445 lines
25 KiB
TeX
445 lines
25 KiB
TeX
\chapter{Fault Tolerant QEC}
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\section{Fault Tolerance}
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\section{Noise Models}
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\subsection{Depolarizing Channel}
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\subsection{Phenomenological Noise}
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\subsection{Circuit-Level Noise}
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\section{Detector Error Models}
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\subsection{Measurement Syndrome Matrix}
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\subsection{Detector Error Matrix}
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\subsection{Detector Error Models}
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\section{Practical Considerations}
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\subsection{Practical Methodology}
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\subsection{Stim}
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\begin{figure}[t]
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\centering
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\newcommand{\xerr}{\gate[style={fill=KITblue!50}]{\phantom{1}}}
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\newcommand{\xyzerr}{\gate[style={
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draw=black,
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fill=KITred,
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path picture={
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% tex-fmt: off
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\fill[KITblue!60]
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($(path picture bounding box.south west)+(0,0)$)
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-- ($(path picture bounding box.north west)+(0,0)$)
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-- ($(path picture bounding box.north west)+(0.28,0)$)
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-- cycle;
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\fill[KITorange!60]
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($(path picture bounding box.north east)+(0,0)$)
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-- ($(path picture bounding box.south east)+(0,0)$)
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-- ($(path picture bounding box.south east)+(-0.28,0)$)
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-- cycle;
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\fill[KITred!60]
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($(path picture bounding box.north east)+(0,0)$)
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-- ($(path picture bounding box.south east)+(-0.28,0)$)
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-- ($(path picture bounding box.south west)+(0,0)$)
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-- ($(path picture bounding box.north west)+(0.28,0)$)
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-- cycle;
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% tex-fmt: on
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}
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}]{\phantom{1}}}
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\begin{subfigure}{0.48\textwidth}
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\centering
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% tex-fmt: off
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\begin{quantikz}[row sep=4mm, column sep=4mm]
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\lstick[3]{$\ket{\psi}$} & \xyzerr & \ctrl{3} & & & & & \\
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& \xyzerr & & \ctrl{2} & \ctrl{3} & & & \\
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& \xyzerr & & & & \ctrl{2} & & \\
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\lstick{$\ket{0}_{\text{A}_1}$} & & \targ{} & \targ{} & & & \meter{} & \setwiretype{c} \\
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\lstick{$\ket{0}_{\text{A}_2}$} & & & & \targ{} & \targ{} & \meter{} & \setwiretype{c}
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\end{quantikz}
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% tex-fmt: on
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\caption{Depolarizing channel.}
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\end{subfigure}%
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\begin{subfigure}{0.48\textwidth}
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\centering
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% tex-fmt: off
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\begin{quantikz}[row sep=4mm, column sep=4mm]
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\lstick[3]{$\ket{\psi}$} & \xerr & \ctrl{3} & & & & & & \\
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& \xerr & & \ctrl{2} & \ctrl{3} & & & & \\
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& \xerr & & & & \ctrl{2} & & & \\
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\lstick{$\ket{0}_{\text{A}_1}$} & & \targ{} & \targ{} & & & \xerr & \meter{} & \setwiretype{c} \\
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\lstick{$\ket{0}_{\text{A}_2}$} & & & & \targ{} & \targ{} & \xerr & \meter{} & \setwiretype{c}
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\end{quantikz}
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% tex-fmt: on
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\caption{Phenomenological noise.}
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\end{subfigure}
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\vspace*{3mm}
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\centering
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\begin{subfigure}[c]{0.75\textwidth}
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\centering
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% tex-fmt: off
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\begin{quantikz}[row sep=4mm, column sep=2mm]
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\lstick[3]{$\ket{\psi}$} & \xyzerr & \ctrl{3} & \xyzerr \wire[d][3]{q} & & & & & & & & & \\
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& \xyzerr & & & \ctrl{2} & \xyzerr \wire[d][2]{q} & \ctrl{3} & \xyzerr \wire[d][3]{q} & & & & & \\
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& \xyzerr & & & & & & & \ctrl{2} & \xyzerr \wire[d][2]{q} & & & \\
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\lstick{$\ket{0}_{\text{A}_1}$} & \xyzerr & \targ{} & \xyzerr & \targ{} & \xyzerr & & & & & \xerr & \meter{} & \setwiretype{c} \\
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\lstick{$\ket{0}_{\text{A}_2}$} & \xyzerr & & & & & \targ{} & \xyzerr & \targ{} & \xyzerr & \xerr & \meter{} & \setwiretype{c}
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\end{quantikz}
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% tex-fmt: on
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\caption{Circuit-level noise.}
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\end{subfigure}%
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\hfill%
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\begin{subfigure}[c]{0.25\textwidth}
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\centering
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% tex-fmt: off
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\begin{quantikz}[row sep=4mm, column sep=2mm]
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\setwiretype{n} & \xerr & \gate[style={right, draw=none, xshift=-15mm}]{\text{X error}} \\
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\setwiretype{n} & \xyzerr & \gate[style={right, draw=none, xshift=-15mm}]{\text{X,Z, or Y error}} \\
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\setwiretype{n} & \gate{\phantom{1}}\wire[d][1]{q} & \gate[style={right, draw=none, xshift=-15mm},2]{\text{Correlated error}} \\
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\setwiretype{n} & \gate{\phantom{1}} &
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\end{quantikz}
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% tex-fmt: on
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\end{subfigure}
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\caption{Types of noise models.}
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\label{fig:Types of noise models.}
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\end{figure}
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\begin{figure}[t]
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\centering
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\begin{tikzpicture}
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\node[rectangle, draw, fill=orange!20, minimum
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height=2cm, minimum width=2.5cm, align=left] at (0,0)
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(internal) {Internal\\ Errors};
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\node[signal, draw, fill=orange!20, minimum height=2cm,
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minimum width=2.5cm, align=left, signal pointer angle=140]
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at (-2.45, 0) (input) {Input\\ Errors};
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\node at (1.97,0) {\huge =};
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\node[rectangle, draw, fill=orange!20, minimum height=2cm,
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minimum width=2.5cm, align=left] at (4,0) (output)
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{Output\\ Errors};
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\node[above] at (input.north) {\small Input State};
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\node[above] at (internal.north) {\small QEC};
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\node[above] at (output.north) {\small Output State};
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\end{tikzpicture}
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\caption{Sources of error in a fault-tolerant \ac{qec} system.}
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\end{figure}
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\begin{figure}[t]
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\centering
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\begin{minipage}{0.5\textwidth}
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\begin{align*}
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\bm{H} =
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\begin{pmatrix}
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1 & 1 & 0 \\
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0 & 1 & 1
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\end{pmatrix}
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\end{align*}
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\end{minipage}%
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\begin{minipage}{0.5\textwidth}
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% tex-fmt: off
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\begin{quantikz}%[row sep=4mm, column sep=4mm]
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\lstick[3]{$\ket{\psi}$} & \ctrl{3} & & & & & \\
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& & \ctrl{2} & \ctrl{3} & & & \\
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& & & & \ctrl{2} & & \\
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\lstick{$\ket{0}_{\text{A}_1}$} & \targ{} & \targ{} & & & \meter{} & \setwiretype{c} \\
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\lstick{$\ket{0}_{\text{A}_2}$} & & & \targ{} & \targ{} & \meter{} & \setwiretype{c}
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\end{quantikz}
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% tex-fmt: on
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\end{minipage}%
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\caption{
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Syndrome extraction circuit for the three-qubit repetition
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code under bit-flip noise.
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}
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\label{fig:}
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\end{figure}
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\begin{figure}[t]
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\centering
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\begin{minipage}{0.3\textwidth}
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\centering
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\begin{tikzpicture}
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\node{$%
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\bm{\Omega} =
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\begin{pmatrix}
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1 & 1 & 0 \\
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0 & 1 & 1 \\
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1 & 1 & 0 \\
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0 & 1 & 1 \\
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1 & 1 & 0 \\
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0 & 1 & 1
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\end{pmatrix}%
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$
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};
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\draw [
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line width=1pt,
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decorate,
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decoration={brace,mirror,amplitude=3mm,raise=5mm}
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]
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(1,0.55) -- (1,1.4)
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node[midway,right,xshift=10mm]{$\text{SE}_1$};
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\draw [
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line width=1pt,
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decorate,
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decoration={brace,mirror,amplitude=3mm,raise=5mm}
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]
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(1,-0.4) -- (1,0.45)
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node[midway,right,xshift=10mm]{$\text{SE}_2\hspace{2mm},$};
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\draw [
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line width=1pt,
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decorate,
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decoration={brace,mirror,amplitude=3mm,raise=5mm}
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]
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(1,-1.38) -- (1,-0.5)
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node[midway,right,xshift=10mm]{$\text{SE}_3$};
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\end{tikzpicture}
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\end{minipage}%
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\begin{minipage}{0.3\textwidth}
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\centering
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\vspace*{-6mm}
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\begin{gather*}
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\bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\}
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\end{gather*}
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\end{minipage}
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\newcommand{\preperr}[1]{
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\gate[style={fill=blue!20}]{\scriptstyle #1}
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}
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\vspace*{5mm}
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\begin{quantikz}[
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row sep=4mm, column sep=4mm,
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wire types={q,q,q,q,q,n,n,n,n},
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execute at end picture={
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\draw [
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line width=1pt,
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decorate,
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decoration={brace,amplitude=3mm,raise=9mm}
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]
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(\tikzcdmatrixname-4-19.north east)
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--
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(\tikzcdmatrixname-5-19.south east)
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node[midway,right,xshift=14mm]{$\text{SE}_1$};
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\draw [
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line width=1pt,
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decorate,
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decoration={brace,amplitude=3mm,raise=9mm}
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]
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(\tikzcdmatrixname-6-19.north east)
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--
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(\tikzcdmatrixname-7-19.south east)
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node[midway,right,xshift=14mm]{$\text{SE}_2$};
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\draw [
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line width=1pt,
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decorate,
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decoration={brace,amplitude=3mm,raise=9mm}
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]
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(\tikzcdmatrixname-8-19.north east)
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--
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(\tikzcdmatrixname-9-19.south east)
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node[midway,right,xshift=14mm]{$\text{SE}_3$};
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}
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]
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% tex-fmt: off
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\lstick[3]{$\ket{\psi}$} & \preperr{E_0} & \ctrl{3} & & & & & & \ctrl{5} & & & & & & \ctrl{7} & & & & & \\
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& \preperr{E_1} & & \ctrl{2} & \ctrl{3} & & & & & \ctrl{4} & \ctrl{5} & & & & & \ctrl{6} & \ctrl{7} & & & \\
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& \preperr{E_2} & & & & \ctrl{2} & & & & & & \ctrl{4} & & & & & & \ctrl{6} & & \\
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\lstick{$\ket{0}_{\text{A}_1}$} & & \targ{} & \targ{} & & & & & & & & & & & & & & & \meter{} & \setwiretype{c} \\
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\lstick{$\ket{0}_{\text{A}_2}$} & & & & \targ{} & \targ{} & & & & & & & & & & & & & \meter{} & \setwiretype{c} \\
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& & & & & & \lstick{$\ket{0}_{\text{A}_3}$} & \setwiretype{q} & \targ{} & \targ{} & & & & & & & & & \meter{} & \setwiretype{c} \\
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& & & & & & \lstick{$\ket{0}_{\text{A}_4}$} & \setwiretype{q} & & & \targ{} & \targ{} & & & & & & & \meter{} & \setwiretype{c} \\
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& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_5}$} & \setwiretype{q} & \targ{} & \targ{} & & & \meter{} & \setwiretype{c} \\
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& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_6}$} & \setwiretype{q} & & & \targ{} & \targ{} & \meter{} & \setwiretype{c}
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% tex-fmt: on
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\end{quantikz}
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\caption{
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Repeated syndrome extraction circuit for the three-qubit
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repetition code under bit flip noise.
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}
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\end{figure}
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\begin{figure}[t]
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\begin{gather*}
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\hspace*{-33.3mm}%
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\begin{array}{c}
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E_6 \\
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\downarrow
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\end{array}
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\end{gather*}
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\vspace*{-8mm}
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\begin{gather*}
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\bm{\Omega} =
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\left(
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\begin{array}{
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cccccc%
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>{\columncolor{red!20}}c%
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cccccccc
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}
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1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0
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& 0 & 0 & 0 & 0 & 0 \\
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0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0
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& 0 & 0 & 0 & 0 & 0 \\
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1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0
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& 0 & 0 & 0 & 0 & 0 \\
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0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1
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& 0 & 0 & 0 & 0 & 0 \\
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1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0
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& 1 & 1 & 0 & 1 & 0 \\
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0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0
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& 0 & 1 & 1 & 0 & 1
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\end{array}
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\right),
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\hspace*{7mm}
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\bm{s} \in \text{span} \mleft\{
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\bm{\Omega} \mright\}
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\end{gather*}
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\vspace*{5mm}
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\newcommand{\preperr}[1]{
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\gate[style={fill=blue!20}]{\scriptstyle #1}
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}
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\newcommand{\measerr}{\gate[style={fill=blue!20}]{\phantom{1}}}
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\newcommand{\noise}{
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\gate[style={noisy}]{\text{\small X}}%
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\setwiretype{n}%
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\wire[l][1]{q}
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}
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\newcommand{\redwire}[1]{
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\wire[r][#1][style={draw=red, line width=1.5pt}]{q}
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}
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\newcommand{\redtarg}{
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\targ[style={draw=red}]{}%
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\setwiretype{n}%
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\wire[l][1]{q}
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}
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\newcommand{\redctrl}[1]{
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\ctrl[style={draw=red,fill=red,line width=1.5pt}]{#1}
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}
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\newcommand{\redmeter}{\meter[style={draw=red,fill=red!20}]{}}
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\tikzset{
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noisy/.style={
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starburst,
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starburst point height=2.5mm,
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fill=red!25, draw=red!85!black,
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line width=1.5pt,
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inner xsep=-2pt, inner ysep=-2pt
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},
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}
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\centering
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% tex-fmt: off
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\begin{quantikz}[row sep=4mm, column sep=3mm, wire types={q,q,q,q,q,n,n,n,n}]
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\lstick[3]{$\ket{\psi}$} & \preperr{E_0} & \ctrl{3} & & & & \preperr{E_5} & & \ctrl{5} & & & & \preperr{E_{10}} & & \ctrl{7} & & & & & & \\
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& \preperr{E_1} & & \ctrl{2} & \ctrl{3} & & \noise\redwire{14} & & & \redctrl{4} & \redctrl{5} & & \preperr{E_{11}} & & & \redctrl{6} & \redctrl{7} & & & & \\
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& \preperr{E_2} & & & & \ctrl{2} & \preperr{E_7} & & & & & \ctrl{4} & \preperr{E_{12}} & & & & & \ctrl{6} & & & \\
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\lstick{$\ket{0}_{\text{A}_1}$} & & \targ{} & \targ{} & & & & & & & & & & & & & & & \preperr{E_3} & \meter{} & \setwiretype{c} \\
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\lstick{$\ket{0}_{\text{A}_2}$} & & & & \targ{} & \targ{} & & & & & & & & & & & & & \preperr{E_4} & \meter{} & \setwiretype{c} \\
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& & & & & & \lstick{$\ket{0}_{\text{A}_3}$} & \setwiretype{q} & \targ{} & \redtarg\redwire{10} & & & & & & & & & \preperr{E_8} & \redmeter\wire[r][1][style={draw=red,double, line width=1.5pt}]{q} & \setwiretype{n} \\
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& & & & & & \lstick{$\ket{0}_{\text{A}_4}$} & \setwiretype{q} & & & \redtarg\redwire{9} & \targ{} & & & & & & & \preperr{E_9} & \redmeter\wire[r][1][style={draw=red,double, line width=1.5pt}]{q} & \setwiretype{n} \\
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& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_5}$} & \setwiretype{q} & \targ{} & \redtarg\redwire{4} & & & \preperr{E_{13}} & \redmeter\wire[r][1][style={draw=red,double, line width=1.5pt}]{q} & \setwiretype{n} \\
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& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_6}$} & \setwiretype{q} & & & \redtarg\redwire{3} & \targ{} & \preperr{E_{14}} & \redmeter\wire[r][1][style={draw=red,double, line width=1.5pt}]{q} & \setwiretype{n}
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\end{quantikz}
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% tex-fmt: on
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|
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\caption{
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Repeated syndrome extraction circuit for the three-qubit
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repetition code under phenomenological noise.
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}
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\end{figure}
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\begin{figure}[t]
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\centering
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\hspace*{-5mm}
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\begin{minipage}{0.42\textwidth}
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\newcommand{\redwire}[1]{
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\wire[r][#1][style={draw=red, line width=1.5pt, double}]{q}
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}
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\newcommand{\inwire}{
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\wire[l][1][style={draw=red, line width=1.5pt}]{q}
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}
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\newcommand{\redtarg}{
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\targ[style={draw=red,line width=1.5pt}]{}%
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\setwiretype{n}%
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}
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\newcommand{\redctrl}[1]{
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\ctrl[style={draw=red,fill=red, line width=1.5pt}]{0}%
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\wire[d][#1][style={draw=red, line width=1.5pt, double}]{q}
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}
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\newcommand{\redmeter}{\meter[style={draw=red,fill=red!20}]{}}
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\newcommand{\redgate}[1]{\gate[style={draw=red,fill=red!20}]{\textcolor{red}{#1}}}
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% tex-fmt: off
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\begin{quantikz}[row sep=4mm, column sep=3mm, wire types={n,n,n,n,n,n}]
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& \meter{}\wire[l][1]{q}\wire[r][1]{c} & \setwiretype{c} & & & \ctrl[vertical wire=c]{2} & & \gate{D_1} \\
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& \meter{}\wire[l][1]{q}\wire[r][1]{c} & \setwiretype{c} & & & & \ctrl[vertical wire=c]{2} & \gate{D_2} \\
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& \redmeter{}\inwire\redwire{6} & & \redctrl{2} & & \targ{} & & \redgate{D_3} \\
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& \redmeter{}\inwire\redwire{6} & & & \redctrl{2} & & \targ{} & \redgate{D_4} \\
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& \redmeter{}\inwire\redwire{2} & & \redtarg\wire[r][4]{c} & & & & \gate{D_5} \\
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& \redmeter{}\inwire\redwire{3} & & & \redtarg\wire[r][3]{c} & & & \gate{D_6}
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\end{quantikz}
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% tex-fmt: on
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\end{minipage}%
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\begin{minipage}{0.56\textwidth}
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\newcommand\cc{\cellcolor{orange!20}}
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|
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\begin{align*}
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\bm{H} =
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% tex-fmt: off
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|
\left(\begin{array}{ccccccccccccccc}
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1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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\cc{0} & \cc{0} & \cc{0} & \cc{1} & \cc{0} & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
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\cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{1} & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
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\cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{1} & \cc{0} & 1 & 1 & 0 & 1 & 0 \\
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\cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{1} & 0 & 1 & 1 & 0 & 1
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\end{array}\right)
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% tex-fmt: on
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|
\end{align*}
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|
\end{minipage}
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|
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\caption{Construction of detectors from the measurements of a
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|
three-qubit repetition code.}
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|
\label{fig:Construction of the detectors from the measurements}
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|
\end{figure}
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|
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\begin{figure}[t]
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|
\centering
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|
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|
\tikzset{
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gate/.style={
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draw, %line width=1pt,
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|
minimum height=2cm,
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|
}
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|
}
|
|
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|
% tex-fmt: off
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|
\begin{quantikz}[row sep=2mm, column sep=4mm, wire types={q,q,q,n,n,n}]
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|
& \gate[3]{\text{SE}_1} & & \gate[3]{\text{SE}_2} & & \gate[3]{\text{SE}_3} & & \gate[3]{\text{SE}_4} & \\
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\lstick{$\ket{\psi}$} & & & & & & & & & \setwiretype{n} & \ldots \\
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& \wire[d][3]{c} & & \wire[d][1]{c} & & \wire[d][1]{c} & & \wire[d][1]{c} & \\
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|
& \ctrl[wire=c]{0}\wire[r][1]{c} & \wire[d][1]{c} & \ctrl[vertical wire=c]{1}\wire[r][1]{c} & \wire[d][1]{c} & \ctrl[vertical wire=c]{1}\wire[r][1]{c} & \wire[d][1]{c} & \ctrl[vertical wire=c]{1}\wire[r][1]{c} & \\
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& & \wire[r][1]{c} & \targ{}\wire[d][1]{c} & \wire[r][1]{c} & \targ{}\wire[d][1]{c} & \wire[r][1]{c} & \targ{}\wire[d][1]{c} & \\
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& \gate[1]{\bm{D}_1} & & \gate[1]{\bm{D}_2} & & \gate[1]{\bm{D}_3} & & \gate[1]{\bm{D}_4} & \\
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|
\end{quantikz}
|
|
% tex-fmt: on
|
|
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|
\caption{Construction of detectors from measurements in the general case.}
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|
\end{figure}
|