slidesC.tex

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\newtoggle{confidences}
\newtoggle{LsAndGermsSet}
\newtoggle{debuggingaidsappendix}
\newtoggle{pixelplotsappendix}
\newtoggle{whackamoleappendix}
\togglefalse{confidences}
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\togglefalse{debuggingaidsappendix}
\togglefalse{pixelplotsappendix}
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\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 & $2\Delta\log(\mathcal{L})$ & $k$ & $2\Delta\log(\mathcal{L})-k$ & $\sqrt{2k}$ & $P$ & $N_s$ & $N_p$ & Rating \\ \hline
0 & 53.74042 & 61 & -7.259581 & 11.04536 & 0.73 & 92 & 31 & $\bigstar\bigstar\bigstar\bigstar\bigstar$ \\ \hline
1 & 53.74027 & 61 & -7.259733 & 11.04536 & 0.73 & 92 & 31 & $\bigstar\bigstar\bigstar\bigstar\bigstar$ \\ \hline
2 & 106.549 & 137 & -30.451 & 16.55295 & 0.97 & 168 & 31 & $\bigstar\bigstar\bigstar\bigstar\bigstar$ \\ \hline
4 & 354.2974 & 410 & -55.70262 & 28.63564 & 0.98 & 441 & 31 & $\bigstar\bigstar\bigstar\bigstar\bigstar$ \\ \hline
8 & 691.0505 & 786 & -94.94949 & 39.64846 & 0.99 & 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=4.000000in,height=3.000000in,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|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 & \begin{tabular}{c}Frobenius\\Distance\end{tabular} \\ \hline
Gi & 0.039931 & 0.042033 & 0.048064 & 0.101842 \\ \hline
Gx & -0.102707 & 0.925912 & 1.404888 & 2.136226 \\ \hline
Gy & 0.010043 & 0.522965 & 0.683333 & 1.124471 \\ \hline
\end{tabular}

\\
\vspace{2em}
\\
\begin{tabular}[l]{|c|c|}
\hline
Gate & Error Generator \\ \hline
Gi & $ \left(\!\!\begin{array}{cccc}
0 & 0 & 0 & 0 \\ 
0.0204 & -0.0558 & -0.0099 & -0.0056 \\ 
-0.0345 & 0.0008 & -0.0559 & 0.0016 \\ 
0.0151 & 0.0002 & 0.0025 & -0.0524
 \end{array}\!\!\right) $
 \\ \hline
Gx & $ \left(\!\!\begin{array}{cccc}
0 & 0 & 0 & 0 \\ 
-0.1417 & -0.0034 & -0.0942 & 0.2589 \\ 
1.3661 & -0.407 & 0.482 & -0.4731 \\ 
0.6238 & -0.1413 & -0.469 & -0.6339
 \end{array}\!\!\right) $
 \\ \hline
Gy & $ \left(\!\!\begin{array}{cccc}
0 & 0 & 0 & 0 \\ 
-0.7914 & 0.181 & 0.0471 & 0.2597 \\ 
0.0308 & -0.4321 & -0.0846 & -0.1081 \\ 
0.1674 & 0.2612 & -0.1412 & -0.2513
 \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}
0.9487 \\ 
0.9458e^{i0.0} \\ 
0.9458e^{-i0.0} \\ 
1
 \end{array} $
 & $ \begin{array}{c}
0.6131 \\ 
0.2737 \\ 
-0.3701 \\ 
0.1593
 \end{array} $
 & $ \begin{array}{c}
0 \\ 
0.8838 \\ 
-0.0211 \\ 
-0.4673
 \end{array} $
 & 0.051276 & 0.054221 \\ \hline
Gx & $ \begin{array}{c}
0.9491e^{i1.6} \\ 
0.9491e^{-i1.6} \\ 
0.9505 \\ 
1
 \end{array} $
 & $ \begin{array}{c}
0.6128 \\ 
0.2743 \\ 
-0.3701 \\ 
0.1585
 \end{array} $
 & $ \begin{array}{c}
0 \\ 
0.9766 \\ 
0.1799 \\ 
-0.1177
 \end{array} $
 & 0.049515 & 0.050938 \\ \hline
Gy & $ \begin{array}{c}
0.9489e^{i1.6} \\ 
0.9489e^{-i1.6} \\ 
0.9512 \\ 
1
 \end{array} $
 & $ \begin{array}{c}
0.6129 \\ 
0.2742 \\ 
-0.3701 \\ 
0.1585
 \end{array} $
 & $ \begin{array}{c}
0 \\ 
-0.091 \\ 
0.9937 \\ 
0.0657
 \end{array} $
 & 0.048792 & 0.051087 \\ \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 & 0.000627$\pi$ &  & 0.132677$\pi$ & 0.5$\pi$ \\ \hline
Gx & 0.500356$\pi$ & 0.132677$\pi$ &  & 0.473816$\pi$ \\ \hline
Gy & 0.50007$\pi$ & 0.5$\pi$ & 0.473816$\pi$ &  \\ \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 \\ 
0.02 & 0.9457 & -0.0093 & -0.0053 \\ 
-0.0335 & 0.0007 & 0.9457 & 0.0015 \\ 
0.0146 & 0.0002 & 0.0024 & 0.9489
 \end{array}\!\!\right) $
 \\ \hline
Gx & $ \left(\!\!\begin{array}{cccc}
1 & 0 & 0 & 0 \\ 
-0.1747 & 1.0115 & -0.1855 & 0.2225 \\ 
-0.1679 & 0.0145 & 0.4695 & -0.6146 \\ 
1.7013 & -0.5075 & 1.7847 & -0.5326
 \end{array}\!\!\right) $
 \\ \hline
Gy & $ \left(\!\!\begin{array}{cccc}
1 & 0 & 0 & 0 \\ 
0.0386 & 0.2862 & -0.115 & 0.8171 \\ 
0.1997 & -0.4731 & 0.9176 & -0.1464 \\ 
0.8525 & -1.2252 & -0.0326 & -0.2529
 \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 & Hilbert-Schmidt vector (Pauli basis) & Matrix \\ \hline
$\rho_{0}$ & $ \begin{array}{c}
0.9503 \\ 
0.4194 \\ 
-0.5638 \\ 
0.2879
 \end{array} $
 & $ \left(\!\!\begin{array}{cc}
0.8756 & 0.4969e^{i0.9} \\ 
0.4969e^{-i0.9} & 0.4684
 \end{array}\!\!\right) $
 \\ \hline
$E_{0}$ & $ \begin{array}{c}
3.9955 \\ 
-1.3065 \\ 
0.4888 \\ 
-10.011
 \end{array} $
 & $ \left(\!\!\begin{array}{cc}
-4.2536 & 0.9864e^{-i2.8} \\ 
0.9864e^{i2.8} & 9.9042
 \end{array}\!\!\right) $
 \\ \hline
\end{tabular}

\\
\vspace{2em}
\\
\begin{tabular}[l]{|c|c|c|}
\hline
 & $E_{0}$ & $E_{-1}$ \\ \hline
$\rho_{0}$ & 0.090934 & 0.909571 \\ \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}
0.9601 & 0.005e^{-i0.0} & 0.0085e^{-i0.2} & 0.0044e^{-i0.6} \\ 
0.005e^{i0.0} & 0.0128 & 0.0042e^{-i1.0} & 0.0085e^{i1.7} \\ 
0.0085e^{i0.2} & 0.0042e^{i1.0} & 0.0128 & 0.0051e^{-i1.8} \\ 
0.0044e^{i0.6} & 0.0085e^{-i1.7} & 0.0051e^{i1.8} & 0.0144
 \end{array}\!\!\right) $
 & $ \begin{array}{c}
0.0024 \\ 
0.0135 \\ 
0.024 \\ 
0.9602
 \end{array} $
 \\ \hline
Gx & $ \left(\!\!\begin{array}{cccc}
0.4871 & 0.6014e^{-i1.6} & 0.1873e^{i1.3} & 0.4283e^{-i0.1} \\ 
0.6014e^{i1.6} & 0.5186 & 0.4275e^{-i1.5} & 0.0827e^{i2.6} \\ 
0.1873e^{-i1.3} & 0.4275e^{i1.5} & 0.2477 & 0.2958e^{i3.0} \\ 
0.4283e^{i0.1} & 0.0827e^{-i2.6} & 0.2958e^{-i3.0} & -0.2534
 \end{array}\!\!\right) $
 & $ \begin{array}{c}
-0.6977 \\ 
0.0042 \\ 
0.3854 \\ 
1.308
 \end{array} $
 \\ \hline
Gy & $ \left(\!\!\begin{array}{cccc}
0.4877 & 0.03e^{-i1.2} & 0.513e^{i1.7} & 0.2312e^{i0.4} \\ 
0.03e^{i1.2} & 0.1554 & 0.2589e^{-i1.0} & 0.1136e^{-i2.7} \\ 
0.513e^{-i1.7} & 0.2589e^{i1.0} & 0.4711 & 0.0458e^{-i0.2} \\ 
0.2312e^{-i0.4} & 0.1136e^{i2.7} & 0.0458e^{i0.2} & -0.1142
 \end{array}\!\!\right) $
 & $ \begin{array}{c}
-0.3273 \\ 
0.0066 \\ 
0.2625 \\ 
1.0583
 \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 & Hilbert-Schmidt vector (Pauli basis) & Matrix \\ \hline
$\rho_{0}$ & $ \begin{array}{c}
0.7071 \\ 
0 \\ 
0 \\ 
0.7071
 \end{array} $
 & $ \left(\!\!\begin{array}{cc}
1 & 0 \\ 
0 & 0
 \end{array}\!\!\right) $
 \\ \hline
$E_{0}$ & $ \begin{array}{c}
0.7071 \\ 
0 \\ 
0 \\ 
-0.7071
 \end{array} $
 & $ \left(\!\!\begin{array}{cc}
0 & 0 \\ 
0 & 1
 \end{array}\!\!\right) $
 \\ \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
Gram singular vals & \small$ \begin{array}{c}
0.0075 \\ 
0.0158 \\ 
0.0261 \\ 
0.6631 \\ 
0.6791 \\ 
1.0608 \\ 
3.4605
 \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}
\includegraphics[width=4.000000in,height=2.500000in,keepaspectratio]{slidesC_files/directLongSeqGSTLogLBoxes.pdf}
\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}
\includegraphics[width=4.000000in,height=2.500000in,keepaspectratio]{slidesC_files/directLongSeqGSTDeviationBoxes.pdf}
\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}
\includegraphics[width=4.000000in,height=2.500000in,keepaspectratio]{slidesC_files/smallEigvalErrRateBoxes.pdf}
\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}