diff --git a/src/thesis/chapters/4_decoding_under_dems.tex b/src/thesis/chapters/4_decoding_under_dems.tex index 3b38ec3..ff6a07c 100644 --- a/src/thesis/chapters/4_decoding_under_dems.tex +++ b/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 diff --git a/src/thesis/chapters/5_conclusion_and_outlook.tex b/src/thesis/chapters/5_conclusion_and_outlook.tex index e7375a7..bdfd6f4 100644 --- a/src/thesis/chapters/5_conclusion_and_outlook.tex +++ b/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.