slidesC.tex

\documentclass{beamer}
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\usepackage{adjustbox}
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\usepackage{units}
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\usepackage{pdfcomment}

\newcommand{\rrangle}{\rangle\!\rangle} \newcommand{\llangle}{\langle\!\langle}
\newcommand{\ket}[1]{\ensuremath{\left|#1\right\rangle}}
\newcommand{\bra}[1]{\ensuremath{\left\langle#1\right|}}
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%Command used for python automatic substitution
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\newtoggle{confidences}
\newtoggle{LsAndGermsSet}
\newtoggle{debuggingaidsappendix}
\newtoggle{pixelplotsappendix}
\newtoggle{whackamoleappendix}
\togglefalse{confidences}
\toggletrue{LsAndGermsSet}
\togglefalse{debuggingaidsappendix}
\togglefalse{pixelplotsappendix}
\togglefalse{whackamoleappendix}


\hypersetup{
  pdfinfo={ Author={pyGSTi},
Title={GST on $\mathcal{D}$},
Keywords={GST},
pyGSTi_Version={0.9.3},
opt_long_tables={False},
opt_table_class={pygstiTbl},
opt_template_path={None},
opt_latex_cmd={pdflatex},
L_germ_tuple_base_string_dict={Dict[[0, Gx]: {}, [0, Gy]: {}, [0, Gi]: {}, [0, GxGy]: {}, [0, GxGyGi]: {}, [0, GxGiGy]: {}, [0, GxGiGi]: {}, [0, GyGiGi]: {}, [0, GxGxGiGy]: {}, [0, GxGyGyGi]: {}, [0, GxGxGyGxGyGy]: {}, [1, Gx]: [Gx], [1, Gy]: [Gy], [1, Gi]: [Gi], [1, GxGy]: {}, [1, GxGyGi]: {}, [1, GxGiGy]: {}, [1, GxGiGi]: {}, [1, GyGiGi]: {}, [1, GxGxGiGy]: {}, [1, GxGyGyGi]: {}, [1, GxGxGyGxGyGy]: {}, [2, Gx]: [Gx]2, [2, Gy]: [Gy]2, [2, Gi]: [Gi]2, [2, GxGy]: [GxGy], [2, GxGyGi]: {}, [2, GxGiGy]: {}, [2, GxGiGi]: {}, [2, GyGiGi]: {}, [2, GxGxGiGy]: {}, [2, GxGyGyGi]: {}, [2, GxGxGyGxGyGy]: {}, [4, Gx]: [Gx]4, [4, Gy]: [Gy]4, [4, Gi]: [Gi]4, [4, GxGy]: [GxGy]2, [4, GxGyGi]: [GxGyGi], [4, GxGiGy]: [GxGiGy], [4, GxGiGi]: [GxGiGi], [4, GyGiGi]: [GyGiGi], [4, GxGxGiGy]: [GxGxGiGy], [4, GxGyGyGi]: [GxGyGyGi], [4, GxGxGyGxGyGy]: {}, [8, Gx]: [Gx]8, [8, Gy]: [Gy]8, [8, Gi]: [Gi]8, [8, GxGy]: [GxGy]4, [8, GxGyGi]: [GxGyGi]2, [8, GxGiGy]: [GxGiGy]2, [8, GxGiGi]: [GxGiGi]2, [8, GyGiGi]: [GyGiGi]2, [8, GxGxGiGy]: [GxGxGiGy]2, [8, GxGyGyGi]: [GxGyGyGi]2, [8, GxGxGyGxGyGy]: [GxGxGyGxGyGy]]},
cptpPenaltyFactor={0},
defaultBasename={None},
defaultDirectory={None},
distributeMethod={deriv},
fiducial_pairs={None},
gaugeOptParams={Dict[]},
hessianProjection={optimal gate CIs},
linlogPercentile={5},
max_length_list={[0, 1, 2, 4, 8]},
memLimit={None},
minProbClip={1e-06},
minProbClipForWeighting={0.0001},
objective={logl},
probClipInterval={[-1000000.0, 1000000.0]},
radius={0.0001},
weights={None}  }
}


\title{GST on $\mathcal{D}$}
\subtitle{A projector-friendly look at your GST results}
\author{PyGSTi}
\institute{Sandia National Labs}
\date{\today}

\begin{document}

\section{Title}
\begin{frame}
\titlepage
\end{frame}

\section{Contents}
\begin{frame}
\frametitle{Outline}
\tableofcontents
\end{frame}


\iftoggle{LsAndGermsSet}{
\section{Goodness-of-model}

\subsection{$\log{\mathcal{L}}$ progress}
\begin{frame}
\frametitle{$\log{\mathcal{L}}$ vs.~GST iteration}

\begin{itemize}
\item $N_S$ is the number of gate strings
\item $N_p$ is the number of parameters
\end{itemize}

\begin{table}[h]
\begin{center}
\adjustbox{max height=\dimexpr\textheight-5.5cm\relax, max width=\textwidth}{
\begin{tabular}[l]{|c|c|c|c|c|c|c|c|c|}
\hline
L & $2\Delta\log(\mathcal{L})$ & $k$ & $2\Delta\log(\mathcal{L})-k$ & $\sqrt{2k}$ & $N_\sigma$ & $N_s$ & $N_p$ & Rating \\ \hline
0 & 53.75 & 61 & -7.251 & 11.05 & -0.66 & 92 & 31 & $\bigstar\bigstar\bigstar\bigstar\bigstar$ \\ \hline
1 & 53.75 & 61 & -7.251 & 11.05 & -0.66 & 92 & 31 & $\bigstar\bigstar\bigstar\bigstar\bigstar$ \\ \hline
2 & 106.8 & 137 & -30.25 & 16.55 & -1.83 & 168 & 31 & $\bigstar\bigstar\bigstar\bigstar\bigstar$ \\ \hline
4 & 439.5 & 410 & 29.48 & 28.64 & 1.03 & 441 & 31 & $\bigstar\bigstar\bigstar\bigstar$ \\ \hline
8 & 691.3 & 786 & -94.72 & 39.65 & -2.39 & 817 & 31 & $\bigstar\bigstar\bigstar\bigstar\bigstar$ \\ \hline
\end{tabular}
}
\caption{Comparison between the computed and expected $2\Delta\log{\mathcal{L}}$ for different values of $L$}
\end{center}
\end{table}
\end{frame}


\subsection{Color box plot}
\begin{frame}
\frametitle{Detailed $\log{\mathcal{L}}$ analysis}

\begin{figure}
\begin{center}
\includegraphics[width=4.00in,height=3.00in,keepaspectratio]{slidesC_files/bestLogLBoxes.pdf}
\end{center}
\end{figure}
\end{frame}

}{}

\section{Best gate set estimate}

\subsection{Estimate vs.~target}
\begin{frame}
\frametitle{GST Estimate vs.~target}

%\begin{itemize}
%\item Error generator is defined by the equation $\hat{G} = G_{\mathrm{target}}e^{\mathbb{L}}$.
%\end{itemize} 

\begin{table}[h]
\begin{center}

\adjustbox{max height=\dimexpr\textheight-5.5cm\relax, max width=\textwidth}{
\begin{tabular}{c}
\begin{tabular}[l]{|c|c|c|c|}
\hline
Gate & \begin{tabular}{c}Process\\Infidelity\end{tabular} & \begin{tabular}{c}$\nicefrac{1}{2}$ Trace\\Distance\end{tabular} & $\nicefrac{1}{2}$ $\Diamond$-Norm \\ \hline
Gi & 0.04 & 0.04 & 0.04 \\ \hline
Gx & 0.038 & 0.038 & 0.038 \\ \hline
Gy & 0.038 & 0.038 & 0.038 \\ \hline
\end{tabular}

\\
\vspace{2em}
\\
\begin{tabular}[l]{|c|c|}
\hline
Gate & Error Generator \\ \hline
Gi & $ \left(\!\!\begin{array}{cccc}
0 & 0 & 0 & 0 \\ 
-0.001 & -0.057 & -0.007 & -0.009 \\ 
4\e{-4} & 0.001 & -0.054 & 0.004 \\ 
0.001 & 2\e{-5} & 0.001 & -0.053
 \end{array}\!\!\right) $
 \\ \hline
Gx & $ \left(\!\!\begin{array}{cccc}
0 & 0 & 0 & 0 \\ 
3\e{-4} & -0.051 & -0.005 & -0.006 \\ 
2\e{-4} & 3\e{-4} & -0.052 & -0.002 \\ 
-2\e{-4} & -3\e{-4} & -7\e{-5} & -0.052
 \end{array}\!\!\right) $
 \\ \hline
Gy & $ \left(\!\!\begin{array}{cccc}
0 & 0 & 0 & 0 \\ 
2\e{-4} & -0.052 & 0.007 & 5\e{-4} \\ 
3\e{-4} & 0.001 & -0.05 & 0.001 \\ 
2\e{-4} & 5\e{-5} & -0.004 & -0.053
 \end{array}\!\!\right) $
 \\ \hline
\end{tabular}

\end{tabular}
}
\end{center}
\end{table}

\end{frame}


\subsection{Gate decomposition}
\begin{frame}
\frametitle{GST Estimate decomposition}

\begin{table}[h]
\begin{center}

\adjustbox{max height=\dimexpr\textheight-5.5cm\relax, max width=\textwidth}{
\begin{tabular}{c}
\begin{tabular}[l]{|c|c|c|c|c|c|}
\hline
Gate & Eigenvalues & Fixed pt & Rotn. axis & Diag. decay & Off-diag. decay \\ \hline
Gi & $ \begin{array}{c}
0.949 \\ 
0.946e^{i0.00} \\ 
0.946e^{-i0.00} \\ 
1
 \end{array} $
 & $ \begin{array}{c}
0.999 \\ 
-0.023 \\ 
0.009 \\ 
0.021
 \end{array} $
 & $ \begin{array}{c}
0 \\ 
0.933 \\ 
-0.151 \\ 
-0.326
 \end{array} $
 & 0.051 & 0.054 \\ \hline
Gx & $ \begin{array}{c}
0.949e^{i1.57} \\ 
0.949e^{-i1.57} \\ 
0.951 \\ 
1
 \end{array} $
 & $ \begin{array}{c}
1 \\ 
0.007 \\ 
-4\e{-6} \\ 
2\e{-4}
 \end{array} $
 & $ \begin{array}{c}
0 \\ 
-1 \\ 
1\e{-6} \\ 
-3\e{-4}
 \end{array} $
 & 0.049 & 0.051 \\ \hline
Gy & $ \begin{array}{c}
0.949e^{i1.57} \\ 
0.949e^{-i1.57} \\ 
0.951 \\ 
1
 \end{array} $
 & $ \begin{array}{c}
1 \\ 
-3\e{-5} \\ 
0.006 \\ 
-2\e{-4}
 \end{array} $
 & $ \begin{array}{c}
0 \\ 
-0.005 \\ 
1 \\ 
-0.001
 \end{array} $
 & 0.049 & 0.051 \\ \hline
\end{tabular}

\\
\vspace{2em}
\\
\begin{tabular}[l]{|c|c|c|c|c|}
\hline
\multirow{2}{*}{Gate} & \multirow{2}{*}{Angle} & \multicolumn{3}{c|}{Angle between Rotation Axes} \\ \cline{3-5}
 & & Gi & Gx & Gy \\ \hline
Gi & 0.001$\pi$ &  & 0.883$\pi$ & 0.55$\pi$ \\ \hline
Gx & 0.5$\pi$ & 0.883$\pi$ &  & 0.498$\pi$ \\ \hline
Gy & 0.5$\pi$ & 0.55$\pi$ & 0.498$\pi$ &  \\ \hline
\end{tabular}

\end{tabular}
}
\caption{Eigen-decomposition of estimated gates}
\end{center}
\end{table}

\end{frame}


\subsection{Raw Gates}
\begin{frame}
\frametitle{Raw GST Estimate: Gates}

\begin{table}[h]
\begin{center}
\adjustbox{max height=\dimexpr\textheight-5.5cm\relax, max width=\textwidth}{
\begin{tabular}[l]{|c|c|}
\hline
Gate & Superoperator (Pauli basis) \\ \hline
Gi & $ \left(\!\!\begin{array}{cccc}
1 & 0 & 0 & 0 \\ 
-0.001 & 0.945 & -0.006 & -0.009 \\ 
4\e{-4} & 0.001 & 0.947 & 0.004 \\ 
0.001 & 2\e{-5} & 0.001 & 0.949
 \end{array}\!\!\right) $
 \\ \hline
Gx & $ \left(\!\!\begin{array}{cccc}
1 & 0 & 0 & 0 \\ 
3\e{-4} & 0.951 & -0.005 & -0.005 \\ 
2\e{-4} & 3\e{-4} & 7\e{-5} & -0.949 \\ 
2\e{-4} & 3\e{-4} & 0.949 & -0.002
 \end{array}\!\!\right) $
 \\ \hline
Gy & $ \left(\!\!\begin{array}{cccc}
1 & 0 & 0 & 0 \\ 
2\e{-4} & 4\e{-5} & -0.004 & 0.949 \\ 
3\e{-4} & 0.001 & 0.951 & 0.001 \\ 
-2\e{-4} & -0.949 & -0.006 & -5\e{-4}
 \end{array}\!\!\right) $
 \\ \hline
\end{tabular}
}
\caption{GST estimate of the logic gate operations}
\end{center}
\end{table}

\end{frame}


\subsection{Raw SPAM}
\begin{frame}
\frametitle{Raw GST Estimate: SPAM}

\begin{table}[h]
\begin{center}
\adjustbox{max height=\dimexpr\textheight-5.5cm\relax, max width=\textwidth}{
\begin{tabular}{c}
\begin{tabular}[l]{|c|c|c|}
\hline
Operator & Matrix & Hilbert-Schmidt vector (Pauli basis) \\ \hline
$\rho_{0}$ & $ \left(\!\!\begin{array}{cc}
0.95 & 0.001e^{-i0.17} \\ 
0.001e^{i0.17} & 0.05
 \end{array}\!\!\right) $
 & $ \begin{array}{c}
0.707 \\ 
0.002 \\ 
3\e{-4} \\ 
0.637
 \end{array} $
 \\ \hline
$E_{0}$ & $ \left(\!\!\begin{array}{cc}
0.046 & 0.002e^{i1.61} \\ 
0.002e^{-i1.61} & 0.955
 \end{array}\!\!\right) $
 & $ \begin{array}{c}
0.708 \\ 
-1\e{-4} \\ 
-0.003 \\ 
-0.643
 \end{array} $
 \\ \hline
\end{tabular}

\\
\vspace{2em}
\\
\begin{tabular}[l]{|c|c|c|}
\hline
 & $E_{0}$ & $E_C$ \\ \hline
$\rho_{0}$ & 0.091 & 0.909 \\ \hline
\end{tabular}

\end{tabular}
}
\caption{The GST estimate of the SPAM operations and dot-product probabilities}
\end{center}
\end{table}

\end{frame}


\subsection{Choi Matrices}
\begin{frame}
\frametitle{Raw GST Choi Matrices}
\begin{table}[h]
\begin{center}
\adjustbox{max height=\dimexpr\textheight-5.5cm\relax, max width=\textwidth}{
\begin{tabular}[l]{|c|c|c|}
\hline
Gate & Choi matrix (Pauli basis) & Eigenvalues \\ \hline
Gi & $ \left(\!\!\begin{array}{cccc}
0.96 & 0.001e^{-i1.88} & 0.002e^{-i1.52} & 0.002e^{i1.43} \\ 
0.001e^{i1.88} & 0.012 & 0.001e^{-i2.94} & 0.002e^{i3.09} \\ 
0.002e^{i1.52} & 0.001e^{i2.94} & 0.013 & 0.001e^{i0.22} \\ 
0.002e^{-i1.43} & 0.002e^{-i3.09} & 0.001e^{-i0.22} & 0.014
 \end{array}\!\!\right) $
 & $ \begin{array}{c}
0.011 \\ 
0.013 \\ 
0.017 \\ 
0.96
 \end{array} $
 \\ \hline
Gx & $ \left(\!\!\begin{array}{cccc}
0.487 & 0.475e^{i1.57} & 0.001e^{-i1.54} & 0.001e^{i1.53} \\ 
0.475e^{-i1.57} & 0.488 & 0.001e^{-i3.09} & 0.001e^{i3.10} \\ 
0.001e^{i1.54} & 0.001e^{i3.09} & 0.013 & 9\e{-5}e^{-i1.85} \\ 
0.001e^{-i1.53} & 0.001e^{-i3.10} & 9\e{-5}e^{i1.85} & 0.012
 \end{array}\!\!\right) $
 & $ \begin{array}{c}
0.011 \\ 
0.013 \\ 
0.014 \\ 
0.962
 \end{array} $
 \\ \hline
Gy & $ \left(\!\!\begin{array}{cccc}
0.488 & 0.002e^{-i1.54} & 0.474e^{i1.57} & 0.001e^{i1.61} \\ 
0.002e^{i1.54} & 0.012 & 0.001e^{i3.07} & 1\e{-4}e^{i2.20} \\ 
0.474e^{-i1.57} & 0.001e^{-i3.07} & 0.488 & 0.001e^{-i3.10} \\ 
0.001e^{-i1.61} & 1\e{-4}e^{-i2.20} & 0.001e^{i3.10} & 0.012
 \end{array}\!\!\right) $
 & $ \begin{array}{c}
0.011 \\ 
0.012 \\ 
0.015 \\ 
0.962
 \end{array} $
 \\ \hline
\end{tabular}
}
\caption{Choi matrix representation of the GST estimated gateset.}
\end{center}
\end{table}
\end{frame}



\section{Inputs to GST}

\subsection{Target gate set}
\begin{frame}
\frametitle{Target SPAM}

\begin{table}[h]
\begin{center}
\adjustbox{max height=\dimexpr\textheight-5.5cm\relax, max width=\textwidth}{
\begin{tabular}[l]{|c|c|c|}
\hline
Operator & Matrix & Hilbert-Schmidt vector (Pauli basis) \\ \hline
$\rho_{0}$ & $ \left(\!\!\begin{array}{cc}
1 & 0 \\ 
0 & 0
 \end{array}\!\!\right) $
 & $ \begin{array}{c}
0.707 \\ 
0 \\ 
0 \\ 
0.707
 \end{array} $
 \\ \hline
$E_{0}$ & $ \left(\!\!\begin{array}{cc}
0 & 0 \\ 
0 & 1
 \end{array}\!\!\right) $
 & $ \begin{array}{c}
0.707 \\ 
0 \\ 
0 \\ 
-0.707
 \end{array} $
 \\ \hline
\end{tabular}
}
\caption{State preparation and measurement targets}
\end{center}
\end{table}

\end{frame}


\begin{frame}
\frametitle{Target Gates}

\begin{table}[h]
\begin{center}
\adjustbox{max height=\dimexpr\textheight-5.5cm\relax, max width=\textwidth}{
\begin{tabular}[l]{|c|c|c|c|}
\hline
Gate & Superoperator (Pauli basis) & Rotation axis & Angle \\ \hline
Gi & $ \left(\!\!\begin{array}{cccc}
1 & 0 & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 1
 \end{array}\!\!\right) $
 & $ \begin{array}{c}
0 \\ 
1 \\ 
0 \\ 
0
 \end{array} $
 & 0$\pi$ \\ \hline
Gx & $ \left(\!\!\begin{array}{cccc}
1 & 0 & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & -1 \\ 
0 & 0 & 1 & 0
 \end{array}\!\!\right) $
 & $ \begin{array}{c}
0 \\ 
1 \\ 
0 \\ 
0
 \end{array} $
 & 0.5$\pi$ \\ \hline
Gy & $ \left(\!\!\begin{array}{cccc}
1 & 0 & 0 & 0 \\ 
0 & 0 & 0 & 1 \\ 
0 & 0 & 1 & 0 \\ 
0 & -1 & 0 & 0
 \end{array}\!\!\right) $
 & $ \begin{array}{c}
0 \\ 
0 \\ 
1 \\ 
0
 \end{array} $
 & 0.5$\pi$ \\ \hline
\end{tabular}
}
\caption{Summary of target gates}
\end{center}
\end{table}

\end{frame}


\iftoggle{LsAndGermsSet}{
\subsection{Fiducials and Germs}
\begin{frame}
\frametitle{Fiducial and Germ Gate Strings}

\begin{table}[h]
\footnotesize
\begin{center}
\begin{minipage}[b]{0.40\linewidth}
\centering
\adjustbox{max width=\linewidth}{
\begin{tabular}[l]{|c|c|c|}
\hline
 & \multicolumn{2}{c|}{Fiducials} \\ \hline
\# & Prep. & Measure \\ \hline
1 & $$ & $$ \\ \hline
2 & $\mbox{Gx}$ & $\mbox{Gx}$ \\ \hline
3 & $\mbox{Gy}$ & $\mbox{Gy}$ \\ \hline
4 & $\mbox{Gx}\cdot\mbox{Gx}$ & $\mbox{Gx}\cdot\mbox{Gx}$ \\ \hline
5 & $\mbox{Gx}\cdot\mbox{Gx}\cdot\mbox{Gx}$ & $\mbox{Gx}\cdot\mbox{Gx}\cdot\mbox{Gx}$ \\ \hline
6 & $\mbox{Gy}\cdot\mbox{Gy}\cdot\mbox{Gy}$ & $\mbox{Gy}\cdot\mbox{Gy}\cdot\mbox{Gy}$ \\ \hline
\end{tabular}

}
\end{minipage}
\begin{minipage}[b]{0.40\linewidth}
\adjustbox{max width=\linewidth}{
\begin{tabular}[l]{|c|c|}
\hline
\# & Germ \\ \hline
1 & $\mbox{Gx}$ \\ \hline
2 & $\mbox{Gy}$ \\ \hline
3 & $\mbox{Gi}$ \\ \hline
4 & $\mbox{Gx}\cdot\mbox{Gy}$ \\ \hline
5 & $\mbox{Gx}\cdot\mbox{Gy}\cdot\mbox{Gi}$ \\ \hline
6 & $\mbox{Gx}\cdot\mbox{Gi}\cdot\mbox{Gy}$ \\ \hline
7 & $\mbox{Gx}\cdot\mbox{Gi}\cdot\mbox{Gi}$ \\ \hline
8 & $\mbox{Gy}\cdot\mbox{Gi}\cdot\mbox{Gi}$ \\ \hline
9 & $\mbox{Gx}\cdot\mbox{Gx}\cdot\mbox{Gi}\cdot\mbox{Gy}$ \\ \hline
10 & $\mbox{Gx}\cdot\mbox{Gy}\cdot\mbox{Gy}\cdot\mbox{Gi}$ \\ \hline
11 & $\mbox{Gx}\cdot\mbox{Gx}\cdot\mbox{Gy}\cdot\mbox{Gx}\cdot\mbox{Gy}\cdot\mbox{Gy}$ \\ \hline
\end{tabular}

}
\end{minipage}
\end{center}
\end{table}

\end{frame}

}{}


\subsection{Dataset}
\begin{frame}
\frametitle{Dataset Overview}

\begin{table}[h]
\begin{center}
\adjustbox{max height=\dimexpr\textheight-5.5cm\relax, max width=\textwidth}{
\begin{tabular}[l]{|c|c|}
\hline
Quantity & Value \\ \hline
Number of strings & 817 \\ \hline
Gate labels & Gx, Gy, Gi \\ \hline
SPAM labels & plus, minus \\ \hline
Counts per string & 1000 \\ \hline
\begin{tabular}{c}Gram singular values\\ (right column gives the values\\when using the target gate set)\end{tabular} & \small$ \begin{array}{cc}
0.003 & 0 \\ 
0.02 & 0 \\ 
0.663 & 1 \\ 
0.679 & 1 \\ 
0.747 & 1 \\ 
3.007 & 3
 \end{array} $
 \\ \hline
\end{tabular}
}
\caption{General dataset properties}
\end{center}
\end{table}

\end{frame}



\iftoggle{debuggingaidsappendix}{
\section{Debugging Aids}

\begin{frame}
\frametitle{Direct-GST}

\begin{figure}
\begin{center}

\caption{How well direct GST analysis of each base sequence fits the observed data.}
\end{center}
\end{figure}

\end{frame}


\begin{frame}
\frametitle{Direct-GST Deviation}

\begin{figure}
\begin{center}

\caption{For each base sequence, the increase in ``upper bound of fidelity with unitary'' when using the direct-GST result for a gate sequence instead of the process given by the best gateset.}
\end{center}
\end{figure}

\end{frame}


\begin{frame}
\frametitle{Per-gate error rates}

\begin{figure}
\begin{center}

\caption{Extrapolated from the smallest magnitude eigenvalue of the direct GST gate matrix.}
\end{center}
\end{figure}

\end{frame}
}{}

\iftoggle{pixelplotsappendix}{
\section{Pixel Plots}

}{}


\iftoggle{whackamoleappendix}{
\section{Whack-a-mole}

}{}

\end{document}