Wwrite a few paragraphs on the window generation/decoding

src/thesis/acronyms.tex

@@ -23,6 +23,11 @@
     long=belief propagation with guided decimation
 }
 
+\DeclareAcronym{gdg}{
+    short=GDG,
+    long=guided decimation guessing
+}
+
 \DeclareAcronym{nms}{
     short=NMS,
     long=normalized min-sum
@@ -122,3 +127,18 @@
     short=BB,
     long=bivariate bicycle
 }
+
+\DeclareAcronym{hgp}{
+    short=HGP,
+    long=hypergraph product
+}
+
+\DeclareAcronym{lp}{
+    short=LP,
+    long=lifted-product
+}
+
+\DeclareAcronym{bpc}{
+    short=BPC,
+    long=balanced product code
+}

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.
 
@@ -107,35 +107,10 @@ time dimension.
 Each of these windows is then decoded separately.
 
 %%%%%%%%%%%%%%%%
-\subsection{Existing Literature}
-\label{subsec:Existing Literature}
+\subsection{Review of Existing Literature}
+\label{subsec:Review of Existing Literature}
 
-% Review of existing literature
-
-Research on this topic has been ongoing for some time, though mostly
-for topological codes.
-The literature on \ac{qldpc} codes is more limited. Figure
-\Cref{fig:literature} gives an overview of the related body of work.
-
-\red{
-    \begin{itemize}
-        \item \cite{huang_increasing_2024} use BP+OSD,
-            \cite{gong_toward_2024} use BP+GDG
-        \item \cite{huang_improved_2023} use phenomenological noise,
-            \cite{gong_toward_2024} circuit-level noise
-        \item Go into the way the parallel decoding approaches
-            consolidate the overlap regions
-        \item \cite{huang_improved_2023} use hypegraph and lifted
-            product codes, \cite{gong_toward_2024} use BB codes
-        \item \cite{kuo_fault-tolerant_2024} use toric codes, the
-            rest of the topological papers surface codes
-        \item \cite{dennis_topological_2002} call their scheme ``overlap-add''
-        \item QUITS views sliding-window decoding more separately
-        \item Reasons for latency improvement ()
-    \end{itemize}
-}
-
-\begin{figure}[H]
+\begin{figure}[t]
     \centering
 
     \tikzset{
@@ -156,20 +131,22 @@ The literature on \ac{qldpc} codes is more limited. Figure
         }
     }
 
+    \tikzexternaldisable
     \begin{tikzpicture}[node distance = 0mm and 0mm]
         % tex-fmt: off
-        \node[heading, minimum width=15mm, fill=gray!25]              (code)  {Code};
-        \node[heading, below right=1mm and -5mm of code, fill=orange!20]  (top)   {Topological};
-        \node[heading, below right=42mm and -5mm of code, fill=orange!20] (qldpc) {QLDPC};
+        \node[heading, minimum width=15mm, fill=gray!25]                  (code)  {Code};
+        \node[heading, below right=2mm and -5mm of code, fill=orange!20]  (top)   {Topological};
+        \node[heading, below right=45mm and -5mm of code, fill=orange!20] (qldpc) {QLDPC};
 
-        \node[literature, below right=0mm and -12mm of top] (dennis) {\cite{dennis_topological_2002}};
+        \node[literature, below right=1mm and -12mm of top] (dennis) {\cite{dennis_topological_2002}};
         \node[literature, below=of dennis]                  (tan)    {\cite{tan_scalable_2023}};
         \node[literature, below=of tan]                     (skoric) {\cite{skoric_parallel_2023}};
         \node[literature, below=of skoric]                  (bombin) {\cite{bombin_modular_2023}};
         \node[literature, below=of bombin]                  (kuo)    {\cite{kuo_fault-tolerant_2024}};
 
-        \node[literature, below right=0mm and -12mm of qldpc] (huang) {\cite{huang_improved_2023},\cite{huang_increasing_2024}};
+        \node[literature, below right=1mm and -12mm of qldpc] (huang) {\cite{huang_improved_2023},\cite{huang_increasing_2024}};
         \node[literature, below=of huang]                     (gong)  {\cite{gong_toward_2024}};
+        \node[literature, below=of gong]                     (kang)   {\cite{kang_quits_2025}};
 
         \coordinate (code-anchor)  at ($(code.south) + (-2mm,0)$);
         \coordinate (top-anchor)   at ($(top.south) + (-5mm,0)$);
@@ -186,6 +163,7 @@ The literature on \ac{qldpc} codes is more limited. Figure
 
         \draw (qldpc-anchor) |- (huang);
         \draw (qldpc-anchor) |- (gong);
+        \draw (qldpc-anchor) |- (kang);
 
         \draw [
             line width=1pt,
@@ -208,15 +186,113 @@ The literature on \ac{qldpc} codes is more limited. Figure
             decorate,
             decoration={brace,amplitude=2mm,raise=5mm}
         ]
-        (huang.north east) -- (gong.south east)
+        (huang.north east) -- (kang.south east)
         node[midway,right,xshift=10mm]{Sequential};
         % tex-fmt: on
     \end{tikzpicture}
+    \tikzexternalenable
 
     \caption{Overview of literature on sliding-window decoding.}
     \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}
@@ -299,32 +375,63 @@ The literature on \ac{qldpc} codes is more limited. Figure
 %     \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
-    sliding-window part in a general fashion that is compatible with many
-    different decoder parts. Combine this with QUITS modular structure
-for sliding window decoding}
-
 %%%%%%%%%%%%%%%%
-\subsection{Implementation of Sliding-Window Decoding}
-\label{subsec:Implementation of Sliding-Window Decoding}
+\subsection{Algorithm}
+\label{subsec:Algorithm}
 
-We build on the approach taken by \cite{huang_increasing_2024} and
-\cite{gong_toward_2024}.
+In this section, we will examine the methodology by which a detector
+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}.
 
-% High-level overview of Sliding-Window decoding
+% Very high-level overview
 
-\content{Benefits of sliding-window decoding (lower latency due to
-earlier decoding start)}
-\content{Why it works (block diagonal structure $\rightarrow$ ``Done
-with processing'' some VNs)}
+Sliding-window decoding is made possible by the time-like structure
+of the syndrome extraction circuitry.
+This is epecially clearly visible under the \ac{dem} formalism, where
+this manifests as a block-diagonal structure of the detector
+error matrix $\bm{H}$.
+Note that this presupposes a choice of detectors as seen in
+\Cref{subsec:Detector Error Matrix}.
+This block-diagonal structure introduces some locality in the
+interdependence between \acp{vn} and \acp{cn}.
+For each local set of \acp{vn}, there is only a local set of connected \acp{cn}.
+We exploit this fact by cutting the matrix into overlapping windows.
+\Cref{fig:windowing_pcm} depicts this process.
 
-% Detailed explanation of sliding-window decoding
+% High-level overview
 
-\content{We look at rows not columns}
-\content{Define W}
-\content{Define F}
+How the locality is leveraged can be understood by considering the
+decoding process.
+After decoding a window, there is a subset of \acp{cn} that no longer
+contribute to the decoding process, as they do not share any \acp{vn}
+with the \acp{cn} of subsequent windows.\\
+\content{Commit VNs}
+\content{Benefit of this approach (as stated above: earlier decoding start)}
+
+% W and F
+
+There are two degrees of freedom in how we perform the windowing.
+The \emph{window size} $W \in \mathbb{N}$ represents the number of
+syndrome extraction rounds lumped into one window.
+The \emph{step size} $F \in \mathbb{N}$ represents the number of
+syndrome extraction rounds passed over before starting the next window.
+$W$ controls the size of the windows while $F$ controls the overlap
+between windows.
+
+% Why we look at rows, not columns
+
+As illustrated in \Cref{fig:windowing_pcm}, $W$ and $F$ control the
+window dimensions and locactions by defining the related \acp{cn},
+not the \acp{vn}.
+This is because while the number of overall \acp{cn} is only affected
+by the choice of the underlying code and the number of syndrome
+measurement rounds, the number of \acp{vn} depends on the noise model
+and is difficult to predict beforehand.
+
+% How we get the corresponding rows and columns
+
+\content{How we get the rows}
 \content{Explain how we get the columns once we know the rows}
 \content{\textbf{General note}: Mathematical definitions where possible}
 
@@ -337,13 +444,14 @@ with processing'' some VNs)}
 
 % Complete process
 
+\content{(?) Proper algorithm definition?}
 \content{1. Decode window}
 \content{2. Commit VN estimates}
 \content{3. Update syndrome}
 \content{4. Decode next window}
 \content{(?) Explicitly mention we don't reuse existing messages?}
 
-\begin{figure}[H]
+\begin{figure}[t]
     \centering
 
     \hspace*{-114mm}%
@@ -361,7 +469,7 @@ with processing'' some VNs)}
     \begin{tikzpicture}
         \draw[{Latex}-{Latex}, line width=.7pt] (0,   -0.75mm) -- (0,  5mm);
         \draw[line width=1pt]                   (-1mm,-0.75mm) --
-(3mm,-0.75mm);
+        (3mm,-0.75mm);
         \draw[line width=1pt]                   (-1mm,5mm)     -- (3mm,5mm);
         \node[left] at                          (-2mm,2.125mm) {$\sim W$};
 
@@ -379,6 +487,14 @@ with processing'' some VNs)}
     \label{fig:windowing_pcm}
 \end{figure}
 
+% TODO: Do I need this?
+% \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
+%     sliding-window part in a general fashion that is compatible with many
+%     different decoder parts. Combine this with QUITS modular structure
+% for sliding window decoding}
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Warm-Start Sliding-Window Decoding}
 \label{sec:warm_start_bp}
@@ -807,7 +923,7 @@ standard circuit-based depolarizing noise model, etc.)}
             {3/KITred/triangle*,4/KITblue/diamond*,5/KITorange/square*} {
                 \edef\temp{\noexpand
                     \addplot+[mark=\mark, solid, mark
-options={fill=\col}, \col]
+                    options={fill=\col}, \col]
                     table[
                         col sep=comma, x=physical_p,
                         y=LER_per_round,
@@ -888,7 +1004,7 @@ options={fill=\col}, \col]
             {3/KITred/triangle*,4/KITblue/diamond*,5/KITorange/square*} {
                 \edef\temp{\noexpand
                     \addplot+[mark=\mark, solid, mark
-options={fill=\col}, \col]
+                    options={fill=\col}, \col]
                     table[
                         col sep=comma, x=physical_p,
                         y=LER_per_round,
@@ -971,7 +1087,7 @@ options={fill=\col}, \col]
             {3/KITred/triangle*,2/KITblue/diamond*,1/KITorange/square*} {
                 \edef\temp{\noexpand
                     \addplot+[mark=\mark, solid, mark
-options={fill=\col}, \col]
+                    options={fill=\col}, \col]
                     table[
                         col sep=comma, x=physical_p,
                         y=LER_per_round,
@@ -1039,7 +1155,7 @@ options={fill=\col}, \col]
                 {3/KITred/triangle,4/KITblue/diamond,5/KITorange/square} {
                     \edef\temp{\noexpand
                         \addplot+[mark=\mark, densely dashed,
-forget plot, \col]
+                        forget plot, \col]
                         table[
                             col sep=comma, x=max_iter,
                             y=LER_per_round,
@@ -1110,7 +1226,7 @@ forget plot, \col]
                 {3/KITred/triangle,2/KITblue/diamond,1/KITorange/square} {
                     \edef\temp{\noexpand
                         \addplot+[mark=\mark, densely dashed,
-forget plot, \col]
+                        forget plot, \col]
                         table[
                             col sep=comma, x=max_iter,
                             y=LER_per_round,
@@ -1148,7 +1264,7 @@ forget plot, \col]
         under circuit-level noise.
         $12$ rounds of syndrome extraction were performed and
         standard circuit-based depolarizing noise was chosen as the
-noise model.
+        noise model.
         The physical error probabilty was fixed to $0.0025$.
     }
 \end{figure}
@@ -1307,7 +1423,7 @@ noise model.
         information.
         $12$ rounds of syndrome extraction were performed and
         standard circuit-based depolarizing noise was chosen as the
-noise model.
+        noise model.
     }
 \end{figure}
 
@@ -1353,7 +1469,7 @@ noise model.
                 {3/KITred/triangle,4/KITblue/diamond,5/KITorange/square} {
                     \edef\temp{\noexpand
                         \addplot+[mark=\mark, densely dashed,
-forget plot, \col]
+                        forget plot, \col]
                         table[
                             col sep=comma, x=max_iter,
                             y=LER_per_round,
@@ -1424,7 +1540,7 @@ forget plot, \col]
                 {3/KITred/triangle,2/KITblue/diamond,1/KITorange/square} {
                     \edef\temp{\noexpand
                         \addplot+[mark=\mark, densely dashed,
-forget plot, \col]
+                        forget plot, \col]
                         table[
                             col sep=comma, x=max_iter,
                             y=LER_per_round,
@@ -1469,7 +1585,7 @@ forget plot, \col]
         included only the messages on the Tanner graph.
         $12$ rounds of syndrome extraction were performed and
         standard circuit-based depolarizing noise was chosen as the
-noise model.
+        noise model.
         The physical error probabilty was fixed to $0.0025$.
     }
 \end{figure}
@@ -1624,7 +1740,7 @@ noise model.
         included only the messages on the Tanner graph.
         $12$ rounds of syndrome extraction were performed and
         standard circuit-based depolarizing noise was chosen as the
-noise model.
+        noise model.
     }
 \end{figure}
 
@@ -1670,7 +1786,7 @@ noise model.
                 {3/KITred/triangle,4/KITblue/diamond,5/KITorange/square} {
                     \edef\temp{\noexpand
                         \addplot+[mark=\mark, densely dashed,
-forget plot, \col]
+                        forget plot, \col]
                         table[
                             col sep=comma, x=max_iter,
                             y=LER_per_round,
@@ -1741,7 +1857,7 @@ forget plot, \col]
                 {3/KITred/triangle,2/KITblue/diamond,1/KITorange/square} {
                     \edef\temp{\noexpand
                         \addplot+[mark=\mark, densely dashed,
-forget plot, \col]
+                        forget plot, \col]
                         table[
                             col sep=comma, x=max_iter,
                             y=LER_per_round,
@@ -1786,9 +1902,8 @@ forget plot, \col]
         included only the messages on the Tanner graph.
         $12$ rounds of syndrome extraction were performed and
         standard circuit-based depolarizing noise was chosen as the
-noise model.
+        noise model.
         The physical error probabilty was fixed to $0.0025$.
     }
 \end{figure}
 
-

src/thesis/chapters/5_conclusion_and_outlook.tex

@@ -2,4 +2,6 @@
 
 \content{\textbf{Ideas for further research}}
 \content{Softer way of decimating VNs}
+\content{Systematic study on using different inner decoders (AED,
+SED, BPGD, ...)}