diff --git a/src/thesis/chapters/3_fault_tolerant_qec.tex b/src/thesis/chapters/3_fault_tolerant_qec.tex
index 95d6cd0..37d10a8 100644
--- a/src/thesis/chapters/3_fault_tolerant_qec.tex
+++ b/src/thesis/chapters/3_fault_tolerant_qec.tex
@@ -1,21 +1,144 @@
+% TODO: Make all [H] -> [t]
 \chapter{Fault Tolerant QEC}
-\section{Fault Tolerance}
+
+% Intro
+
+\content{Syndrome extraction circuitry itself introduces errors}
+\content{High level explanation of fault tolerance (with figure)}
+\content{Mathematical definition of fault tolerance}
+
+% Practical considerations
+
+\content{We generally need to perform multiple rounds of syndrome extraction}
+\content{The number of rounds of syndrome extraction is usually
+chosen equal to the $d_\text{min}$ of the code}
+\content{One-shot decoding property}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{tikzpicture}
+        \node[rectangle, draw, fill=orange!20, minimum
+        height=2cm, minimum width=2.5cm, align=left] at (0,0)
+        (internal) {Internal\\ Errors};
+
+        \node[signal, draw, fill=orange!20, minimum height=2cm,
+        minimum width=2.5cm, align=left, signal pointer angle=140]
+        at (-2.45, 0) (input) {Input\\ Errors};
+
+        \node at (1.97,0) {\huge =};
+
+        \node[rectangle, draw, fill=orange!20, minimum height=2cm,
+        minimum width=2.5cm, align=left] at (4,0) (output)
+        {Output\\ Errors};
+
+        \node[above] at (input.north) {\small Input State};
+        \node[above] at (internal.north) {\small QEC};
+        \node[above] at (output.north) {\small Output State};
+    \end{tikzpicture}
+
+    \caption{Sources of error in a fault-tolerant \ac{qec} system.}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Noise Models}
-\subsection{Depolarizing Channel}
-\subsection{Phenomenological Noise}
-\subsection{Circuit-Level Noise}
-\section{Detector Error Models}
-\subsection{Measurement Syndrome Matrix}
-\subsection{Detector Error Matrix}
-\subsection{Detector Error Models}
-\section{Practical Considerations}
-\subsection{Practical Methodology}
+\label{sec:Noise Models}
 
-\content{Per-round-LER explanation}
+% Intro
 
-\subsection{Stim}
+\content{Explanation of what a noise model is}
+\content{Mention there are different types of noise models, each with
+different possible error locations}
 
-\begin{figure}[t]
+% Figure intro
+
+\content{\autoref{fig:pure_syndrome_extraction} shows the syndrome
+    extraction circuit of a three-qubit repetition code with stabilizers
+    $Z_1Z_2$ and $Z_2Z_3$. This code is only able to deal with X errors.
+    We will use it as a propotypical model to examine the different types
+of noise models}
+\content{This is now a concrete implementation of the syndrome
+    measurement circuit using CNOT gates, as opposed to the system-level
+view in \autoref{subsec:Stabilizer Codes}}
+\content{\autoref{fig:noise_model_types} shows a number of diffent
+types of noise models}
+
+% Bit-flip noise
+
+\content{Bit-flip noise}
+\content{Introduce \emph{data qubits}}
+\content{Only X errors on data qubits}
+\content{Most similar to classical channel coding}
+\content{\textbf{TODO}: What is this useful for? Just as a first step?}
+
+% Depolarizing channel
+
+\content{Depolarizing channel}
+\content{X/Y/Z errors on data qubits}
+\content{\textbf{TODO}: What does this model? Memory experiment with
+ideal syndrome extraction?}
+\content{\textbf{TODO}: Why is it called depolarizing?}
+
+% Phenomenological noise
+
+\content{Phenomenological noise}
+\content{First noise model that considers errors during syndrome extraction}
+\content{X errors before each syndrome extraction round}
+\content{X errors before measurements}
+\content{\textbf{TODO}: Why is this useful? Derks et al. mentioned
+something about it being useful to derive fault-tolerant circuits}
+\content{\textbf{TODO}: Make sure phenomenological noise is only X errors}
+
+% Circuit-level noise
+
+\content{Circuit-level noise}
+\content{This is generally what we strive to be able to decode under}
+\content{X/Y/Z errors before each syndrome extraction round}
+\content{$n$-qubit Pauli errors after each $n$-qubit Pauli gate}
+\content{Define $n$-qubit Pauli errors}
+\content{X errors right before the measurements}
+\content{Note that the only errors right before the measurements that
+    have any effect on the measurement outcomes are X errors. That is why
+it is enough to consider this type of error at this point in the circuit.}
+
+% Different noise models for circuit-level noise
+
+\content{Comparison from Gidney's paper}
+\content{In this work we only consider standard circuit-based
+depolarizing noise}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{minipage}{0.5\textwidth}
+        \begin{align*}
+            \bm{H} =
+            \begin{pmatrix}
+                1 & 1 & 0 \\
+                0 & 1 & 1
+            \end{pmatrix}
+        \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
+        code under bit-flip noise.
+    }
+    \label{fig:pure_syndrome_extraction}
+\end{figure}
+
+\begin{figure}[H]
     \centering
 
     \newcommand{\xerr}{\gate[style={fill=KITblue!50}]{\phantom{1}}}
@@ -125,67 +248,60 @@
     \end{minipage}
 
     \caption{Types of noise models.}
-    \label{fig:Types of noise models.}
+    \label{fig:noise_model_types}
 \end{figure}
 
-\begin{figure}[t]
-    \centering
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Detector Error Models}
+\label{sec:Detector Error Models}
 
-    \begin{tikzpicture}
-        \node[rectangle, draw, fill=orange!20, minimum
-        height=2cm, minimum width=2.5cm, align=left] at (0,0)
-        (internal) {Internal\\ Errors};
+\content{\textbf{TODO}: Look up how Derks et al. introduce DEMs}
 
-        \node[signal, draw, fill=orange!20, minimum height=2cm,
-        minimum width=2.5cm, align=left, signal pointer angle=140]
-        at (-2.45, 0) (input) {Input\\ Errors};
+% Different ways of implementing fault tolerance
 
-        \node at (1.97,0) {\huge =};
+\content{Ways of implementing fault tolerance different from DEMs}
 
-        \node[rectangle, draw, fill=orange!20, minimum height=2cm,
-        minimum width=2.5cm, align=left] at (4,0) (output)
-        {Output\\ Errors};
+% Core idea
 
-        \node[above] at (input.north) {\small Input State};
-        \node[above] at (internal.north) {\small QEC};
-        \node[above] at (output.north) {\small Output State};
-    \end{tikzpicture}
+\content{Model additional error locations in the code}
+\content{Construct ``circuit code'' from original code}
 
-    \caption{Sources of error in a fault-tolerant \ac{qec} system.}
-\end{figure}
+% Benefits
 
-\begin{figure}[t]
-    \centering
+\content{Benefits of this approach}
 
-    \begin{minipage}{0.5\textwidth}
-        \begin{align*}
-            \bm{H} =
-            \begin{pmatrix}
-                1 & 1 & 0 \\
-                0 & 1 & 1
-            \end{pmatrix}
-        \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}%
+%%%%%%%%%%%%%%%%
+\subsection{Measurement Syndrome Matrix}
+\label{subsec:Measurement Syndrome Matrix}
 
-    \caption{
-        Syndrome extraction circuit for the three-qubit repetition
-        code under bit-flip noise.
-    }
-    \label{fig:}
-\end{figure}
+% Core idea
 
-\begin{figure}[t]
+\content{Core idea: Matrix describes parity checks \\
+    $\rightarrow$ A column shows which parity checks the
+    corresponding VN contributes to \\
+    $\rightarrow$ View columns as syndromes corresponding to error
+    locations in the circuit
+}
+
+% Multiple rounds of syndrome extraction
+
+% TODO: First introduce syndrome measurement matrix, mathematically
+% (consult Derks et al.'s paper). Then use the three-qubit repetition
+% code as an example only
+\autoref{fig:rep_code_multiple_rounds_bit_flip} shows a circuit
+performing three rounds of syndrome extraction for the three-qubit
+repetition code introduced earlier.
+We are only considering bit-flip noise at this point.
+For each syndrome extraction round, we get an additional set of
+syndrome measurements.
+We combine these measurements by stacking them in a new vector $\bm{s}
+\in \mathbb{F}_2^{n_\text{rounds}\cdot(n-k)}$.
+To model this behavior mathematically, we append additional rows to
+the check matrix.
+We call this matrix the \emph{measurement syndrome matrix}
+$\bm{\Omega}$.
+
+\begin{figure}[H]
     \centering
 
     \begin{minipage}{0.3\textwidth}
@@ -293,9 +409,10 @@
         Repeated syndrome extraction circuit for the three-qubit
         repetition code under bit flip noise.
     }
+    \label{fig:rep_code_multiple_rounds_bit_flip}
 \end{figure}
 
-\begin{figure}[t]
+\begin{figure}[H]
     \begin{gather*}
         \hspace*{-33.3mm}%
         \begin{array}{c}
@@ -388,7 +505,52 @@
     }
 \end{figure}
 
-\begin{figure}[t]
+%%%%%%%%%%%%%%%%
+\subsection{Detector Error Matrix}
+\label{subsec:Detector Error Matrix}
+
+% Core idea
+
+% TODO: Make this a proper definition?
+Instead of using the measurements as parity indicators directly, we
+may wish to combine them in some way.
+We call such combinations \emph{detectors}.
+Formally, a detector is a parity constraint on a set of measurement
+outcomes \cite[Def.~2.1]{derks_designing_2025}.
+
+\content{Detector matrix}
+
+\content{Detector error matrix}
+
+\content{One way of defining the detectors is ...}
+
+\begin{figure}[H]
+    \centering
+
+    \tikzset{
+        gate/.style={
+            draw, %line width=1pt,
+            minimum height=2cm,
+        }
+    }
+
+    % tex-fmt: off
+    \begin{quantikz}[row sep=2mm, column sep=4mm, wire types={q,q,q,n,n,n}]
+        \lstick[3]{$\ket{\psi}_\text{L}$} & \gate[3]{\text{SE}_1}          &                & \gate[3]{\text{SE}_2}                   &                & \gate[3]{\text{SE}_3}                   &                & \gate[3]{\text{SE}_4}                   &                            \\
+                                          &                                &                &                                         &                &                                         &                &                                         & & \setwiretype{n} & \ldots \\
+                                          & \wire[d][3]{c}                 &                & \wire[d][1]{c}                          &                & \wire[d][1]{c}                          &                & \wire[d][1]{c}                          &                            \\
+                                          & \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} &                            \\
+                                          &                                & \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}                   &                            \\
+                                          & \gate[1]{\bm{D}_1}             &                & \gate[1]{\bm{D}_2}                      &                & \gate[1]{\bm{D}_3}                      &                & \gate[1]{\bm{D}_4}                      &                            \\
+    \end{quantikz}
+    % tex-fmt: on
+
+    \caption{Construction of detectors from measurements in the general case.}
+\end{figure}
+
+\content{The three-qubit repetition code as an exmaple}
+
+\begin{figure}[H]
     \centering
 
     \hspace*{-5mm}
@@ -444,27 +606,38 @@
     \label{fig:Construction of the detectors from the measurements}
 \end{figure}
 
-\begin{figure}[t]
-    \centering
+%%%%%%%%%%%%%%%%
+\subsection{Detector Error Models}
+\label{subsec:Detector Error Models}
 
-    \tikzset{
-        gate/.style={
-            draw, %line width=1pt,
-            minimum height=2cm,
-        }
-    }
+\content{Combination of detector error matrix and noise model}
+\content{Contains all information necessary for decoding
+\cite[Intro.]{derks_designing_2025}}
+\content{Not only useful for decoding, but also for ... (Derks et al.)}
 
-    % tex-fmt: off
-    \begin{quantikz}[row sep=2mm, column sep=4mm, wire types={q,q,q,n,n,n}]
-        \lstick[3]{$\ket{\psi}_\text{L}$} & \gate[3]{\text{SE}_1}          &                & \gate[3]{\text{SE}_2}                   &                & \gate[3]{\text{SE}_3}                   &                & \gate[3]{\text{SE}_4}                   &                            \\
-                                          &                                &                &                                         &                &                                         &                &                                         & & \setwiretype{n} & \ldots \\
-                                          & \wire[d][3]{c}                 &                & \wire[d][1]{c}                          &                & \wire[d][1]{c}                          &                & \wire[d][1]{c}                          &                            \\
-                                          & \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} &                            \\
-                                          &                                & \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}                   &                            \\
-                                          & \gate[1]{\bm{D}_1}             &                & \gate[1]{\bm{D}_2}                      &                & \gate[1]{\bm{D}_3}                      &                & \gate[1]{\bm{D}_4}                      &                            \\
-    \end{quantikz}
-    % tex-fmt: on
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Practical Considerations}
+\label{sec:Practical Considerations}
 
-    \caption{Construction of detectors from measurements in the general case.}
-\end{figure}
+% Intro
+
+\content{Intro}
+
+%%%%%%%%%%%%%%%%
+\subsection{Practical Methodology}
+\label{subsec:Practical Methodology}
+
+\content{Per-round-LER explanation}
+
+%%%%%%%%%%%%%%%%
+\subsection{Stim}
+\label{subsec:Stim}
+
+\content{Circuit code heavily depends on the exact circuit construction}
+\content{Not easy to predict how errors at different locations
+propagate through the circuit an what detectors they affect}
+
+\content{Stim is a software package that generates DEMs from circuits}
+\content{The user still has to define the circuit themselves, and
+especially the detectors \cite[Sec~2.5]{derks_designing_2025}}
 
diff --git a/src/thesis/chapters/4_decoding_under_dems.tex b/src/thesis/chapters/4_decoding_under_dems.tex
index ad28b8e..acb8edb 100644
--- a/src/thesis/chapters/4_decoding_under_dems.tex
+++ b/src/thesis/chapters/4_decoding_under_dems.tex
@@ -1,3 +1,4 @@
+% TODO: Make all [H] -> [t]
 \chapter{Decoding under Detector Error Models}
 
 % Intro