Write conclusion

src/thesis/chapters/4_decoding_under_dems.tex

@@ -2274,7 +2274,7 @@ In both panels, every curve again exhibits the expected monotonic
 increase of the per-round \ac{ler} with the physical error rate.
 Across both panels and across all parameter choices, the warm-start
 curves lie above the corresponding cold-start curves, i.e.,
-the warm-start variant performsworse than its cold-start counterpart.
+the warm-start variant performs worse than its cold-start counterpart.
 This is the opposite of what we observed for plain \ac{bp}, where
 warm-start improved upon cold-start at every parameter setting.
 The gap between the warm- and cold-start curves additionally widens

src/thesis/chapters/5_conclusion_and_outlook.tex

@@ -1,15 +1,129 @@
 \chapter{Conclusion and Outlook}
 \label{ch:Conclusion}
 
-\content{Takeaway: Warm-start more effective for lower numbers of max
-    iterations (plays into our hands because lower number of iterations
-means lower latency)}
-\content{Warm-start initialization limited to decoding algorithms
-providing relevant soft information}
+% Recap of motivation
 
-\content{\textbf{Ideas for further research}}
-\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}
+This thesis investigated decoding under \acp{dem} for fault-tolerant
+\ac{qec}, with a focus on low-latency decoding methods for \ac{qldpc} codes.
+The repetition of the syndrome measurements, especially under
+consideration of circuit-level noise, leads to a significant increase
+in decoding complexity: in our experiments on the $\llbracket
+144,12,12 \rrbracket$ \ac{bb} code with $12$ syndrome extraction
+rounds, the check matrix grew from 144 \acp{vn} and 72
+\acp{cn} to 9504 \acp{vn} and 1008 \acp{cn}.
+
+% Recap of research gap and own work
+
+Sliding-window decoding addresses the latency constraint by
+exploiting the time-like locality of the syndrome extraction circuit,
+which manifests as a block-diagonal structure in the detector error
+matrix when detectors are defined as the difference of consecutive
+syndrome measurement rounds.
+We drew a comparison to windowed decoding for \ac{sc}-\ac{ldpc}
+codes, but noted that the existing realizations of sliding-window
+decoding discard the soft information produced inside one window
+before moving to the next.
+Building on this observation, we proposed warm-start sliding-window
+decoding, in which the \ac{bp} messages on the edges crossing into
+the overlap region of the previous window are reused to initialise
+the corresponding messages of the next window in place of the
+standard cold-start initialisation.
+
+We formulated the warm start first for plain \ac{bp} and then for
+\ac{bpgd}, the latter being attractive as an inner decoder because it
+addresses the convergence problems caused by short cycles and
+degeneracy in \ac{qldpc} Tanner graphs.
+The decoders were evaluated by Monte Carlo simulation on the
+$\llbracket 144,12,12 \rrbracket$ \ac{bb} code over $12$ syndrome
+extraction rounds under standard circuit-based depolarizing noise.
+We focused on a qualitative analysis, refraining from further
+optimizations such as introducing a normalization parameter for the
+min-sum algorithm.
+
+% Recap of experimental conclusions
+
+For plain min-sum \ac{bp}, the warm start was consistently beneficial
+across the parameter ranges we examined. The size of the gain depended
+on the overlap between consecutive windows: enlarging $W$ or
+shrinking $F$, both of which enlarge the overlap, raised the
+warm-start performance increase.
+We argued that the underlying mechanism is an effective increase in
+the number of \ac{bp} iterations spent on the \acp{vn} in the overlap
+region: each such \ac{vn} is processed by multiple consecutive window
+invocations, and the warm start lets these invocations accumulate
+iterations on the same \acp{vn} rather than restarting from scratch.
+The gain was most pronounced at low numbers of maximum iterations, where
+every additional iteration carries proportionally more information.
+
+For \ac{bpgd}, we noted that more information is available in the
+overlap region of a window: in addition to the \ac{bp} messages,
+there is information about which \acp{vn} were decimated and to what value.
+Passing this decimation information to the next window in addition to
+the messages turned out to worsen the performance considerably, which
+we attributed to a premature hard decision of the \acp{vn} in the
+overlap region.
+Restricting the warm start to the \ac{bp} messages alone, removed this effect.
+The resulting message-only warm start recovered a consistent
+improvement over cold-start that followed the same qualitative
+behaviour as for plain \ac{bp}: larger overlap, achieved by larger
+$W$ or smaller $F$, yielded a larger gain, and the
+performance difference was most pronounced at low numbers of maximum iterations.
+
+% Implications from experimental results
+
+These observations imply that the warm-start modification to
+sliding-window decoding provides a universal improvement, as long as
+some care is taken with specifying the information to be passed to
+the subsequent window.
+Not that this comes at no additional cost to the decoding complexity,
+since the only difference between warm- and cold-start sliding-window
+decoding is the initialization of the \ac{bp} messages.
+We expect similar behavior with other inner decoders that support
+soft information initialization in the overlap region.
+
+% Further research
+
+Several directions for further research emerge from this work.
+The most immediate is an extension of the evaluation to other
+\ac{qldpc} code families, to other circuit-level noise models such as
+SI1000 or EM3, and to a range of code sizes.
+This would clarify the generality of the gain due to the warm-start
+initialization.
+We expect the qualitative findings to carry over, since the
+underlying mechanism is structural rather than code-specific, but
+quantifying the gain across code families and noise models is left to
+future work.
+
+A second direction is a systematic study of inner decoders under the
+warm-start framework.
+We considered plain min-sum \ac{bp} and \ac{bpgd}, but other
+algorithms used for \ac{qldpc} decoding, such as automorphism
+ensemble decoding \cite{koutsioumpas_automorphism_2025} or neural
+\ac{bp} \cite{miao_quaternary_2025} may admit warm-start variants of their own.
+
+A third direction is a softer treatment of the decimation state in \ac{bpgd}.
+Rather than discarding the decimation information of the previous
+window entirely, as in the message-only warm start used here, one
+could encode the decimation decisions as strong but finite biases on
+the channel \acp{llr} of the next window, allowing the new window's parity
+checks to override them if the syndrome calls for it.
+This would interpolate between the two warm-start variants studied here and
+might combine the benefits of both.
+A related question is whether the decimation schedule itself should
+be aware of the window structure, for instance by deferring
+decimation of \acp{vn} in the overlap region until they have been
+visited by the next window.
+
+A final direction is suggested by the structural similarity between
+sliding-window decoding for \acp{dem} and windowed decoding for
+\ac{sc}-\ac{ldpc} codes.
+The current approach to generating the syndrome extraction circuitry
+necessarily leads to a coupling width of one between adjacent
+syndrome measurement rounds.
+A natural question is whether the coupling width could be
+increased, e.g., by interleaving two separate realizations of the
+syndrome measurement circuitry instead of always repeating the same one.
+Work in this direction would also be a step toward bringing
+sliding-window decoding under DEMs within the scope of the analytical
+machinery developed for SC-LDPC codes.