
} %}
{ {
\begin{frame}{Performance: Circuit Length on Graphical Representation} \begin{frame}{Performance: Circuit Length on Graphical Representation}
@ -588,48 +607,48 @@
\end{frame} \end{frame}
} }
{ %{
\begin{frame}{Performance: Circuit Length on Graphical Representation} %\begin{frame}{Performance: Circuit Length on Graphical Representation}
\begin{itemize} % \begin{itemize}
\item{There seem to be three regimes: Low-linear regime, intermediate regime and high-linear regime.} % \item{There seem to be three regimes: Low-linear regime, intermediate regime and high-linear regime.}
\item{In the low-linear regime only few VOPs have to be cleared. % \item{In the low-linear regime only few VOPs have to be cleared.
The length of this regime increases with the number of qbits. % The length of this regime increases with the number of qbits.
} % }
\item{In the high-linear regime the neighbourhoods are big; the probability that VOPs % \item{In the high-linear regime the neighbourhoods are big; the probability that VOPs
must be cleared is high. Clearing VOPs involves many vertices.} % must be cleared is high. Clearing VOPs involves many vertices.}
\item{The intermediate is dominated by growing neighbourhoods $\Rightarrow$ % \item{The intermediate is dominated by growing neighbourhoods $\Rightarrow$
no linear behaviour.} % no linear behaviour.}
\end{itemize} % \end{itemize}
%
%\end{frame}
%}
\end{frame} %{
} %\begin{frame}{Graph in the Low-Linear Regime}
% \includegraphics[width=\textwidth]{../thesis/graphics/graph_low_linear_regime.png}
%\end{frame}
%}
%{
%\begin{frame}{Window in a Graph in the Intermediate Regime}
% \includegraphics[width=\textwidth]{../thesis/graphics/graph_intermediate_regime_cut.png}
%\end{frame}
%}
%{
%\begin{frame}{Window in a Graph in the High-Linear Regime}
% \includegraphics[width=\textwidth]{../thesis/graphics/graph_high_linear_regime_cut.png}
%\end{frame}
%}
{ %{
\begin{frame}{Graph in the Low-Linear Regime} %\begin{frame}{Performance: Circuit Length on Graphical Representation}
\includegraphics[width=\textwidth]{../thesis/graphics/graph_low_linear_regime.png} % \begin{itemize}
\end{frame} % \item{Pauli measurements reduce entanglement entropy.}
} % \item{Pauli measurements reduce the amount of edges in the graph.}
{ % \item{When adding measurement to the random circuits the regimes
\begin{frame}{Window in a Graph in the Intermediate Regime} % do not show up.}
\includegraphics[width=\textwidth]{../thesis/graphics/graph_intermediate_regime_cut.png} % \end{itemize}
\end{frame} %\end{frame}
} %}
{
\begin{frame}{Window in a Graph in the High-Linear Regime}
\includegraphics[width=\textwidth]{../thesis/graphics/graph_high_linear_regime_cut.png}
\end{frame}
}
{
\begin{frame}{Performance: Circuit Length on Graphical Representation}
\begin{itemize}
\item{Pauli measurements reduce entanglement entropy.}
\item{Pauli measurements reduce the amount of edges in the graph.}
\item{When adding measurement to the random circuits the regimes
do not show up.}
\end{itemize}
\end{frame}
}
{ {
\begin{frame}{Performance: Circuit Length on Graphical Representation} \begin{frame}{Performance: Circuit Length on Graphical Representation}
@ -638,4 +657,27 @@
} }
\section{Conclusion and Outlook} \section{Conclusion and Outlook}
{
\begin{frame}{Performance}
\begin{itemize}
\item{Simulation using stabilizers (in particular using the graphical representation)
is polynomial hard.}
\item{Simulation using dense state vectors is exponentially hard.}
\item{The performance of the graphical simulator depends on the state/circuit.}
\end{itemize}
\end{frame}
}
{
\begin{frame}{Physical Applications}
\begin{itemize}
\item{The graphical simulator cannot be used for physical applications because
interesting problems require real (or at least rational) probability amplitudes.}
\item{Stabilizer states do not span a vector space.}
\item{The idea of finding a more efficient representation for a subset of
states can be used in physics.}
\end{itemize}
\end{frame}
}
\end{document} \end{document}

112
presentation/main_long.tex
View File

@ -795,38 +795,120 @@
{ {
\begin{frame}{Clearing VOPs: Example} \begin{frame}{Clearing VOPs: Example}
\includegraphics[width=0.5\textwidth]{graphs/clear_vop_01.png} \noindent\begin{minipage}{0.5\textwidth}
\includegraphics[width=0.5\textwidth]{graphs/clear_vop_02.png} \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_02.png}
\end{minipage} \hfill
\begin{minipage}{0.4\textwidth}
Vertex operator \\
$ 21 = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]$\\
$ 21 = \sqrt{iZ}^2\sqrt{-iX}^2$
\end{minipage}
\end{frame} \end{frame}
} }
{ {
\begin{frame}{Clearing VOPs: Example} \begin{frame}{Clearing VOPs: Example}
\includegraphics[width=0.5\textwidth]{graphs/clear_vop_02.png} \noindent\begin{minipage}{0.5\textwidth}
\includegraphics[width=0.5\textwidth]{graphs/clear_vop_03.png} \includegraphics[width=\textwidth]{graphs/clear_vop_02.png}
\end{minipage} \hfill
\begin{minipage}{0.4\textwidth}
Vertex operator \\
$ 21 = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]$\\
$ 21 = \sqrt{iZ}^2\sqrt{-iX}^2$
\end{minipage}
\noindent\begin{minipage}{0.5\textwidth}
\includegraphics[width=\textwidth]{graphs/clear_vop_03.png}
\end{minipage} \hfill
\begin{minipage}{0.4\textwidth}
Vertex operator \\
$11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
$11 = \sqrt{iZ}^2\sqrt{-iX}$
\end{minipage}
\end{frame} \end{frame}
} }
{ {
\begin{frame}{Clearing VOPs: Example} \begin{frame}{Clearing VOPs: Example}
\includegraphics[width=0.5\textwidth]{graphs/clear_vop_03.png} \noindent\begin{minipage}{0.5\textwidth}
\includegraphics[width=0.5\textwidth]{graphs/clear_vop_04.png} \includegraphics[width=\textwidth]{graphs/clear_vop_03.png}
\end{minipage} \hfill
\begin{minipage}{0.4\textwidth}
Vertex operator \\
$11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\
$11 = \sqrt{iZ}^2\sqrt{-iX}$
\end{minipage}
\noindent\begin{minipage}{0.5\textwidth}
\includegraphics[width=\textwidth]{graphs/clear_vop_04.png}
\end{minipage} \hfill
\begin{minipage}{0.4\textwidth}
Vertex operator \\
$5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$\\
$5 = \sqrt{iZ}^2$
\end{minipage}
\end{frame} \end{frame}
} }
{ {
\begin{frame}{Clearing VOPs: Example} \begin{frame}{Clearing VOPs: Example}
\includegraphics[width=0.5\textwidth]{graphs/clear_vop_04.png} \noindent\begin{minipage}{0.5\textwidth}
\includegraphics[width=0.5\textwidth]{graphs/clear_vop_05.png} \includegraphics[width=\textwidth]{graphs/clear_vop_04.png}
\end{minipage} \hfill
\begin{minipage}{0.4\textwidth}
Vertex operator \\
$5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$\\
$5 = \sqrt{iZ}^2$
\end{minipage}
\noindent\begin{minipage}{0.5\textwidth}
\includegraphics[width=\textwidth]{graphs/clear_vop_05.png}
\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}
\end{frame} \end{frame}
} }
{ {
\begin{frame}{Clearing VOPs: Example} \begin{frame}{Clearing VOPs: Example}
\includegraphics[width=0.5\textwidth]{graphs/clear_vop_05.png} \noindent\begin{minipage}{0.5\textwidth}
\includegraphics[width=0.5\textwidth]{graphs/clear_vop_06.png} \includegraphics[width=\textwidth]{graphs/clear_vop_05.png}
\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} \end{frame}
} }
{ {
\begin{frame}{Clearing VOPs: Example} \begin{frame}{Clearing VOPs: Example}
\includegraphics[width=0.5\textwidth]{graphs/clear_vop_01.png} \noindent\begin{minipage}{0.5\textwidth}
\includegraphics[width=0.5\textwidth]{graphs/clear_vop_06.png} \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} \end{frame}
} }
@ -867,6 +949,12 @@
} }
\section{Implementation and Performance} \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} \begin{frame}{Implementation}