Rewrite DEM subsection; Write first draft of practical considerations

src/thesis/acronyms.tex

@@ -8,6 +8,11 @@
     long=detector error model
 }
 
+\DeclareAcronym{ler}{
+    short=LER,
+    long=logical error rate
+}
+
 \DeclareAcronym{bp}{
     short=BP,
     long=belief propagation

src/thesis/bibliography.bib

@@ -1082,3 +1082,19 @@
 	year = {2018},
 	pages = {53},
 }
+
+@article{chen_exponential_2021,
+	title = {Exponential suppression of bit or phase errors with cyclic error correction},
+	volume = {595},
+	copyright = {2021 The Author(s)},
+	issn = {1476-4687},
+	doi = {10.1038/s41586-021-03588-y},
+	language = {en},
+	number = {7867},
+	journal = {Nature},
+	publisher = {Nature Publishing Group},
+	author = {Chen, Zijun and Satzinger, Kevin J. and Atalaya, Juan and Korotkov, Alexander N. and Dunsworth, Andrew and Sank, Daniel and Quintana, Chris and McEwen, Matt and Barends, Rami and Klimov, Paul V. and Hong, Sabrina and Jones, Cody and Petukhov, Andre and Kafri, Dvir and Demura, Sean and Burkett, Brian and Gidney, Craig and Fowler, Austin G. and Paler, Alexandru and Putterman, Harald and Aleiner, Igor and Arute, Frank and Arya, Kunal and Babbush, Ryan and Bardin, Joseph C. and Bengtsson, Andreas and Bourassa, Alexandre and Broughton, Michael and Buckley, Bob B. and Buell, David A. and Bushnell, Nicholas and Chiaro, Benjamin and Collins, Roberto and Courtney, William and Derk, Alan R. and Eppens, Daniel and Erickson, Catherine and Farhi, Edward and Foxen, Brooks and Giustina, Marissa and Greene, Ami and Gross, Jonathan A. and Harrigan, Matthew P. and Harrington, Sean D. and Hilton, Jeremy and Ho, Alan and Huang, Trent and Huggins, William J. and Ioffe, L. B. and Isakov, Sergei V. and Jeffrey, Evan and Jiang, Zhang and Kechedzhi, Kostyantyn and Kim, Seon and Kitaev, Alexei and Kostritsa, Fedor and Landhuis, David and Laptev, Pavel and Lucero, Erik and Martin, Orion and McClean, Jarrod R. and McCourt, Trevor and Mi, Xiao and Miao, Kevin C. and Mohseni, Masoud and Montazeri, Shirin and Mruczkiewicz, Wojciech and Mutus, Josh and Naaman, Ofer and Neeley, Matthew and Neill, Charles and Newman, Michael and Niu, Murphy Yuezhen and O’Brien, Thomas E. and Opremcak, Alex and Ostby, Eric and Pató, Bálint and Redd, Nicholas and Roushan, Pedram and Rubin, Nicholas C. and Shvarts, Vladimir and Strain, Doug and Szalay, Marco and Trevithick, Matthew D. and Villalonga, Benjamin and White, Theodore and Yao, Z. Jamie and Yeh, Ping and Yoo, Juhwan and Zalcman, Adam and Neven, Hartmut and Boixo, Sergio and Smelyanskiy, Vadim and Chen, Yu and Megrant, Anthony and Kelly, Julian and {Google Quantum AI}},
+	month = jul,
+	year = {2021},
+	pages = {383--387},
+}

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} \\
+                        && \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{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{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}\}
+    \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 $\bm{p}$.
+error matrix $\bm{H}$ and the noise model $\bm{p}$.
-
+\cite[Sec.~6]{derks_designing_2025}.
-\content{Combination of detector error matrix and noise model}
+It serves as an abstract representation of a circuit and can be used
-\content{Contains all information necessary for decoding
+both to transfer information to a decoder but also to aid in the
-\cite[Intro.]{derks_designing_2025}}
+design of fault-tolerant systems.
-\content{Not only useful for decoding, but also for ... (Derks et al.)}
+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}
+\subsection{Per-Round Logical Error Rate}
-\label{subsec:Practical Methodology}
+\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}}