Complete text for first two figures

src/thesis/chapters/4_decoding_under_dems.tex

@@ -1201,82 +1201,23 @@ generated by simulating at least $200$ logical error events.
 \label{subsec:Belief Propagation}
 
 % Local experimental setup
-%   - BP variant
 
 We began our investigation by using \ac{bp} with no further
 modifications as the inner decoder.
 We chose the min-sum variant of \ac{bp} due to its low computational complexity.
 
 % [Thread] Get impression for max gain
-%   - More global = better -> Compare windowed vs. whole
-
-% [Description] Figure 4.8
-%   - Parameters
-%     - # BP iterations
-%     - W,F
-%     - Physical error rates
-%     - Windowed (cold start) vs whole decoding
-%     - (?) Semilog y axis
-%   - Figure description
-%     - TODO:
-
-% [Interpretation] Figure 4.8
-%   - Larger window -> better, because more global decoding
-%   - Diminishing returns as the window becomes larger
-%   - As expected, whole works best
-
-% [Thread] First comparison with warm start
-%    - Compare performance of warm start to cold start
-
-% [Description] Figure 4.9
-%   - Parameters
-%     - # BP iterations
-%     - W,F
-%     - Physical error rates
-%     - Warm vs cold start
-%   - Figure description
-%     - TODO:
-
-% [Interpretation] Figure 4.9
-%   - Generally better performance with warm start, as expected
-%   - It is surprising that warm start performs better than whole
-
-% [Thread] Warm start is better than whole due to more effective iterations
-
-% [Description] Figure 4.10
-%   - Parameters
-%     - # BP iterations
-%     - W,F
-%     - Physical error rates
-%     - Warm vs cold start
-%   - Figure description
-%     - TODO:
-
-% [Interpretation] Figure 4.10
-%   -
 
 We initially wanted to gain an impression for the performance gain we could
 expect from a modification to the sliding-window decoding procedure.
 To this end, we began by analyzing the decoding performance of the
 original process, without our warm-start modification.
 We will call this \emph{cold-start} decoding in the following.
-We examined the decoding performance for different window sizes $W$
+Because we expected more global decoding to work better (the inner
-and compared it against the performance when decoding on the whole
+    decoder then has access to a larger portion of the long-range
-detector error matrix at once, i.e., without windowing.
+    correlations encoded in the detector error matrix before any commit
-
+is made) we initially decided to use decoding on the whole detector
-\Cref{fig:whole_vs_cold} depicts the results of this analysis.
+error matrix as a proxy for the attainable decoding performance.
-\red{[Write more about the experimental setup (200 BP iterations,
-fixed step size, what else?)]}
-\red{[Describe the plot (whole decoding in black, (?) list different
-window sizes and colors/markers, what else?)]}
-We can see that a larger window results in a lower overall error rate.
-This seems sensible, because the lower the window size, the more
-locally the decoding is performed.
-While this allows us to leverage the time-like structure of the
-circuitry more strongly and further reduce the latency, it is
-expected to lower the performance, since \red{[find something to say here]}.
-Decoding the whole detector error matrix globally with no windowing
-provides the best performance.
 
 \begin{figure}[t]
     \centering
@@ -1329,33 +1270,64 @@ provides the best performance.
     \end{tikzpicture}
 
     \caption{
-        Comparison of the decoding performance of the $\llbracket
+        \red{\lipsum[2]}
-        144,12,12 \rrbracket$ \ac{bb} code under min-sum decoding
-        ($200$ iterations) for different window sizes.
-        The step size was fixed to $F=1$, $12$ rounds of syndrome
-        extraction were performed and the noise model is
-        standard circuit-based depolarizing noise.
     }
     \label{fig:whole_vs_cold}
 \end{figure}
 
-% Initial results of warm-start decoding
+% [Experimental parameters] Figure 4.7
 
-As a next step, we additionally generated error rate curves using our
+\Cref{fig:whole_vs_cold} shows the simulation results for this initial
-warm-start modification.
+investigation.
-\red{[Again 200 BP iterations, etc.]}
+The three colored curves correspond to cold-start sliding-window
-\Cref{fig:whole_vs_cold_vs_warm} shows the numerical results from
+decoding with window sizes $W \in \{3, 4, 5\}$, all with the step
-this experiment.
+size fixed to $F = 1$, while the black curve gives the per-round
-The cold-start results from the previous graph are now plotted in
+\ac{ler} obtained when decoding on the whole detector error matrix at once.
-dashed lines, while the new warm-start results are plotted with solid lines.
+In all cases, the inner \ac{bp} decoder was allowed a maximum of
-We can see that the decoding performance has been improved overall.
+$200$ iterations, and the physical error rate was swept from
+$p = 0.001$ to $p = 0.004$ in steps of $0.0005$.
 
-% Unexpected: Warm-start better than whole
+% [Description] Figure 4.7
 
-Additionally, we can see some initially unexpected behavior: The warm-start
+Across the entire range of physical error rates, all curves exhibit
-sliding window decoding with $W=5$ performs better than decoding
+the expected monotonic increase in logical error rate with increasing
-under consideration of the whole detector error matrix at once, even
+physical noise.
-though the process is less global.
+The $W = 3$ decoder consistently yields the highest LER, performing
+roughly an order of magnitude worse than the baseline at low physical
+error rates.
+Increasing the window size to $W = 4$ substantially closes this gap,
+and the $W = 5$ curve nearly coincides with the whole-block decoder
+across the full range of physical error rates.
+
+% [Interpretation] Figure 4.7
+
+This behavior is consistent with the intuition behind sliding-window decoding.
+The detector error matrix encodes correlations between detection
+events that span the full syndrome extraction history, so errors
+lying in the commit region of an early window are in general
+constrained by check nodes that only become visible in subsequent windows.
+Larger windows expose the inner decoder to more of these constraints
+before any commit is made, leading to better-informed decisions and a
+lower per-round \ac{ler}.
+Decoding the whole matrix at once represents the limiting case of
+this trend and, as expected, achieves the strongest performance.
+The fact that the $W = 5$ curve is already very close to the
+whole-block decoder indicates that the marginal benefit of enlarging
+the window saturates after a certain point.
+From a practical standpoint, the choice of $W$ thus represents a
+trade-off between decoding latency and accuracy: larger windows
+delay the start of decoding by requiring more syndrome extraction
+rounds to be collected upfront, while the diminishing returns above
+$W = 4$ suggest that growing the window much further yields little
+additional accuracy in return.
+
+% [Thread] First comparison with warm start
+
+Next, we additionally generated error rate curves for warm-start
+sliding-window decoding to assess how much of the gap between
+cold-start and whole-block decoding can be recovered by our modification.
+We chose the same window sizes as before, so that the warm- and
+cold-start curves can be compared directly at matching values of $W$.
 
 \begin{figure}[t]
     \centering
@@ -1429,53 +1401,107 @@ though the process is less global.
     \end{tikzpicture}
 
     \caption{
-        Comparison of the decoding performance of cold and warm-start
+        \red{\lipsum[2]}
-        decoding under the $\llbracket 144,12,12 \rrbracket$ \ac{bb}
-        code for different window sizes.
-        Decoding was performed using the min-sum algorithm ($200$
-        iterations).
-        The step size was fixed to $F=1$, $12$ rounds of syndrome
-        extraction were performed and the noise model is
-        standard circuit-based depolarizing noise.
     }
     \label{fig:whole_vs_cold_vs_warm}
 \end{figure}
 
+% [Experimental parameters] Figure 4.8
+
+\Cref{fig:whole_vs_cold_vs_warm} extends the previous comparison by
+additionally including the warm-start variant of sliding-window decoding.
+The dashed colored curves reproduce the cold-start results from
+\Cref{fig:whole_vs_cold}, while the solid colored curves show the
+corresponding warm-start runs for the same window sizes
+$W \in \{3, 4, 5\}$.
+The remaining experimental parameters are unchanged:
+the step size is fixed to $F = 1$,
+the inner \ac{bp} decoder is allowed up to $200$ iterations per
+window invocation, the black curve again gives the whole-block
+reference, and the physical error rate is swept from $p = 0.001$ to
+$p = 0.004$ in steps of $0.0005$.
+
+% [Description] Figure 4.8
+
+For each window size, the warm-start variant consistently outperforms
+its cold-start counterpart, with the dashed curves lying above the
+corresponding solid curves across the entire range of physical error rates.
+The performance gap between the two approaches is most pronounced for
+the largest window ($W = 5$) and gradually narrows as the window size decreases.
+Additionally, the gap between the cold- and warm-start curves
+generally widens as the physical error rate decreases.
+
+% [Interpretation] Figure 4.8
+
+The improvement of warm-start over cold-start decoding matches the
+motivation for the modification:
+By reusing already existing messages from the previous window in the
+overlap region, the next window invocation has additional information
+at its disposal about the reliability of the \acp{vn} and \acp{cn}.
+The widening of the gap towards larger window sizes is consistent
+with this picture, since with $F$ fixed to $1$ the overlap between
+consecutive windows spans $W - F = W - 1$ syndrome rounds, so larger
+$W$ implies that more messages are carried over and a larger fraction
+of the next window starts in a warm state.
+% TODO: Possibly insert explanation for higher gain at lowre error rates
+A perhaps surprising observation is that the warm-start curve for
+$W = 5$ actually lies below the whole-block reference across the
+entire range of physical error rates, even though warm-start
+sliding-window decoding is, by construction, more local than
+whole-block decoding.
+A possible explanation for this effect is discussed in the following.
+
+% [Thread] Warm start is better than whole due to more effective iterations
+
+A possible explanation for this surprising behavior lies in the
+number of \ac{bp} iterations effectively spent on the \acp{vn}
+inside the overlap region.
+Each \ac{vn} in such an overlap is processed by multiple consecutive
+window invocations, and because every new window resumes from the
+messages left over by its predecessor, these invocations effectively
+accumulate iterations on the same \acp{vn} rather than restarting
+from scratch.
+The whole-block decoder, by contrast, performs only a single run of
+at most $200$ iterations on the entire detector error matrix, so
+each of its \acp{vn} receives at most that many iterations.
+It seems this larger effective iteration budget on the overlap
+regions can outweigh the loss of globality incurred by windowing.
+
+A natural way to test this hypothesis is to raise the maximum number
+of \ac{bp} iterations of the whole-block decoder until its per-round
+\ac{ler} saturates.
+If the above interpretation is correct, the resulting saturation
+level should constitute a lower bound that no windowed scheme,
+irrespective of the initialization, can beat, since by construction
+whole-block decoding has access to the full set of constraints
+available to any window.
+
+% [Description] Figure 4.9
+%   - Parameters
+%     - # BP iterations
+%     - W,F
+%     - Physical error rates
+%     - Warm vs cold start
+%   - Figure description
+%     - TODO:
+
+% [Interpretation] Figure 4.9
+%   -
+
+% At some later point
+\content{When looking at max iterations: Callback to diminishing
+    returns with growing window size: More iterations more beneficial
+than larger window (+1 for warm-start)}
+
 \begin{figure}[t]
     \centering
     \begin{tikzpicture}
-        \begin{axis}[
+        \def\spyxmin{32}
-                width=\figwidth,
+        \def\spyxmax{512}
-                height=\figheight,
+        \def\spyymin{5e-3}
-                ymode=log,
+        \def\spyymax{7e-2}
-                % xmode=log,
-                legend style={
-                    cells={anchor=west},
-                    cells={align=left},
-                },
-                enlargelimits=false,
-                ymin=1e-3, ymax=1e-1,
-                grid=both,
-                legend pos = north east,
-                xtick={32,512,1024,2048,4096},
-                % xtick={0.001,0.0015,...,0.004},
-                xticklabels =
-                {$32$,$512$,$1{,}024$,,$2{,}048$,,$3{,}072$,,$4{,}096$},
-                xtick={32, 512, 1024, 1536, 2048, 2560, 3072, 3584, 4096},
-                xticklabel style={/pgf/number format/fixed},
-                xticklabel style={/pgf/number format/precision=4},
-                % x tick label style={rotate=45, anchor=north east,
-                % inner sep=1mm},
-                scaled x ticks=false,
-                xlabel={Number of BP iterations},
-                ylabel={Per-round-LER},
-                extra description/.code={
-                    \node[rotate=90, anchor=south]
-                    at ([xshift=10mm]current axis.east)
-                    {Warm s. (---), Cold s. (- - -)};
-                },
-            ]
 
+        \newcommand{\plotcurves}{%
             \foreach \W/\col/\mark in
             {3/KITred/triangle,4/KITblue/diamond,5/KITorange/square} {
                 \edef\temp{\noexpand
@@ -1489,12 +1515,11 @@ though the process is less global.
                 }
                 \temp
             }
-
             \foreach \W/\col/\mark in
             {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, forget plot]
                     table[
                         col sep=comma, x=max_iter,
                         y=LER_per_round,
@@ -1502,33 +1527,176 @@ though the process is less global.
                     {res/sim/max_iter/WindowingSyndromeMinSumDecoder/p_0.0025/pass_soft_info_True/F_1/W_\W/LERs.csv};
                 }
                 \temp
-
-                \addlegendentryexpanded{$W = \W$}
             }
-
+            \addplot+[mark=*, solid, mark options={fill=black}, black,
-            \addplot+[mark=*, solid, mark options={fill=black}, black]
+            forget plot]
-            table[
+            table[col sep=comma, x=max_iter, y=LER_per_round]
-                col sep=comma, x=max_iter,
-                y=LER_per_round,
-            ]
             {res/sim/max_iter/SyndromeMinSumDecoder/p_0.0025/LERs.csv};
+        }
 
+        \begin{axis}[
+                name=main,
+                width=\figwidth,
+                height=\figheight,
+                ymode=log,
+                enlargelimits=false,
+                ymin=1e-3, ymax=1e-1,
+                grid=both,
+                legend pos=north east,
+                xtick={32, 512, 1024, 1536, 2048, 2560, 3072, 3584, 4096},
+                xticklabels={$32$,$512$,$1{,}024$,,$2{,}048$,,$3{,}072$,,$4{,}096$},
+                xticklabel style={/pgf/number format/fixed},
+                scaled x ticks=false,
+                xlabel={Number of BP iterations},
+                ylabel={Per-round-LER},
+                extra description/.code={
+                    \node[rotate=90, anchor=south]
+                    at ([xshift=10mm]current axis.east)
+                    {Warm s. (---), Cold s. (- - -)};
+                },
+            ]
+
+            \plotcurves
+
+            \addlegendimage{KITred, mark=triangle*}
+            \addlegendentry{$W = 3$}
+            \addlegendimage{KITblue, mark=diamond*}
+            \addlegendentry{$W = 4$}
+            \addlegendimage{KITorange, mark=square*}
+            \addlegendentry{$W = 5$}
+            \addlegendimage{black, mark=*}
             \addlegendentry{Whole}
+
+            \node[draw=black, fit={(axis cs:\spyxmin,\spyymin) (axis
+            cs:\spyxmax,\spyymax)}, inner sep=0pt, name=spybox] {};
+
         \end{axis}
+
+        \begin{axis}[
+                name=inset,
+                at={(main.north)},
+                anchor=south,
+                xshift=0mm, yshift=6mm,
+                width=6.5cm, height=4.875cm,
+                ymode=log,
+                enlargelimits=false,
+                xmin=\spyxmin, xmax=\spyxmax,
+                ymin=\spyymin, ymax=\spyymax,
+                xtick={32,128,256, 512},
+                yticklabels={\empty},
+                xticklabels={\empty},
+                grid=both,
+                axis background/.style={fill=white},
+            ]
+
+            \plotcurves
+        \end{axis}
+
+        \draw (spybox.north east) -- (inset.south west);
     \end{tikzpicture}
 
     \caption{
-        Comparison of the decoding performance of cold and warm-start
+        \red{\lipsum[2]}
-        decoding under the $\llbracket 144,12,12 \rrbracket$ \ac{bb}
-        code for different step sizes.
-        Decoding was performed using the min-sum algorithm ($200$
-        iterations).
-        The window size was fixed to $W=5$, $12$ rounds of syndrome
-        extraction were performed and the noise model is
-        standard circuit-based depolarizing noise.
     }
 \end{figure}
 
+% \begin{figure}[t]
+%     \centering
+%     \begin{tikzpicture}[spy using outlines={circle, magnification=2,
+%         connect spies}]
+%
+%         \begin{axis}[
+%                 width=\figwidth,
+%                 height=\figheight,
+%                 ymode=log,
+%                 % xmode=log,
+%                 legend style={
+%                     cells={anchor=west},
+%                     cells={align=left},
+%                 },
+%                 enlargelimits=false,
+%                 ymin=1e-3, ymax=1e-1,
+%                 grid=both,
+%                 legend pos = north east,
+%                 xtick={32,512,1024,2048,4096},
+%                 % xtick={0.001,0.0015,...,0.004},
+%                 xticklabels =
+%                 {$32$,$512$,$1{,}024$,,$2{,}048$,,$3{,}072$,,$4{,}096$},
+%                 xtick={32, 512, 1024, 1536, 2048, 2560, 3072, 3584, 4096},
+%                 xticklabel style={/pgf/number format/fixed},
+%                 xticklabel style={/pgf/number format/precision=4},
+%                 % x tick label style={rotate=45, anchor=north east,
+%                 % inner sep=1mm},
+%                 scaled x ticks=false,
+%                 xlabel={Number of BP iterations},
+%                 ylabel={Per-round-LER},
+%                 extra description/.code={
+%                     \node[rotate=90, anchor=south]
+%                     at ([xshift=10mm]current axis.east)
+%                     {Warm s. (---), Cold s. (- - -)};
+%                 },
+%             ]
+%
+%             \foreach \W/\col/\mark in
+%             {3/KITred/triangle,4/KITblue/diamond,5/KITorange/square} {
+%                 \edef\temp{\noexpand
+%                     \addplot+[mark=\mark, densely dashed,
+%                     forget plot, \col]
+%                     table[
+%                         col sep=comma, x=max_iter,
+%                         y=LER_per_round,
+%                     ]
+%                     {res/sim/max_iter/WindowingSyndromeMinSumDecoder/p_0.0025/pass_soft_info_False/F_1/W_\W/LERs.csv};
+%                 }
+%                 \temp
+%             }
+%
+%             \foreach \W/\col/\mark in
+%             {3/KITred/triangle*,4/KITblue/diamond*,5/KITorange/square*} {
+%                 \edef\temp{\noexpand
+%                     \addplot+[mark=\mark, solid, mark
+%                     options={fill=\col}, \col]
+%                     table[
+%                         col sep=comma, x=max_iter,
+%                         y=LER_per_round,
+%                     ]
+%                     {res/sim/max_iter/WindowingSyndromeMinSumDecoder/p_0.0025/pass_soft_info_True/F_1/W_\W/LERs.csv};
+%                 }
+%                 \temp
+%
+%                 \addlegendentryexpanded{$W = \W$}
+%             }
+%
+%             \addplot+[mark=*, solid, mark options={fill=black}, black]
+%             table[
+%                 col sep=comma, x=max_iter,
+%                 y=LER_per_round,
+%             ]
+%             {res/sim/max_iter/SyndromeMinSumDecoder/p_0.0025/LERs.csv};
+%
+%             \addlegendentry{Whole}
+%
+%             \coordinate (spypoint) at (axis cs:250,1e-2);
+%             \coordinate (magnifyglass) at (axis cs:2048,0.5);
+%
+%         \end{axis}
+%         \spy [black, size=4cm] on (spypoint)
+%         in node[fill=white] at (magnifyglass);
+%
+%     \end{tikzpicture}
+%
+%     \caption{
+%         Comparison of the decoding performance of cold and warm-start
+%         decoding under the $\llbracket 144,12,12 \rrbracket$ \ac{bb}
+%         code for different step sizes.
+%         Decoding was performed using the min-sum algorithm ($200$
+%         iterations).
+%         The window size was fixed to $W=5$, $12$ rounds of syndrome
+%         extraction were performed and the noise model is
+%         standard circuit-based depolarizing noise.
+%     }
+% \end{figure}
+
 \begin{figure}[t]
     \centering
     \begin{subfigure}{0.48\textwidth}
@@ -1674,14 +1842,7 @@ though the process is less global.
     \end{subfigure}
 
     \caption{
-        Comparison of cold and warm-start sliding-window
+        \red{\lipsum[2]}
-        min-sum decoding for the $\llbracket 144, 12, 12 \rrbracket$
-        \ac{bb} code
-        under circuit-level noise.
-        $12$ rounds of syndrome extraction were performed and
-        standard circuit-based depolarizing noise was chosen as the
-        noise model.
-        The physical error probabilty was fixed to $0.0025$.
     }
 \end{figure}
 
@@ -1830,16 +1991,7 @@ though the process is less global.
     \end{subfigure}
 
     \caption{
-        Comparison of the decoding performance of cold and warm-start
+        \red{\lipsum[2]}
-        decoding under the $\llbracket 144,12,12 \rrbracket$ \ac{bb}.
-        Decoding was performed using the \ac{bpgd} algorithm with
-        $T=1$ and no limit on the number of outer iterations.
-        The information used for the warm-start intialization
-        included both the messages on the Tanner graph and decimation
-        information.
-        $12$ rounds of syndrome extraction were performed and
-        standard circuit-based depolarizing noise was chosen as the
-        noise model.
     }
 \end{figure}
 
@@ -1990,19 +2142,7 @@ though the process is less global.
     \end{subfigure}
 
     \caption{
-        Comparison of the decoding performance of cold and warm-start
+        \red{\lipsum[2]}
-        decoding for the $\llbracket 144,12,12 \rrbracket$ \ac{bb}
-        under circuit-level noise.
-        Decoding was performed using the \ac{bpgd} algorithm with
-        $T=1$.
-        The number of iterations refers to the outer \ac{bpgd}
-        iterations, i.e., the number of decimations.
-        The information used for the warm-start intialization
-        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.
-        The physical error probabilty was fixed to $0.0025$.
     }
 \end{figure}
 
@@ -2147,16 +2287,7 @@ though the process is less global.
     \end{subfigure}
 
     \caption{
-        Comparison of the decoding performance of cold and warm-start
+        \red{\lipsum[2]}
-        decoding for the $\llbracket 144,12,12 \rrbracket$ \ac{bb}
-        under circuit-level noise.
-        Decoding was performed using the \ac{bpgd} algorithm with
-        $T=1$ and no limit on the number of outer iterations.
-        The information used for the warm-start intialization
-        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.
     }
 \end{figure}
 
@@ -2307,19 +2438,7 @@ though the process is less global.
     \end{subfigure}
 
     \caption{
-        Comparison of the decoding performance of cold and warm-start
+        \red{\lipsum[2]}
-        decoding for the $\llbracket 144,12,12 \rrbracket$ \ac{bb}
-        under circuit-level noise.
-        Decoding was performed using the \ac{bpgd} algorithm with
-        $T=1$.
-        The number of iterations refers to the outer \ac{bpgd}
-        iterations, i.e., the number of decimations.
-        The information used for the warm-start intialization
-        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.
-        The physical error probabilty was fixed to $0.0025$.
     }
 \end{figure}
 

src/thesis/chapters/5_conclusion_and_outlook.tex

@@ -4,4 +4,5 @@
 \content{Softer way of decimating VNs}
 \content{Systematic study on using different inner decoders (AED,
 SED, BPGD, ...)}
+\content{Investigate SC-LDPC window decoding wave-like effects}
 

src/thesis/main.tex

@@ -28,6 +28,7 @@
 \usepackage{nicematrix}
 \usepackage{colortbl}
 \usepackage{cleveref}
+\usepackage{lipsum}
 
 \usetikzlibrary{calc, positioning, arrows, fit}
 \usetikzlibrary{external}