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src/thesis/chapters/4_decoding_under_dems.tex

@@ -72,7 +72,7 @@ problem into many smaller ones that can be solved more efficiently.
 % warm-start decoding. Or just go into warm-start decoding)
 We will start by briefly reviewing the existing work related to
 sliding-window decoding,
-before focusing on one specific incarnation.
+before focusing on one specific realization.
 We will then introduce a modification to the existing algorithm and
 perform numerical simulations to evaluate it.
 
@@ -110,100 +110,6 @@ Each of these windows is then decoded separately.
 \subsection{Review of Existing Literature}
 \label{subsec:Review of Existing Literature}
 
-% Description of the figure
-
-\Cref{fig:literature} gives an overview over the existing body of work
-related to sliding-window decoding.
-The papers \cite{huang_improved_2023} and \cite{huang_increasing_2024} are
-lumped together, as they share the same content;
-one is simply preprint published earlier.
-We will only refer to \cite{huang_increasing_2024} in the following.
-\cite{kang_quits_2025} is somewhat special in that the authors focus
-more on the introduction of a new simluator framework they call
-QUITS, rather than the performance of sliding-window decoding itself.
-\cite{gong_toward_2024} and \cite{kang_quits_2025} have made their
-software freely available online%
-\footnote{
-    https://github.com/mkangquantum/quits
-}%
-\footnote{
-    https://github.com/gongaa/SlidingWindowDecoder
-}.
-A final thing to note is that \cite{dennis_topological_2002} never
-explicitly mention sliding windows, they call their scheme
-``overlapping recovery''.
-
-% Topological vs QLDPC
-
-Research has focused on two categories of \ac{qec} codes, topological
-and \ac{qldpc} codes.
-Most of the work on topological codes has treated surface codes,
-with the exception of \cite{kuo_fault-tolerant_2024} where toric
-codes were considered.
-With regard to \ac{qldpc} codes, in \cite{huang_increasing_2024}
-they examine \emph{hypergraph product} (\acs{hgp}) and
-\emph{lifted-product} (\acs{lp}) codes.
-HGP codes are constructed from the product of two classical codes,
-while LP codes generalize this construction by additionally applying
-a lift to reduce the qubit overhead.
-In \cite{kang_quits_2025}, \emph{balanced product codes} (\acs{bpc})
-are additionally considered.
-Finally, in \cite{gong_toward_2024} the authors explore \ac{bb} codes.
-
-% Sequential vs parallel
-
-After having divided the whole circuit into separate windows, the question
-arises of how exactly to realize the decoding.
-There are two main approaches, with differing mechanisms of reducing
-the latency.
-Some papers decode the sliding windows in a parallel fashion.
-The benefit in this case is the option to more effectively utilize
-classical hardware for decoding.
-Others choose a sequential approach.
-Here, decoding can start earlier, as there is no need to wait for the
-syndrome measurements of all windows before beginning with the decoding.
-With the exception of \cite{dennis_topological_2002}, literature
-treating topological codes has mostly focused on parallel decoding
-while literature treating \ac{qldpc} codes has wholely considered
-sequential decoding.
-
-% Deep-dive into QLDPC methods
-
-\renewcommand{\arraystretch}{1.1}
-\setlength{\tabcolsep}{12pt}
-\begin{table}[t]
-    \centering
-    \caption{Experimental conditions for papers related to \ac{qldpc} codes.}
-    \vspace*{3mm}
-    \label{table:experimental_conditions}
-    \begin{tabular}{l|ccc}
-        % tex-fmt: off
-        Publication                                                            &    Code                       & Noise Model                     & Decoder \\ \hline
-        \hspace{-2.5mm}\cite{huang_improved_2023},\cite{huang_increasing_2024} & \acs{hgp}, \acs{lp}           & Phenomenological noise & \acs{bp} + \acs{osd} \\
-        \hspace{-2.5mm}\cite{gong_toward_2024}                                 & \acs{bb}                      & Circuit-level noise             & \acs{bp} + \acs{gdg} \\
-        \hspace{-2.5mm}\cite{kang_quits_2025}                                  & \acs{hgp}, \acs{lp}, \acs{bpc}  & Circuit-level noise             & \acs{bp} + \ac{osd}
-        % tex-fmt: on
-    \end{tabular}
-\end{table}
-
-For this work, the publications treating \ac{qldpc} codes are
-especially interesting.
-The experimental conditions for these are summarized in
-\Cref{table:experimental_conditions}.
-As we noted above, \ac{hgp} and \ac{lp} codes are considered in
-\cite{huang_increasing_2024},
-\ac{hgp}, \ac{lp} and \ac{bpc} codes are considered in \cite{kang_quits_2025},
-and \ac{bb} codes are considered in \cite{gong_toward_2024}.
-The employed noise models also differ;
-\cite{huang_increasing_2024} use phenomenological noise, while
-\cite{gong_toward_2024} and \cite{kang_quits_2025} use circuit-level noise.
-Finally, \cite{gong_toward_2024} introduce their own variation of
-\ac{bpgd}, \ac{bp} with \ac{gdg}, while \cite{huang_increasing_2024}
-and \cite{kang_quits_2025} use \ac{bp} + \ac{osd}.
-We would additionally like to note that only in
-\cite{gong_toward_2024} and \cite{kang_quits_2025} do the authors
-explicitly work with the \ac{dem} formalism.
-
 \begin{figure}[t]
     \centering
 
@@ -290,6 +196,103 @@ explicitly work with the \ac{dem} formalism.
     \label{fig:literature}
 \end{figure}
 
+% Some general notes
+
+\Cref{fig:literature} gives an overview over the existing body of work
+related to sliding-window decoding.
+The papers \cite{huang_improved_2023} and \cite{huang_increasing_2024} are
+lumped together, as they share the same content;
+one is simply preprint published earlier.
+We will only refer to \cite{huang_increasing_2024} in the following.
+\cite{kang_quits_2025} is somewhat special in that the authors focus
+more on the introduction of a new simluator framework they call
+QUITS, rather than the performance of sliding-window decoding itself.
+\cite{gong_toward_2024} and \cite{kang_quits_2025} have made their
+software freely available online%
+\footnote{
+    https://github.com/mkangquantum/quits
+}%
+\footnote{
+    https://github.com/gongaa/SlidingWindowDecoder
+}.
+A final thing to note is that \cite{dennis_topological_2002} never
+explicitly mention sliding windows, they call their scheme
+``overlapping recovery''.
+
+% Topological vs QLDPC
+
+Research has focused on two categories of \ac{qec} codes, topological
+and \ac{qldpc} codes.
+Most of the work on topological codes has treated surface codes,
+with the exception of \cite{kuo_fault-tolerant_2024} where toric
+codes were considered.
+With regard to \ac{qldpc} codes, in \cite{huang_increasing_2024}
+they examine \emph{hypergraph product} (\acs{hgp}) and
+\emph{lifted-product} (\acs{lp}) codes.
+HGP codes are constructed from the product of two classical codes,
+while LP codes generalize this construction by additionally applying
+a lift to reduce the qubit overhead.
+In \cite{kang_quits_2025}, \emph{balanced product codes} (\acs{bpc})
+are additionally considered.
+Like HGP codes, BPC codes are derived from a product construction,
+but exploit an additional symmetry to yield fewer physical qubits for
+the same code parameters.
+Finally, in \cite{gong_toward_2024} the authors explore \ac{bb} codes.
+
+% Sequential vs parallel
+
+After having divided the whole circuit into separate windows, the question
+arises of how exactly to realize the decoding.
+There are two main approaches, with differing mechanisms of reducing
+the latency.
+Some papers decode the sliding windows in a parallel fashion.
+The benefit in this case is the option to more effectively utilize
+classical hardware for decoding.
+Others choose a sequential approach.
+Here, decoding can start earlier, as there is no need to wait for the
+syndrome measurements of all windows before beginning with the decoding.
+With the exception of \cite{dennis_topological_2002}, literature
+treating topological codes has mostly focused on parallel decoding
+while literature treating \ac{qldpc} codes has wholely considered
+sequential decoding.
+
+% Deep-dive into QLDPC methods
+
+For this work, the publications treating \ac{qldpc} codes are
+especially interesting.
+The experimental conditions for these are summarized in
+\Cref{table:experimental_conditions}.
+As we noted above, \ac{hgp} and \ac{lp} codes are considered in
+\cite{huang_increasing_2024},
+\ac{hgp}, \ac{lp} and \ac{bpc} codes are considered in \cite{kang_quits_2025},
+and \ac{bb} codes are considered in \cite{gong_toward_2024}.
+The employed noise models also differ;
+\cite{huang_increasing_2024} use phenomenological noise, while
+\cite{gong_toward_2024} and \cite{kang_quits_2025} use circuit-level noise.
+Finally, \cite{gong_toward_2024} introduce their own variation of
+\ac{bpgd}, \ac{bp} with \ac{gdg}, while \cite{huang_increasing_2024}
+and \cite{kang_quits_2025} use \ac{bp} + \ac{osd}.
+We would additionally like to note that only in
+\cite{gong_toward_2024} and \cite{kang_quits_2025} do the authors
+explicitly work with the \ac{dem} formalism.
+
+\renewcommand{\arraystretch}{1.1}
+\setlength{\tabcolsep}{12pt}
+\begin{table}[t]
+    \centering
+    \caption{Experimental conditions for papers related to \ac{qldpc} codes.}
+    \vspace*{3mm}
+    \label{table:experimental_conditions}
+    \begin{tabular}{l|ccc}
+        % tex-fmt: off
+        Publication                                                            &    Code                       & Noise Model                     & Decoder \\ \hline
+        \hspace{-2.5mm}\cite{huang_improved_2023},\cite{huang_increasing_2024} & \acs{hgp}, \acs{lp}           & Phenomenological noise & \acs{bp} + \acs{osd} \\
+        \hspace{-2.5mm}\cite{gong_toward_2024}                                 & \acs{bb}                      & Circuit-level noise             & \acs{bp} + \acs{gdg} \\
+        \hspace{-2.5mm}\cite{kang_quits_2025}                                  & \acs{hgp}, \acs{lp}, \acs{bpc}  & Circuit-level noise             & \acs{bp} + \ac{osd}
+        % tex-fmt: on
+    \end{tabular}
+\end{table}
+
 % \red{
 %     Existing work
 %     \begin{itemize}
@@ -381,12 +384,6 @@ error matrix is divided into overlapping windows.
 The algorithm detailed here follows \cite{kang_quits_2025}, whose
 work is in turn based on \cite{huang_increasing_2024}.
 
-\red{
-    \begin{itemize}
-        \item QUITS views sliding-window decoding more separately
-    \end{itemize}
-}
-
 \content{Possibly go into the fact that current sliding-window
     approaches don't differentiate clearly between the sliding-window
     part and the decoder part. This work aims to extend the
@@ -394,9 +391,6 @@ work is in turn based on \cite{huang_increasing_2024}.
     different decoder parts. Combine this with QUITS modular structure
 for sliding window decoding}
 
-We build on the approach taken by \cite{huang_increasing_2024} and
-\cite{gong_toward_2024}.
-
 % High-level overview of Sliding-Window decoding
 
 \content{Benefits of sliding-window decoding (lower latency due to
@@ -427,7 +421,7 @@ with processing'' some VNs)}
 \content{4. Decode next window}
 \content{(?) Explicitly mention we don't reuse existing messages?}
 
-\begin{figure}[H]
+\begin{figure}[t]
     \centering
 
     \hspace*{-114mm}%