Rewrite DEM subsection; Write first draft of practical considerations

src/thesis/chapters/3_fault_tolerant_qec.tex

@@ -838,10 +838,10 @@ violate the same set of detectors, i.e.,
 \begin{align*}
     \hspace{-15mm}
     % tex-fmt: off
-                       && \bm{H} \bm{e}_1^\text{T} & \neq \bm{H} \bm{e}_2^\text{T} \\
-   \iff \hspace{-33mm} && \bm{H} \left( \bm{e}_1 - \bm{e}_2 \right)^\text{T} & \neq 0 \\
-   \iff \hspace{-33mm} && \bm{D} \bm{\Omega} \left( \bm{e}_1 - \bm{e}_2 \right)^\text{T} & \neq 0 \\
-   \iff \hspace{-33mm} && \bm{\Omega} \left( \bm{e}_1 - \bm{e}_2 \right)^\text{T} & \notin \text{kern} \{\bm{D}\}
+                        && \bm{H} \bm{e}_1^\text{T} & \neq \bm{H} \bm{e}_2^\text{T} \\
+    \iff \hspace{-33mm} && \bm{H} \left( \bm{e}_1 - \bm{e}_2 \right)^\text{T} & \neq 0 \\
+    \iff \hspace{-33mm} && \bm{D} \bm{\Omega} \left( \bm{e}_1 - \bm{e}_2 \right)^\text{T} & \neq 0 \\
+    \iff \hspace{-33mm} && \bm{\Omega} \left( \bm{e}_1 - \bm{e}_2 \right)^\text{T} & \notin \text{kern} \{\bm{D}\}
     % tex-fmt: on
     .%
 \end{align*}
@@ -998,18 +998,27 @@ identical to obtain this structure.
 \label{subsec:Detector Error Models}
 
 A \emph{detector error model} is the combination of the detector
-error matric $\bm{H}$ and the noise model $\bm{p}$.
-
-\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.)}
+error matrix $\bm{H}$ and the noise model $\bm{p}$.
+\cite[Sec.~6]{derks_designing_2025}.
+It serves as an abstract representation of a circuit and can be used
+both to transfer information to a decoder but also to aid in the
+design of fault-tolerant systems.
+E.g., it can be used to investigate the properties of a circuit with
+respect to fault tolerance.
+It contains all information necessary for the decoding process.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Practical Considerations}
 \label{sec:Practical Considerations}
 
-% Practical simulation aspects
+The previous sections give \red{[theoretical overview of noise models
+and DEMs]}.
+In order to apply them successfully in practice, we must consider a
+few further aspects.
+
+%%%%%%%%%%%%%%%%
+\subsection{Choice of Noise Model}
+\label{subsec:Choice of Noise Model}
 
 While these types of noise models give us some constraints on the
 types and locations of errors, the question of how exactly to choose
@@ -1031,19 +1040,74 @@ We thus set the error probabilities of all error locations in the
 circuit-level noise model to the same value, the physical error rate $p$.
 
 %%%%%%%%%%%%%%%%
-\subsection{Practical Methodology}
-\label{subsec:Practical Methodology}
+\subsection{Per-Round Logical Error Rate}
+\label{subsec:Per-Round Logical Error Rate}
 
-\content{Per-round-LER explanation}
+% Per-round LER
+
+\content{Introduce logical error rate}
+
+% TODO: Introduce the logical error rate
+Another aspect that is important to consider is the meaning of the
+logical error rate in the context of a \ac{qec} system with multiple
+rounds of syndrome measurements.
+In order to facilitate the comparability of results obtained from
+simulations with different numbers of syndrome extraction rounds, we
+use the \emph{per-round-\ac{ler}}.
+
+The simplest way of calculating the per-round \ac{ler} is by modeling
+each round as an independent experiment.
+For each experiment, an error might occur with a certain probability
+$p_\text{round}$.
+The overall probability of error is thus
+\begin{align}
+    \hspace{-12mm}
+    p_\text{total} &= 1 - (1 - p_\text{round})^{n_\text{rounds}} \nonumber\\
+    \label{eq:per_round_ler}
+    \implies \hspace{3mm} p_\text{round} &=
+    1 - (1 - p_\text{total})^{1 / n_\text{rounds}}
+    .%
+    \hspace{12mm}
+\end{align}
+We approximate $p_\text{total}$ using a Monte Carlo simulation and
+compute the per-round-\ac{ler} using \autoref{eq:per_round_ler}.
+This is a common approach taken in the literature
+\cite{gong_toward_2024}\cite{wang_fully_2025}.
+
+Another common approach \cite{chen_exponential_2021}%
+\cite{bausch_learning_2024}\cite{maan_decoding_2025}\cite{cao_exact_2025}%
+\cite{beni_tesseract_2025} is to assume a exponential decay for the
+decoder's \emph{fidelity} \red{explain what this is}
+\cite[Eq.~2]{bausch_learning_2024}
+\begin{align*}
+    F_\text{total} = (F_\text{round})^{n_\text{rounds}}
+    .%
+\end{align*}
+As the fidelity is related to the error rate through $F = 1 - 2p$, we obtain
+\begin{align}
+    (1 - 2p_\text{total}) &= (1 - 2p_\text{round})^{n_\text{rounds}} \nonumber\\
+    \implies \hspace{15mm} p_\text{total} &= \frac{1}{2}
+    \left[ 1 - (1 - 2p_\text{round})^{1/n_\text{rounds}} \right]
+    .%
+\end{align}
+
+\content{We choose the first approach}
 
 %%%%%%%%%%%%%%%%
 \subsection{Stim}
 \label{subsec:Stim}
 
+As we noted in \autoref{subsec:Measurement Syndrome Matrix}, to
+obtain a measurement syndrome matrix we must propagate Pauli frames
+through the circuit.
+\red{[This is where stim comes into play]}
+
 \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{Merging of error mechanisms}
+
 \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}}