Move 3-qubit repetition code check matrix; Rewrite DEM intro

src/thesis/chapters/3_fault_tolerant_qec.tex

@@ -160,9 +160,22 @@ different error locations in the circuit.
 
 We will illustrate the most widely used types of error models on the
 example of the three-qubit repetition code for $X$ errors.
-This code has stabilizers $Z_1Z_2$ and $Z_2Z_3$.
+This is a code with check matrix
-\autoref{fig:pure_syndrome_extraction} shows the respective
+\begin{align*}
-check matrix and syndrome extraction circuit.
+    \bm{H} =
+    \left[
+        \begin{array}{ccc|ccc}
+            0 & 0 & 0 & 0 & 0 & 0 \\
+            0 & 0 & 0 & 0 & 0 & 0 \\
+            0 & 0 & 0 & 1 & 1 & 0 \\
+            0 & 0 & 0 & 0 & 1 & 1
+        \end{array}
+    \right]
+    .
+\end{align*}
+We can see that it has stabilizers $Z_1Z_2$ and $Z_2Z_3$.
+\autoref{fig:pure_syndrome_extraction} shows the corresponding
+syndrome extraction circuit.
 We refer to the qubits carrying the logical state
 $\ket{\psi}_\text{L}$ as \emph{data qubits}.
 Note that this is a concrete implementation using CNOT gates, as
@@ -247,30 +260,15 @@ error locations.
 \begin{figure}[t]
     \centering
 
-    \begin{minipage}{0.5\textwidth}
+    % tex-fmt: off
-        \begin{align*}
+    \begin{quantikz}%[row sep=4mm, column sep=4mm]
-            \bm{H} =
+        \lstick[3]{$\ket{\psi}_\text{L}$} & \ctrl{3} &          &          &          &           &                 \\
-            \left[
+                                          &          & \ctrl{2} & \ctrl{3} &          &           &                 \\
-                \begin{array}{ccc|ccc}
+                                          &          &          &          & \ctrl{2} &           &                 \\
-                    0 & 0 & 0 & 0 & 0 & 0 \\
+        \lstick{$\ket{0}_{\text{A}_1}$}   & \targ{}  & \targ{}  &          &          &  \meter{} & \setwiretype{c} \\
-                    0 & 0 & 0 & 0 & 0 & 0 \\
+        \lstick{$\ket{0}_{\text{A}_2}$}   &          &          & \targ{}  & \targ{}  &  \meter{} & \setwiretype{c}
-                    0 & 0 & 0 & 1 & 1 & 0 \\
+    \end{quantikz}
-                    0 & 0 & 0 & 0 & 1 & 1
+    % tex-fmt: on
-                \end{array}
-            \right]
-        \end{align*}
-    \end{minipage}%
-    \begin{minipage}{0.5\textwidth}
-        % tex-fmt: off
-        \begin{quantikz}%[row sep=4mm, column sep=4mm]
-            \lstick[3]{$\ket{\psi}_\text{L}$} & \ctrl{3} &          &          &          &           &                 \\
-                                              &          & \ctrl{2} & \ctrl{3} &          &           &                 \\
-                                              &          &          &          & \ctrl{2} &           &                 \\
-            \lstick{$\ket{0}_{\text{A}_1}$}   & \targ{}  & \targ{}  &          &          &  \meter{} & \setwiretype{c} \\
-            \lstick{$\ket{0}_{\text{A}_2}$}   &          &          & \targ{}  & \targ{}  &  \meter{} & \setwiretype{c}
-        \end{quantikz}
-        % tex-fmt: on
-    \end{minipage}%
 
     \caption{
         Syndrome extraction circuit for the three-qubit repetition
@@ -400,16 +398,29 @@ error locations.
 \section{Detector Error Models}
 \label{sec:Detector Error Models}
 
-\emph{Detector error models} constitue a standardized framework for
+\emph{Detector error models} (\acsp{dem}) constitue a standardized framework for
-passing information about the circuit used for \ac{qec} to a decoder.
+passing information about a circuit used for \ac{qec} to a decoder.
-They are also useful in the design of fault-tolerant \ldots such as
+They are also useful as a theoretical tool to aid in the design of
-fault-tolerant quantum computing schemes \cite[Sec.~1]{derks_designing_2025}.
+fault-tolerant \ac{qec} schemes.
-% While alternate ways of considering fault tolerance exist, detector
+E.g., they can be used to easily determine whether a measurement
-% error models
+schedule is fault-tolerant \cite[Example~12]{derks_designing_2025}.
-% benefit from the fact that
-\content{Benefits of this approach \cite[Sec.~4.2]{derks_designing_2025}}
 
-\content{Where they were introduced originally}
+Other approaches of implementing fault tolerance exist, such as
+flag error correction, which uses additional ancilla qubits to detect
+potentially damaging high-weight errors \cite[Sec.~1]{chamberland_flag_2018}.
+However, \acp{dem} offer some unique advantages
+\cite[Sec.~4.2]{derks_designing_2025}:
+\begin{itemize}
+    \item They distinguish between errors based on their effect on
+        the measurements, not based on their location in the circuit.
+        This allows for merging equivalent errors, which decreases
+        decoding complexity.
+    \item Errors on the data qubits and on the measurements are
+        treated in a unified manner. This leads to a more powerful
+        description of the overall circuit.
+\end{itemize}
+In this work, we only consider the process of decoding under the
+\ac{dem} framework.
 
 % Core idea