Write first couple of pages of chapter 3

src/thesis/bibliography.bib

@@ -1058,3 +1058,15 @@
 	month = dec,
 	year = {2023},
 }
+
+@article{gidney_fault-tolerant_2021,
+	title = {A {Fault}-{Tolerant} {Honeycomb} {Memory}},
+	volume = {5},
+	issn = {2521-327X},
+	doi = {10.22331/q-2021-12-20-605},
+	journal = {Quantum},
+	author = {Gidney, Craig and Newman, Michael and Fowler, Austin and Broughton, Michael},
+	month = dec,
+	year = {2021},
+	pages = {605},
+}

src/thesis/chapters/3_fault_tolerant_qec.tex

@@ -3,41 +3,121 @@
 
 % Intro
 
-\content{Syndrome extraction circuitry itself introduces errors}
-\content{High level explanation of fault tolerance (with figure)}
-\content{Mathematical definition of fault tolerance}
+An important challenge of \ac{qec} that was recognized early on is
+the fact that the error correction machinery itself may introduce new
+errors \cite[Sec.~III]{shor_scheme_1995}.
+Specifically for stabilizer codes, errors may happen during the
+syndrome extraction process, since it is implemented in quantum hardware itself.
+We call the errors the \ac{qec} procedure is supposed to correct
+\emph{input errors} and the errors introduced by the procedure itself
+\emph{internal errors}.
+In order to be \emph{fault-tolerant}, the procedure must be able to
+address both types of errors.
+
+% Definition of fault tolerance
+
+% TODO: Different variable name for N?
+We model the possible occurrence of errors during any processing
+stage as different \emph{error locations} $E_i,~i\in \{1,\ldots,N\}$
+in the circuit.
+$N \in \mathbb{N}$ is the total number of error locations.
+The \emph{circuit error vector} $\bm{e} \in \{0,1\}^N$ is a vector
+indicating which errors occurred, with
+\begin{align*}
+    e_i :=
+    \begin{cases}
+        1, & \text{Error $E_i$ occurred} \\
+        0, & \text{otherwise}
+    \end{cases}
+    .%
+\end{align*}
+\autoref{fig:fault_tolerance_overview} illustrates the flow of errors.
+Specifically for \ac{css} codes, a \ac{qec} procedure is deemed
+\emph{fault-tolerant}, if \cite[Def.~4.2]{derks_designing_2025}
+\begin{gather*}
+    \lVert \bm{e}_{\text{output},X} \rVert \le t \hspace{5mm}
+    \forall\,  \bm{e}_\text{input}, \bm{e}_\text{internal} \in \{0,1\}^N:
+    \lVert \bm{e}_\text{input} \rVert + \lVert \bm{e}_\text{internal}
+    \rVert \le t \\[1mm]
+    \text{and} \\[1mm]
+    \lVert \bm{e}_{\text{output},Z} \rVert \le t \hspace{5mm}
+    \forall\,  \bm{e}_\text{input}, \bm{e}_\text{internal} \in \{0,1\}^N:
+    \lVert \bm{e}_\text{input} \rVert + \lVert \bm{e}_\text{internal}
+    \rVert \le t
+    ,%
+\end{gather*}
+where $t = \lfloor (d_\text{min} -1)/2 \rfloor$ is the number of
+errors the code is able to correct.
+The vectors $\bm{e}_{\text{output},X}$ and $\bm{e}_{\text{output},Z}$
+denote only $X$ and $Z$ errors respectively.
+
+In order to deal with errors during the syndrome extraction that flip
+single syndrome bits, multiple rounds of syndrome measurements must
+be performed.
+Typically, the number of syndrome extraction rounds is chosen as $d_\text{min}$.
+
+% % This is the definition of a fault-tolerant QEC gadget
+% A \ac{qec} procedure is deemed fault tolerant if
+% \cite[Def.~5]{gottesman_introduction_2009}
+% \begin{align*}
+%     % tex-fmt: off
+%     \text{A)}
+%     % tex-fmt: on
+%     \hspace{5mm} & \lVert \bm{e}_\text{output} \rVert
+%     \le \lVert \bm{e}_\text{internal} \rVert
+%     \hspace{5mm} \forall\,
+%     \bm{e}_\text{input}, \bm{e}_\text{internal} \in \{0,1\}^N :
+%     \lVert \bm{e}_\text{internal} \rVert \le t \\
+%     % tex-fmt: off
+%     \text{B)}
+%     % tex-fmt: on
+%     \hspace{5mm} & \lVert \bm{e}_\text{output} \rVert = 0
+%     \hspace{19.3mm} \forall\,
+%     \bm{e}_\text{input}, \bm{e}_\text{internal} \in \{0,1\}^N :
+%     \lVert \bm{e}_\text{input} \rVert + \lVert \bm{e}_\text{internal}
+%     \rVert \le t
+%     ,
+% \end{align*}
+% where $t = \lfloor (d_\text{min} -1)/2 \rfloor$ is the number of
+% errors the original code is able to correct.
+% Condition A limits the spread of input errors during the error
+% correction process.
+% Condition B means that as long as there are few enough internal and
+% input errors, the scheme should be able to correct all of them.
 
 % 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}
+% TODO: Are the fault-tolerant QEC procedures where we don't perform
+% multiple measurement rounds?
 
-\begin{figure}[H]
+\begin{figure}[t]
     \centering
 
     \begin{tikzpicture}
         \node[rectangle, draw, fill=orange!20, minimum
-        height=2cm, minimum width=2.5cm, align=left] at (0,0)
-        (internal) {Internal\\ Errors};
+        height=2cm, minimum width=2.5cm, align=center] at (0,0)
+        (internal) {Internal Errors\\ $\bm{e}_\text{internal}$};
 
         \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};
+        minimum width=2.5cm, align=center, signal pointer angle=140]
+        at (-2.8, 0) (input) {Input Errors \\ $\bm{e}_\text{input}$};
 
-        \node at (1.97,0) {\huge =};
+        \node at (1.99,0) {\huge =};
 
         \node[rectangle, draw, fill=orange!20, minimum height=2cm,
-        minimum width=2.5cm, align=left] at (4,0) (output)
-        {Output\\ Errors};
+        minimum width=2.5cm, align=center] at (4,0) (output)
+        {Output Errors\\ $\bm{e}_\text{output}$};
 
         \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.}
+    \caption{
+        Sources of error in a fault-tolerant \ac{qec} system.
+        Adapted from \cite[Figure~2]{derks_designing_2025}.
+    }
+    \label{fig:fault_tolerance_overview}
 \end{figure}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -46,77 +126,112 @@ chosen equal to the $d_\text{min}$ of the code}
 
 % Intro
 
-\content{Explanation of what a noise model is}
-\content{Mention there are different types of noise models, each with
-different possible error locations}
+We collect the probabilities of error at each location in the
+\emph{noise model}, a vector $\bm{p} \in [0,1]^N$, where $N \in
+\mathbb{N}$ is the number of possible error locations.
+There are different types of noise models, each allowing for
+different error locations in the circuit.
 
 % 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}
+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 $Z_2Z_3$.
+\autoref{fig:pure_syndrome_extraction} shows the respective
+check matrix and 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
+opposed to the system-level view introduced in
+\autoref{subsec:Stabilizer Codes}.
+We visualize the different types of noise models in
+\autoref{fig:noise_model_types}.
 
 % 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?}
+The simplest type of noise model is \emph{bit-flip} noise.
+This corresponds to the classical \ac{bsc}, i.e., only $X$ errors on the
+data qubits are possible \cite[Appendix~A]{gidney_new_2023}.
+Note that we cannot use bit-flip noise to develop fault-tolerant
+systems, as it doesnt't account for errors during the syndrome extraction.
+This type of noise model is shown in \autoref{subfig:bit_flip}.
 
 % 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?}
+Extending bit-flip noise to consider $X,Z$ or $Y$ instead of just $X$
+errors, we obtain the \emph{depolarizing channel}
+\cite[Sec.~7.6]{gottesman_stabilizer_1997}, depicted in
+\autoref{subfig:depolarizing}.
+It is well-suited for modeling memory experiments, where data qubits
+are stored idly for some period of time and errors accumulate due to
+decoherence.
+Bit-flip noise and the depolarizing channel are sometimes referred to
+as \emph{code capacity noise models}.
 
 % 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}
+The \emph{phenomenological noise model} is the first type of noise model we
+examine that accounts for faults during the syndrome extraction.
+Here, we consider multiple rounds of syndrome measurements with a
+depolarizing channel before each round.
+Additionally, we allow for measurement errors by having $X$ error
+locations right before each measurement \cite[Appendix~A]{gidney_new_2023}.
+Note that it is enough to only consider $X$ errors at this point,
+since that is the only type of error directly affecting the
+measurement outcomes.
+While not fully capturing all possible error mechanisms,
+phenomenological noise is already expressive enough to be used for
+the design of fault-tolerant circuitry \cite[Sec.~4.2]{derks_designing_2025}.
+This model is depicted in \autoref{subfig:phenomenological}.
 
 % 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.}
+The most general type of noise model is \emph{circuit-level noise}.
+Here we not only consider noise inbetween syndrome extraction rounds
+and at the measurements, but at each gate.
+Specifically, we allow arbitrary $n$-qubit Pauli errors after each
+$n$-qubit gate \cite[Def.~2.5]{derks_designing_2025}.
+An $n$-qubit Pauli error is simply a series of correlated Pauli
+errors on each related individual qubit.
+While phenomenological noise is useful for some design aspects of
+fault tolerant circuitry, for simulations, circuit-level noise should
+always be used \cite[Sec.~4.2]{derks_designing_2025}.
+This type of noise model is shown in \autoref{subfig:circuit_level}.
 
-% Different noise models for circuit-level noise
+% Practical simulation aspects
 
-\content{Comparison from Gidney's paper}
-\content{In this work we only consider standard circuit-based
-depolarizing noise}
+While these types of noise models give us some constraints on the
+types and locations of errors, the question of how exactly to choose
+their probabilities remains.
+When performing simulations, we typically sweep a range of values of
+the \emph{physical error rate} $p$, and there are different ways of
+obtaining the noise model $\bm{p}$ from this physical error rate.
+For the depolarizing channel in particular, we usually choose the
+same error rate $p/3$ for $X$, $Z$, and $Y$ errors.
+For circuit-level noise, various options exist, such as the \emph{SI1000}
+(superconducting inspired) or the \emph{EM3} (entangling
+measurements) models \cite[Sec.~2.1]{gidney_fault-tolerant_2021}.
+These differ in the way they compute individual error probabilities
+from the physical error rate.
+In this work we only consider \emph{standard circuit-based depolarizing
+noise}, as this is the standard approach in the literature.
+We thus set the error probabilities of all error locations in the
+circuit-level noise model to the same value, the physical error rate $p$.
 
-\begin{figure}[H]
+\begin{figure}[t]
     \centering
 
     \begin{minipage}{0.5\textwidth}
         \begin{align*}
             \bm{H} =
-            \begin{pmatrix}
-                1 & 1 & 0 \\
-                0 & 1 & 1
-            \end{pmatrix}
+            \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*}
     \end{minipage}%
     \begin{minipage}{0.5\textwidth}
@@ -138,7 +253,7 @@ depolarizing noise}
     \label{fig:pure_syndrome_extraction}
 \end{figure}
 
-\begin{figure}[H]
+\begin{figure}[t]
     \centering
 
     \newcommand{\xerr}{\gate[style={fill=KITblue!50}]{\phantom{1}}}
@@ -181,6 +296,7 @@ depolarizing noise}
             % tex-fmt: on
 
             \subcaption{Bit-flip noise.}
+            \label{subfig:bit_flip}
         \end{minipage}
 
         \vspace*{5mm}
@@ -198,6 +314,7 @@ depolarizing noise}
             % tex-fmt: on
 
             \subcaption{Depolarizing channel.}
+            \label{subfig:depolarizing}
         \end{minipage}
 
         \vspace*{5mm}
@@ -206,15 +323,16 @@ depolarizing noise}
             \centering
             % tex-fmt: off
             \begin{quantikz}[row sep=4mm, column sep=4mm]
-                \lstick[3]{$\ket{\psi}_\text{L}$} & \xerr & \ctrl{3} &           &          &          &       &           & \xerr            & & \setwiretype{n} &                                       \\
-                                                  & \xerr &          & \ctrl{2}  & \ctrl{3} &          &       &           & \xerr            & & \setwiretype{n} & \gate[style={left,draw=none}]{\cdots} \\
-                                                  & \xerr &          &           &          & \ctrl{2} &       &           & \xerr            & & \setwiretype{n} &                                       \\
-                \lstick{$\ket{0}_{\text{A}_1}$}   &       & \targ{}  & \targ{}   &          &          & \xerr &  \meter{} & \setwiretype{c}                                                              \\
-                \lstick{$\ket{0}_{\text{A}_2}$}   &       &          &           & \targ{}  & \targ{}  & \xerr &  \meter{} & \setwiretype{c}
+                \lstick[3]{$\ket{\psi}_\text{L}$} & \xyzerr & \ctrl{3} &           &          &          &       &           & \xyzerr        & & \setwiretype{n} &                                       \\
+                                                  & \xyzerr &          & \ctrl{2}  & \ctrl{3} &          &       &           & \xyzerr         & & \setwiretype{n} & \gate[style={left,draw=none}]{\cdots} \\
+                                                  & \xyzerr &          &           &          & \ctrl{2} &       &           & \xyzerr         & & \setwiretype{n} &                                       \\
+                \lstick{$\ket{0}_{\text{A}_1}$}   &         & \targ{}  & \targ{}   &          &          & \xerr &  \meter{} & \setwiretype{c}                                                              \\
+                \lstick{$\ket{0}_{\text{A}_2}$}   &         &          &           & \targ{}  & \targ{}  & \xerr &  \meter{} & \setwiretype{c}
             \end{quantikz}
             % tex-fmt: on
 
             \subcaption{Phenomenological noise.}
+            \label{subfig:phenomenological}
         \end{minipage}
 
         \vspace*{5mm}
@@ -232,6 +350,7 @@ depolarizing noise}
             % tex-fmt: on
 
             \subcaption{Circuit-level noise.}
+            \label{subfig:circuit_level}
         \end{minipage}
     \end{minipage}%
     \hfill%
@@ -241,7 +360,7 @@ depolarizing noise}
         \begin{quantikz}[row sep=4mm, column sep=2mm]
             \setwiretype{n} & \xerr                            & \gate[style={right, draw=none, xshift=-15mm}]{\text{X error}}            \\
             \setwiretype{n} & \xyzerr                          & \gate[style={right, draw=none, xshift=-15mm}]{\text{X,Z, or Y error}}    \\
-            \setwiretype{n} & \gate{\phantom{1}}\wire[d][1]{q} & \gate[style={right, draw=none, xshift=-15mm},2]{\text{Correlated error}} \\
+            \setwiretype{n} & \gate{\phantom{1}}\wire[d][1]{q} & \gate[style={right, draw=none, xshift=-15mm},2]{\text{Two-qubit error}} \\
             \setwiretype{n} & \gate{\phantom{1}}               &
         \end{quantikz}
         % tex-fmt: on
@@ -255,53 +374,81 @@ depolarizing noise}
 \section{Detector Error Models}
 \label{sec:Detector Error Models}
 
-\content{\textbf{TODO}: Look up how Derks et al. introduce DEMs}
-
-% Different ways of implementing fault tolerance
-
-\content{Ways of implementing fault tolerance different from DEMs}
+\emph{Detector error models} constitue a standardized framework for
+passing information about the circuit used for \ac{qec} to a decoder.
+They are also useful in the design of fault-tolerant \ldots such as
+fault-tolerant quantum computing schemes \cite[Sec.~1]{derks_designing_2025}.
+% While alternate ways of considering fault tolerance exist, detector
+% error models
+% benefit from the fact that
+\content{Benefits of this approach \cite[Sec.~4.2]{derks_designing_2025}}
 
 % Core idea
 
-\content{Model additional error locations in the code}
-\content{Construct ``circuit code'' from original code}
-
-% Benefits
-
-\content{Benefits of this approach}
+The goal we strive to achieve is to consider the possible error
+locations in the syndrome extraction circuitry during decoding.
+The core idea behind detector error models is to do this by defining
+a new \emph{circuit code} that describes the circuit.
+Each \ac{vn} of this new code corresponds to an error location in the
+circuit and each corresponds to a \ac{cn} to a syndrome measurement.
+This circuit code, combined with the prior probabilities of error
+given by the noise model, incorporates all information necessary for decoding.
 
 %%%%%%%%%%%%%%%%
 \subsection{Measurement Syndrome Matrix}
 \label{subsec:Measurement Syndrome Matrix}
 
-% Core idea
+% Mathematical definition
 
-\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
-}
+We describe the circuit code using the \emph{measurement syndrome
+matrix} matrix $\bm{\Omega} \in \mathbb{F}_2^{m\times N}$, with
+\begin{align*}
+    \Omega_{j,i} =
+    \begin{cases}
+        1, & \text{Error $i$ flips measurement $j$}\\
+        0, & \text{otherwise}
+    \end{cases}
+    .%
+\end{align*}
+This matrix thus defines the code based on which error mechanism
+flips which measurement, rather than the Pauli type and location of
+each error \cite[Sec.~1.4.3]{higgott_practical_2024}.
+To obtain $\bm{\Omega}$, we must propagate Pauli errors through the
+circuit, tracking which measurements they affect
+\cite[Sec.~2.4]{derks_designing_2025}.
 
-% Multiple rounds of syndrome extraction
+% Example
 
-% 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
+% TODO: Fix syndrome dimension notation
+We turn to our example of the three-qubit repetition code to
+illustrate the construction of the syndrome measurement matrix.
+We begin by replicating the syndrome extraction circuitry, three
+times in this case, as can be seen in
+\autoref{fig:rep_code_multiple_rounds_bit_flip}.
+We consider only bit flip noise at this stage.
+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}
-$\bm{\Omega}$.
+To accomodate the additional syndrome bits, we extend the
+matrix $\bm{\Omega}$ representing the circuit by replicating the rows as well.
+Recall that $\bm{\Omega}$ describes which \ac{vn} is connected to
+which parity check and the syndrome indicates which parity checks
+are violated.
+This means that if an error exists at only a single \ac{vn}, we can
+read off the syndrome in the corresponding column.
 
-\begin{figure}[H]
+% Expand to phenomenological
+
+We now whish to expand the error model to phenomenological noise, though
+only considering $X$ errors in this case.
+We introduce new error locations at the respective positions,
+arriving at the circuit depicted in
+\autoref{fig:rep_code_multiple_rounds_phenomenological}.
+For each additional error location, we extend $\bm{\Omega}$ by
+appending the corresponding syndrome vector as a column.
+
+\begin{figure}[t]
     \centering
 
     \begin{minipage}{0.3\textwidth}
@@ -412,7 +559,7 @@ $\bm{\Omega}$.
     \label{fig:rep_code_multiple_rounds_bit_flip}
 \end{figure}
 
-\begin{figure}[H]
+\begin{figure}[t]
     \begin{gather*}
         \hspace*{-33.3mm}%
         \begin{array}{c}
@@ -503,6 +650,7 @@ $\bm{\Omega}$.
         Repeated syndrome extraction circuit for the three-qubit
         repetition code under phenomenological noise.
     }
+    \label{fig:rep_code_multiple_rounds_phenomenological}
 \end{figure}
 
 %%%%%%%%%%%%%%%%
@@ -517,14 +665,86 @@ 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}.
+Chaning the perspective in this way does not change the theoretical
+error correcting capabilities of the circuit, but it may change the
+decoding performance when using a practical decoder.
+\red{[Possibly a few more words on this (maybe a mathematical
+proof/intuition?)]}
 
-\content{Detector matrix}
+% The detector matrix
 
-\content{Detector error matrix}
+% TODO: Fix the notation mess
+We describe the relationship between measurements and detectors using
+the \emph{detector matrix} $\bm{D} \in \mathbb{F}_2^{d\times m}$
+\cite[Def.~2.2]{derks_designing_2025}.
+Similar to the way a \ac{pcm} connects bits with parity checks, the
+detector matrix links measurements and detectors.
+Each column corresponds to a measurement, while the rows correspond
+to the detectors.
+We should note at this point that the combination of measurements
+into detectors has no bearing on the actual construction of the
+syndrome extraction circuitry.
+It is something that happens ``virtually'' after the fact and only
+affects the decoder.
 
-\content{One way of defining the detectors is ...}
+% The detector error matrix
 
-\begin{figure}[H]
+We now know how the errors at different locations in the circuit
+affect the measurements ($\bm{\Omega}$), and we know how the
+measurements relate to the detectors ($\bm{D}$).
+For decoding, we are interested in the effect of the errors on the
+detectors directly.
+We thus construct the \emph{detector error matrix} $\bm{H} \in
+\mathbb{F}_2^{d\times N}$ \cite[Def.~2.9]{derks_designing_2025} as
+\begin{align*}
+    \bm{H} := \bm{D}\bm{\Omega}
+    .%
+\end{align*}
+Note that, in particular when $d=m$, this is equivalent to performing row
+additions on the matrix $\bm{\Omega}$.
+
+% How to choose the detectors
+
+% TODO: Give results from current and previous stage mathematical names
+We still have a degree of freedom in how we choose the detectors.
+\ldots
+There is, however, one way of defining the detectors that will prove useful
+at a later stage.
+To the measurement results from each syndrome extraction round, we
+can add the results from the previous round, as illustrated in
+\autoref{fig:detectors_from_measurements_general}.
+\red{[Mathematical notation for measurement combination]}
+
+We again turn our attention to the three-qubit repetition code.
+In \autoref{fig:rep_code_multiple_rounds_phenomenological} we can see
+that $E_6$ has occurred and has subsequently tripped the last four measurements.
+We now take those measurements and combine them according to
+\red{[Reference mathematical notation above]}.
+We can see this process graphically in
+\autoref{fig:detectors_from_measurements_rep_code}
+To understand why this way of defining the detectors is useful, we
+note that the error $E_6$ in
+\autoref{fig:rep_code_multiple_rounds_phenomenological} has not only
+tripped the measurements in the syndrome extraction round immediately
+afterwards, but all subsequent ones as well.
+To only see errors in the rounds immediately following them, we
+consider our newly defined detectors instead of the measurements,
+that effectively compute the difference between the measurements.
+
+Each error can only trip syndrome bits that follow it.
+We can see this in the triangular structure of $\bm{\Omega}$ in
+\autoref{fig:rep_code_multiple_rounds_phenomenological}.
+Combining the measurements into detectors according to
+\red{[Reference mathematical notation above]}, we are performing row
+additions in such a way as to clear the bottom left of the matrix.
+This yields a block-diagonal structure for the detector error matrix
+$\bm{H}$, as in the example in
+\autoref{fig:detectors_from_measurements_rep_code}.
+Note that we exploit the fact that each syndrome measurement round is
+identical to obtain this structure.
+
+% TODO: Change notation (\bm{D})
+\begin{figure}[t]
     \centering
 
     \tikzset{
@@ -546,11 +766,10 @@ outcomes \cite[Def.~2.1]{derks_designing_2025}.
     % tex-fmt: on
 
     \caption{Construction of detectors from measurements in the general case.}
+    \label{fig:detectors_from_measurements_general}
 \end{figure}
 
-\content{The three-qubit repetition code as an exmaple}
-
-\begin{figure}[H]
+\begin{figure}[t]
     \centering
 
     \hspace*{-5mm}
@@ -603,7 +822,7 @@ outcomes \cite[Def.~2.1]{derks_designing_2025}.
 
     \caption{Construction of detectors from the measurements of a
     three-qubit repetition code.}
-    \label{fig:Construction of the detectors from the measurements}
+    \label{fig:detectors_from_measurements_rep_code}
 \end{figure}
 
 %%%%%%%%%%%%%%%%

src/thesis/chapters/4_decoding_under_dems.tex

@@ -13,6 +13,7 @@
 
 \content{Callback to previous chapter}
 \content{(Maybe even historical) overview of the literature}
+\content{Better yet: A proper (at least as proper as possible) review}
 
 % High-level overview of Sliding-Window decoding