Polish per-round logical error rate subsection

src/thesis/chapters/3_fault_tolerant_qec.tex

@@ -1059,7 +1059,7 @@ 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
+The overall probability of error is then
 \begin{align}
     \hspace{-12mm}
     p_\text{total} &= 1 - (1 - p_\text{round})^{n_\text{rounds}} \nonumber\\
@@ -1075,15 +1075,17 @@ 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{bausch_learning_2024}\cite{beni_tesseract_2025} is to assume an
+exponential decay for the decoder's \emph{logical fidelity}
 \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
+The logical fidelity is a measure of the quality of a logical state
+\cite[Appendix~E]{postler_demonstration_2024}.
+As it is related to the error rate through $F = 1 - 2p$, we obtain
+\cite[Eq.~4]{bausch_learning_2024}
 \begin{align}
     (1 - 2p_\text{total}) &= (1 - 2p_\text{round})^{n_\text{rounds}} \nonumber\\
     \implies \hspace{15mm} p_\text{total} &= \frac{1}{2}
@@ -1091,7 +1093,9 @@ As the fidelity is related to the error rate through $F = 1 - 2p$, we obtain
     .%
 \end{align}
 
-\content{We choose the first approach}
+We have chosen to use the first approach, i.e.,
+\autoref{eq:per_round_ler}, as the related literature is closer in
+topic to our own work.
 
 %%%%%%%%%%%%%%%%
 \subsection{Stim}
@@ -1108,6 +1112,8 @@ propagate through the circuit an what detectors they affect}
 
 \content{Merging of error mechanisms}
 
+\content{Stim is were DEMs were first introduced}
+
 \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}}