Add paragraphs exp. params, description, and interpretation for fig. 4.9
This commit is contained in:
@@ -1476,23 +1476,6 @@ irrespective of the initialization, can beat, since by construction
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whole-block decoding has access to the full set of constraints
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available to any window.
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% [Description] Figure 4.9
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% - Parameters
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% - # BP iterations
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% - W,F
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% - Physical error rates
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% - Warm vs cold start
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% - Figure description
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% - TODO:
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% [Interpretation] Figure 4.9
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% -
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% At some later point
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\content{When looking at max iterations: Callback to diminishing
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returns with growing window size: More iterations more beneficial
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than larger window (+1 for warm-start)}
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\begin{figure}[t]
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\centering
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\begin{tikzpicture}
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@@ -1598,104 +1581,86 @@ than larger window (+1 for warm-start)}
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\caption{
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\red{\lipsum[2]}
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}
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\label{fig:bp_w_over_iter}
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\end{figure}
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% \begin{figure}[t]
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% \centering
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% \begin{tikzpicture}[spy using outlines={circle, magnification=2,
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% connect spies}]
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%
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% \begin{axis}[
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% width=\figwidth,
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% height=\figheight,
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% ymode=log,
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% % xmode=log,
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% legend style={
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% cells={anchor=west},
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% cells={align=left},
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% },
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% enlargelimits=false,
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% ymin=1e-3, ymax=1e-1,
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% grid=both,
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% legend pos = north east,
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% xtick={32,512,1024,2048,4096},
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% % xtick={0.001,0.0015,...,0.004},
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% xticklabels =
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% {$32$,$512$,$1{,}024$,,$2{,}048$,,$3{,}072$,,$4{,}096$},
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% xtick={32, 512, 1024, 1536, 2048, 2560, 3072, 3584, 4096},
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% xticklabel style={/pgf/number format/fixed},
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% xticklabel style={/pgf/number format/precision=4},
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% % x tick label style={rotate=45, anchor=north east,
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% % inner sep=1mm},
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% scaled x ticks=false,
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% xlabel={Number of BP iterations},
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% ylabel={Per-round-LER},
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% extra description/.code={
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% \node[rotate=90, anchor=south]
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% at ([xshift=10mm]current axis.east)
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% {Warm s. (---), Cold s. (- - -)};
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% },
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% ]
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%
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% \foreach \W/\col/\mark in
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% {3/KITred/triangle,4/KITblue/diamond,5/KITorange/square} {
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% \edef\temp{\noexpand
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% \addplot+[mark=\mark, densely dashed,
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% forget plot, \col]
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% table[
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% col sep=comma, x=max_iter,
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% y=LER_per_round,
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% ]
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% {res/sim/max_iter/WindowingSyndromeMinSumDecoder/p_0.0025/pass_soft_info_False/F_1/W_\W/LERs.csv};
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% }
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% \temp
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% }
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%
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% \foreach \W/\col/\mark in
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% {3/KITred/triangle*,4/KITblue/diamond*,5/KITorange/square*} {
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% \edef\temp{\noexpand
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% \addplot+[mark=\mark, solid, mark
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% options={fill=\col}, \col]
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% table[
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% col sep=comma, x=max_iter,
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% y=LER_per_round,
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% ]
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% {res/sim/max_iter/WindowingSyndromeMinSumDecoder/p_0.0025/pass_soft_info_True/F_1/W_\W/LERs.csv};
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% }
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% \temp
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%
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% \addlegendentryexpanded{$W = \W$}
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% }
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%
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% \addplot+[mark=*, solid, mark options={fill=black}, black]
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% table[
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% col sep=comma, x=max_iter,
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% y=LER_per_round,
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% ]
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% {res/sim/max_iter/SyndromeMinSumDecoder/p_0.0025/LERs.csv};
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%
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% \addlegendentry{Whole}
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%
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% \coordinate (spypoint) at (axis cs:250,1e-2);
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% \coordinate (magnifyglass) at (axis cs:2048,0.5);
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%
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% \end{axis}
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% \spy [black, size=4cm] on (spypoint)
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% in node[fill=white] at (magnifyglass);
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%
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% \end{tikzpicture}
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%
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% \caption{
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% Comparison of the decoding performance of cold and warm-start
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% decoding under the $\llbracket 144,12,12 \rrbracket$ \ac{bb}
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% code for different step sizes.
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% Decoding was performed using the min-sum algorithm ($200$
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% iterations).
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% The window size was fixed to $W=5$, $12$ rounds of syndrome
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% extraction were performed and the noise model is
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% standard circuit-based depolarizing noise.
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% }
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% \end{figure}
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% [Experimental parameters] Figure 4.9
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\Cref{fig:bp_w_over_iter} shows the per-round \ac{ler} as a function
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of the maximum number of \ac{bp} iterations granted to the inner decoders.
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The dashed colored curves correspond to cold-start sliding-window
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decoding for $W \in \{3, 4, 5\}$, the solid colored curves to the
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corresponding warm-start sliding-window decoding, and the black curve
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to the whole-block reference.
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The physical error rate is fixed at $p = 0.0025$, the step size at
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$F = 1$, and the iteration budget is swept over
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$n_\text{iter} \in \{32, 128, 256, 512, 1024, 2048, 4096\}$.
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The enlarged plot magnifies the low-iteration regime
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$n_\text{iter} \in [32, 512]$.
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% [Description] Figure 4.9
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All curves decrease monotonically with the iteration budget, but
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contrary to our expectation, none of them appears to fully saturate
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within the swept range: even at $n_\text{iter} = 4096$, every curve
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still exhibits a noticeable downward slope.
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At $n_\text{iter} = 32$, the whole-block curve lies below both the
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$W=4$ and $W=5$ sliding-window curves.
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At $n_\text{iter} = 128$ the whole-block curve already performs
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better than the $W=4$ sliding-window curve
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and at $n_\text{iter} = 512$ the whole-block and warm-start $W = 5$
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curves also cross.
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From this point onwards, the whole-block decoder lies strictly below
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all windowed schemes, this difference becoming more pronounced as the
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iteration budget grows further.
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Within the magnified plot, the gap between the warm-start and
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cold-start curves at fixed $W$ is largest for the smallest iteration
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counts and shrinks rapidly as $n_\text{iter}$ grows, and at fixed
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$n_\text{iter}$ the size of this gap grows with the window size,
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mirroring the behavior already observed in \Cref{fig:whole_vs_cold_vs_warm}.
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% [Interpretation] Figure 4.9
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These observations are largely consistent with the effective-iterations
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hypothesis put forward above.
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The whole-block decoder eventually overtaking every windowed scheme
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matches the prediction made there: with a sufficiently large
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iteration budget, the whole-block decoder reaches an error rate
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that nonone of the windowed schemes can beat, because of the more global
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nature of the considered constraints.
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Furthermore, the pronounced advantage of warm- over cold-start decoding at low
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numbers of iterations makes sense if we consider the overall trend of the plots.
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At low iteration budgets, each additional iteration is worth more
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than at high budgets.
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As the number of permitted iteration increases, the benefit of
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the additional ``free'' iterations gained due to the the warm-start
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initialization diminishes, and the curves approach each other.
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The fact that no curve clearly saturates within the swept range is
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itself worth noting.
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We know that \ac{bp} on \ac{qldpc} codes suffers from poor
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convergence due to the short cycles in the underlying Tanner graph,
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so even after several thousand iterations the
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decoder may continue to slowly refine its message estimates rather
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than settle into a stable fixed point.
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This is one of the core motivations for moving from plain \ac{bp} to
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the guided-decimation variant studied in
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\Cref{subsec:Belief Propagation with Guided Decimation}.
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Another thing to note is that setting the per-invocation iteration
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budget of the inner decoder equal to the iteration budget of the
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whole-block decoder is not a fair comparison in terms of total
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computational effort.
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The sliding-window scheme processes each \ac{vn} in an overlap region
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multiple times and therefore spends more iterations overall.
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In the context of \ac{qec}, however, the relevant figure of merit is
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not total compute but decoding latency, and in terms of latency the
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sliding-window approach is still at an advantage.
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% At some later point
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\content{When looking at max iterations: Callback to diminishing
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returns with growing window size: More iterations more beneficial
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than larger window (+1 for warm-start)}
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\begin{figure}[t]
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\centering
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@@ -1,8 +1,11 @@
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\chapter{Conclusion and Outlook}
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\content{Takeaway: Warm-start more effective for lower numbers of max
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iterations (plays into our hands because lower number of iterations
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means lower latency)}
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\content{\textbf{Ideas for further research}}
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\content{Softer way of decimating VNs}
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\content{Systematic study on using different inner decoders (AED,
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SED, BPGD, ...)}
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\content{Investigate SC-LDPC window decoding wave-like effects}
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