From 9e9900f31162374e3ef5e03400873861c734a788 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= Date: Wed, 1 Apr 2020 17:03:55 +0200 Subject: [PATCH] Presentation should work like that --- presentation/main.tex | 158 +++++++++++++++++++++++-------------- presentation/main_long.tex | 112 +++++++++++++++++++++++--- 2 files changed, 200 insertions(+), 70 deletions(-) diff --git a/presentation/main.tex b/presentation/main.tex index 49caacc..d268007 100644 --- a/presentation/main.tex +++ b/presentation/main.tex @@ -306,6 +306,7 @@ \begin{itemize} \item{ Choose a finite Abelian subgroup $S$ of $P_n$ with $-I \notin S$. + One can show that all elements of $S$ are hermitian. } \item{ 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: 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 $CZ_{a,b}$ - just toggles the edge $\{a,b\}$ in $E$: $E' = E \Delta \{\{a,b\}\}$. - With the symmetric set difference $\Delta$.} + just toggles the edge $\{a,b\}$ in $E$.%: $E' = E \Delta \{\{a,b\}\}$. + %With the symmetric set difference $\Delta$. + } \item{If the vertices $a, b$ are isolated the resulting state after applying $CZ_{a,b}$ can be precomputed. } \item{When none of the above is possible one can clear at least one @@ -453,14 +455,16 @@ \end{minipage} \hfill \begin{minipage}{0.4\textwidth} Vertex operator \\ - $19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$ + $19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\ + $19 = \sqrt{iZ}^2\sqrt{-iX}^3$ \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 = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right]$\\ + $ 21 = \sqrt{iZ}^2\sqrt{-iX}^2$ \end{minipage} \end{frame} } @@ -471,14 +475,16 @@ \end{minipage} \hfill \begin{minipage}{0.4\textwidth} 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} \noindent\begin{minipage}{0.5\textwidth} \includegraphics[width=\textwidth]{graphs/clear_vop_03.png} \end{minipage} \hfill \begin{minipage}{0.4\textwidth} Vertex operator \\ - $11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$ + $11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\ + $11 = \sqrt{iZ}^2\sqrt{-iX}$ \end{minipage} \end{frame} } @@ -489,14 +495,16 @@ \end{minipage} \hfill \begin{minipage}{0.4\textwidth} Vertex operator \\ - $11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$ + $11 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2} i}{2}\\\frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\ + $11 = \sqrt{iZ}^2\sqrt{-iX}$ \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 \\ - $5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$ + $5 = \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$\\ + $5 = \sqrt{iZ}^2$ \end{minipage} \end{frame} } @@ -507,14 +515,16 @@ \end{minipage} \hfill \begin{minipage}{0.4\textwidth} 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} \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 = \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} } @@ -525,14 +535,15 @@ \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 = \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 \\ - $2 = \left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]$ + $2 = \left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]$\\ \end{minipage} \end{frame} } @@ -543,7 +554,8 @@ \end{minipage} \hfill \begin{minipage}{0.4\textwidth} Vertex operator \\ - $19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$ + $19 = \left[\begin{matrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2} i}{2}\\- \frac{\sqrt{2} i}{2} & - \frac{\sqrt{2}}{2}\end{matrix}\right]$\\ + $19 = \sqrt{iZ}^2\sqrt{-iX}^3$ \end{minipage} \noindent\begin{minipage}{0.5\textwidth} \includegraphics[width=\textwidth]{graphs/clear_vop_06.png} @@ -556,6 +568,13 @@ } \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{itemize} @@ -576,11 +595,11 @@ \includegraphics[width=\textwidth]{../performance/scaling_qbits_linear.png} \end{frame} } -{ -\begin{frame}{Performance: Dense Vector vs. Graphical Representation} - \includegraphics[width=\textwidth]{../performance/scaling_qbits_log.png} -\end{frame} -} +%{ +%\begin{frame}{Performance: Dense Vector vs. Graphical Representation} +% \includegraphics[width=\textwidth]{../performance/scaling_qbits_log.png} +%\end{frame} +%} { \begin{frame}{Performance: Circuit Length on Graphical Representation} @@ -588,48 +607,48 @@ \end{frame} } -{ -\begin{frame}{Performance: Circuit Length on Graphical Representation} - \begin{itemize} - \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. - 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 - must be cleared is high. Clearing VOPs involves many vertices.} - \item{The intermediate is dominated by growing neighbourhoods $\Rightarrow$ - no linear behaviour.} - \end{itemize} - -\end{frame} -} +%{ +%\begin{frame}{Performance: Circuit Length on Graphical Representation} +% \begin{itemize} +% \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. +% 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 +% must be cleared is high. Clearing VOPs involves many vertices.} +% \item{The intermediate is dominated by growing neighbourhoods $\Rightarrow$ +% no linear behaviour.} +% \end{itemize} +% +%\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} +% \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}{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{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} @@ -638,4 +657,27 @@ } \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} diff --git a/presentation/main_long.tex b/presentation/main_long.tex index 391f7a9..bb5ba31 100644 --- a/presentation/main_long.tex +++ b/presentation/main_long.tex @@ -795,38 +795,120 @@ { \begin{frame}{Clearing VOPs: Example} - \includegraphics[width=0.5\textwidth]{graphs/clear_vop_01.png} - \includegraphics[width=0.5\textwidth]{graphs/clear_vop_02.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_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} } { \begin{frame}{Clearing VOPs: Example} - \includegraphics[width=0.5\textwidth]{graphs/clear_vop_02.png} - \includegraphics[width=0.5\textwidth]{graphs/clear_vop_03.png} + \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} + \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} } { \begin{frame}{Clearing VOPs: Example} - \includegraphics[width=0.5\textwidth]{graphs/clear_vop_03.png} - \includegraphics[width=0.5\textwidth]{graphs/clear_vop_04.png} + \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} + \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} } { \begin{frame}{Clearing VOPs: Example} - \includegraphics[width=0.5\textwidth]{graphs/clear_vop_04.png} - \includegraphics[width=0.5\textwidth]{graphs/clear_vop_05.png} + \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} + \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} } { \begin{frame}{Clearing VOPs: Example} - \includegraphics[width=0.5\textwidth]{graphs/clear_vop_05.png} - \includegraphics[width=0.5\textwidth]{graphs/clear_vop_06.png} + \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} + \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}