an.tsouchlos/ma-thesis
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases 12 Wiki Activity

Consistently capitalize character after semicolon

Browse Source
This commit is contained in:
Andreas Tsouchlos
2026-05-04 20:21:21 +02:00
parent 8d6df8a79d
commit 56e3a0e5ca
4 changed files with 27 additions and 27 deletions
Download Patch File Download Diff File Expand all files Collapse all files

2
src/thesis/chapters/2_fundamentals.tex
View File

@@ -193,7 +193,7 @@ decoding \cite[Preface]{ryan_channel_2009}. The iterative decoding algorithms
are generally defined in terms of message passing on the are generally defined in terms of message passing on the
\textit{Tanner graph} of a code. The Tanner graph is a bipartite \textit{Tanner graph} of a code. The Tanner graph is a bipartite
graph that constitutes an alternative representation of the \ac{pcm}. graph that constitutes an alternative representation of the \ac{pcm}.
We define two types of nodes: \acp{vn}, corresponding to codeword We define two types of nodes: \Acp{vn}, corresponding to codeword
bits, and \acp{cn}, corresponding to individual parity checks. bits, and \acp{cn}, corresponding to individual parity checks.
We then construct the Tanner graph by connecting each \ac{cn} to We then construct the Tanner graph by connecting each \ac{cn} to
the \acp{vn} that make up the corresponding parity check the \acp{vn} that make up the corresponding parity check

40
src/thesis/chapters/4_decoding_under_dems.tex
View File

@@ -470,7 +470,7 @@ model and is difficult to predict beforehand.
The block-diagonal structure reflects the time-like locality The block-diagonal structure reflects the time-like locality
of the syndrome extraction circuit, with each block of the syndrome extraction circuit, with each block
corresponding to one syndrome measurement round. corresponding to one syndrome measurement round.
Two consecutive windows are highlighted: the window size $W Two consecutive windows are highlighted: The window size $W
\in \mathbb{N}$ controls the number of syndrome rounds \in \mathbb{N}$ controls the number of syndrome rounds
included in each window, while the step size $F \in included in each window, while the step size $F \in
\mathbb{N}$ controls how many rounds separate the start of \mathbb{N}$ controls how many rounds separate the start of
@@ -1013,8 +1013,8 @@ the most reliable \ac{vn}, meaning we perform a hard decision and
remove it from the following decoding process. remove it from the following decoding process.
This means that when moving from one window to the next, we now have This means that when moving from one window to the next, we now have
more information available: not just the \ac{bp} messages but also the more information available: Not just the \ac{bp} messages but also the
information about what \acp{vn} were decimated and to what values. Information about what \acp{vn} were decimated and to what values.
We call this \emph{decimation information} in the following. We call this \emph{decimation information} in the following.
We can extend \Cref{alg:warm_start_bp} by additionally passing the We can extend \Cref{alg:warm_start_bp} by additionally passing the
decimation information after initializing the \ac{cn} to \ac{vn} messages. decimation information after initializing the \ac{cn} to \ac{vn} messages.
@@ -1404,7 +1404,7 @@ The fact that the $W = 5$ curve is already very close to the
whole-block decoder indicates that the marginal benefit of enlarging whole-block decoder indicates that the marginal benefit of enlarging
the window saturates after a certain point. the window saturates after a certain point.
Thus, from a practical standpoint, the choice of $W$ represents a Thus, from a practical standpoint, the choice of $W$ represents a
trade-off between decoding latency and accuracy: larger windows trade-off between decoding latency and accuracy: Larger windows
delay the start of decoding by requiring more syndrome extraction delay the start of decoding by requiring more syndrome extraction
rounds to be collected upfront, while the diminishing returns above rounds to be collected upfront, while the diminishing returns above
$W = 4$ suggest that growing the window much further yields little $W = 4$ suggest that growing the window much further yields little
@@ -1511,7 +1511,7 @@ The dashed colored curves reproduce the cold-start results from
corresponding warm-start runs for the same window sizes corresponding warm-start runs for the same window sizes
$W \in \{3, 4, 5\}$. $W \in \{3, 4, 5\}$.
The remaining experimental parameters are unchanged: The remaining experimental parameters are unchanged:
the step size is fixed to $F = 1$, The step size is fixed to $F = 1$,
the inner \ac{bp} decoder is allowed up to $200$ iterations per the inner \ac{bp} decoder is allowed up to $200$ iterations per
window invocation, the black curve again gives the whole-block window invocation, the black curve again gives the whole-block
reference, and the physical error rate is swept from $p = 0.001$ to reference, and the physical error rate is swept from $p = 0.001$ to
@@ -1707,7 +1707,7 @@ $n_\text{iter} \in [32, 512]$.
All curves decrease monotonically with the iteration budget, but All curves decrease monotonically with the iteration budget, but
contrary to our expectation, none of them appears to fully saturate contrary to our expectation, none of them appears to fully saturate
within the swept range: even at $n_\text{iter} = 4096$, every curve within the swept range: Even at $n_\text{iter} = 4096$, every curve
still exhibits a noticeable downward slope. still exhibits a noticeable downward slope.
At $n_\text{iter} = 32$, the whole-block curve lies below both the At $n_\text{iter} = 32$, the whole-block curve lies below both the
$W=4$ and $W=5$ sliding-window curves. $W=4$ and $W=5$ sliding-window curves.
@@ -1729,7 +1729,7 @@ mirroring the behavior already observed in \Cref{fig:whole_vs_cold_vs_warm}.
These observations are largely consistent with the effective-iterations These observations are largely consistent with the effective-iterations
hypothesis put forward above. hypothesis put forward above.
The whole-block decoder eventually overtaking every windowed scheme The whole-block decoder eventually overtaking every windowed scheme
matches the prediction made there: with a sufficiently large matches the prediction made there: With a sufficiently large
iteration budget, the whole-block decoder reaches an error rate iteration budget, the whole-block decoder reaches an error rate
that none of the windowed schemes can beat, because of the more global that none of the windowed schemes can beat, because of the more global
nature of the considered constraints. nature of the considered constraints.
@@ -1767,7 +1767,7 @@ sliding-window approach is still at an advantage.
Having examined the effect of the window size $W$, we next turn to Having examined the effect of the window size $W$, we next turn to
the second windowing parameter, the step size $F$. the second windowing parameter, the step size $F$.
We carry out an investigation analogous to the one above: We carry out an investigation analogous to the one above:
we first compare warm- and cold-start decoding across the full range We first compare warm- and cold-start decoding across the full range
of physical error rates at a fixed iteration budget, and then we of physical error rates at a fixed iteration budget, and then we
examine the dependence on the iteration budget at a fixed physical examine the dependence on the iteration budget at a fixed physical
error rate. error rate.
@@ -1994,7 +1994,7 @@ At fixed $F$, the warm-start approach lies below
cold-start across the entire sweep, and at fixed cold-start across the entire sweep, and at fixed
warm or cold start, smaller $F$ produces a lower \ac{ler}. warm or cold start, smaller $F$ produces a lower \ac{ler}.
Both gaps grow as the physical error rate decreases: Both gaps grow as the physical error rate decreases:
the curves at $F = 1$ separate further from those at $F = 2$ and $F = 3$, The curves at $F = 1$ separate further from those at $F = 2$ and $F = 3$,
and the warm-start curves separate further from the cold-start ones. and the warm-start curves separate further from the cold-start ones.
In \Cref{fig:bp_f_over_iter}, all six curves again decrease In \Cref{fig:bp_f_over_iter}, all six curves again decrease
monotonically with the iteration budget, with no clear saturation monotonically with the iteration budget, with no clear saturation
@@ -2016,7 +2016,7 @@ With $W$ held fixed, decreasing $F$ enlarges the overlap between
consecutive windows from $W - F$ to $W - F + 1$ syndrome measurement rounds, so consecutive windows from $W - F$ to $W - F + 1$ syndrome measurement rounds, so
a smaller step size is beneficial for the same reason that a larger a smaller step size is beneficial for the same reason that a larger
window size is: window size is:
each \ac{vn} in an overlap region participates in more window Each \ac{vn} in an overlap region participates in more window
invocations, and the warm-start modification effectively accumulates invocations, and the warm-start modification effectively accumulates
iterations on it across these invocations. iterations on it across these invocations.
The widening of the warm/cold gap towards low iteration counts and The widening of the warm/cold gap towards low iteration counts and
@@ -2281,7 +2281,7 @@ This is the opposite of what we observed for plain \ac{bp}, where
warm-start improved upon cold-start at every parameter setting. warm-start improved upon cold-start at every parameter setting.
The gap between the warm- and cold-start curves additionally widens The gap between the warm- and cold-start curves additionally widens
as the physical error rate decreases: as the physical error rate decreases:
at the lowest sampled rate $p = 0.001$, the per-round \ac{ler} of the At the lowest sampled rate $p = 0.001$, the per-round \ac{ler} of the
warm-start runs is more than two orders of magnitude above that of warm-start runs is more than two orders of magnitude above that of
the corresponding cold-start runs. the corresponding cold-start runs.
In \Cref{fig:bpgd_w}, larger window sizes yield lower per-round In \Cref{fig:bpgd_w}, larger window sizes yield lower per-round
@@ -2300,13 +2300,13 @@ than its cold-start counterpart is surprising in light of the results
for plain \ac{bp}, where the warm-start modification was uniformly beneficial. for plain \ac{bp}, where the warm-start modification was uniformly beneficial.
The dependence on the window size in \Cref{fig:bpgd_w} is, on its own, The dependence on the window size in \Cref{fig:bpgd_w} is, on its own,
consistent with the same explanation that we gave for consistent with the same explanation that we gave for
\Cref{fig:whole_vs_cold}: larger windows expose the inner decoder to \Cref{fig:whole_vs_cold}: Larger windows expose the inner decoder to
a larger fraction of the constraints encoded in the detector error a larger fraction of the constraints encoded in the detector error
matrix at the time of decoding, and this benefits both warm- and matrix at the time of decoding, and this benefits both warm- and
cold-start decoding. cold-start decoding.
The dependence on the step size in \Cref{fig:bpgd_f}, however, is the The dependence on the step size in \Cref{fig:bpgd_f}, however, is the
opposite of the corresponding dependence under plain \ac{bp} opposite of the corresponding dependence under plain \ac{bp}
(\Cref{fig:bp_f_over_p}): for warm-start, smaller $F$ now degrades performance (\Cref{fig:bp_f_over_p}): For warm-start, smaller $F$ now degrades performance
rather than helps, even though smaller $F$ implies a larger overlap rather than helps, even though smaller $F$ implies a larger overlap
in both cases. in both cases.
@@ -2564,7 +2564,7 @@ the warm-start curves now show a clear reordering as $n_\text{iter}$
grows. grows.
At low iteration budgets the warm-start ordering matches the At low iteration budgets the warm-start ordering matches the
cold-start ordering, with $F = 1$ best and $F = 3$ worst, but at the cold-start ordering, with $F = 1$ best and $F = 3$ worst, but at the
largest iteration budget this ordering is fully inverted: warm-start largest iteration budget this ordering is fully inverted: Warm-start
$F = 1$ is now the worst and $F = 3$ the best. $F = 1$ is now the worst and $F = 3$ the best.
% [Interpretation] Figure 4.11 % [Interpretation] Figure 4.11
@@ -2596,7 +2596,7 @@ decoding performance.
The same mechanism explains the inversion of the step-size ordering The same mechanism explains the inversion of the step-size ordering
in \Cref{fig:bpgd_iter_F}. in \Cref{fig:bpgd_iter_F}.
At low iteration budgets, the ordering is set by the same overlap At low iteration budgets, the ordering is set by the same overlap
argument as for plain \ac{bp}: smaller $F$ implies a larger overlap argument as for plain \ac{bp}: Smaller $F$ implies a larger overlap
between consecutive windows, more shared messages, and therefore between consecutive windows, more shared messages, and therefore
better warm-start performance. better warm-start performance.
At large iteration budgets, the ordering is set by the premature hard At large iteration budgets, the ordering is set by the premature hard
@@ -2777,7 +2777,7 @@ since the decimation decisions were made based on the messages themselves.
\Cref{fig:bpgd_msg} repeats the experiment of \Cref{fig:bpgd_wf} \Cref{fig:bpgd_msg} repeats the experiment of \Cref{fig:bpgd_wf}
with the modified warm-start procedure that carries over only the with the modified warm-start procedure that carries over only the
\ac{bp} messages. \ac{bp} messages.
All other experimental parameters are unchanged: the maximum number All other experimental parameters are unchanged: The maximum number
of inner \ac{bp} iterations is $n_\text{iter} = 5000$, and the of inner \ac{bp} iterations is $n_\text{iter} = 5000$, and the
physical error rate is swept from $p = 0.001$ to $p = 0.004$ in steps physical error rate is swept from $p = 0.001$ to $p = 0.004$ in steps
of $0.0005$. of $0.0005$.
@@ -2810,7 +2810,7 @@ the warm-start regression observed in \Cref{fig:bpgd_wf},
and warm-start now consistently outperforms cold-start. and warm-start now consistently outperforms cold-start.
The dependence on the window size and the step size also recovers The dependence on the window size and the step size also recovers
the qualitative behavior we observed for plain \ac{bp} in the qualitative behavior we observed for plain \ac{bp} in
\Cref{fig:whole_vs_cold_vs_warm,fig:bp_f_over_p}: a larger overlap \Cref{fig:whole_vs_cold_vs_warm,fig:bp_f_over_p}: A larger overlap
between consecutive windows, achieved either by enlarging $W$ or by between consecutive windows, achieved either by enlarging $W$ or by
decreasing $F$, both improves the absolute decoding performance and decreasing $F$, both improves the absolute decoding performance and
increases the warm-start advantage over cold-start. increases the warm-start advantage over cold-start.
@@ -2994,7 +2994,7 @@ cold-start curves across the entire range of $n_\text{iter}$ available to us.
\Cref{fig:bpgd_msg_iter} repeats the experiment of \Cref{fig:bpgd_msg_iter} repeats the experiment of
\Cref{fig:bpgd_iter} with the modified warm-start procedure that \Cref{fig:bpgd_iter} with the modified warm-start procedure that
carries over only the \ac{bp} messages. carries over only the \ac{bp} messages.
All other experimental parameters are unchanged: the physical error All other experimental parameters are unchanged: The physical error
rate is fixed at $p = 0.0025$ and the iteration budget is swept over rate is fixed at $p = 0.0025$ and the iteration budget is swept over
$n_\text{iter} \in \{32, 128, 256, 512, 1024, 1536, 2048, 2560, $n_\text{iter} \in \{32, 128, 256, 512, 1024, 1536, 2048, 2560,
3072, 3584, 4096\}$. 3072, 3584, 4096\}$.
@@ -3026,7 +3026,7 @@ initialization no longer freezes any \acp{vn} in the next window.
The dependence of this benefit on $W$ and $F$ also recovers the The dependence of this benefit on $W$ and $F$ also recovers the
pattern observed for plain \ac{bp} in pattern observed for plain \ac{bp} in
\Cref{fig:whole_vs_cold_vs_warm,fig:bp_f_over_p}: \Cref{fig:whole_vs_cold_vs_warm,fig:bp_f_over_p}:
larger overlap, achieved by larger $W$ or smaller $F$, yields more Larger overlap, achieved by larger $W$ or smaller $F$, yields more
effective extra iterations and therefore a larger warm-start gain. effective extra iterations and therefore a larger warm-start gain.
% BPGD conclusion % BPGD conclusion
@@ -3048,7 +3048,7 @@ cold-start that follows the same behavior as for plain \ac{bp} with
regard to overlap. regard to overlap.
A second observation specific to \ac{bpgd} is that its iteration A second observation specific to \ac{bpgd} is that its iteration
requirements are substantially larger than those of plain \ac{bp}: requirements are substantially larger than those of plain \ac{bp}:
the per-round \ac{ler} drops sharply only once the iteration budget The per-round \ac{ler} drops sharply only once the iteration budget
is on the order of the number of \acp{vn} in each window. is on the order of the number of \acp{vn} in each window.
Future work could include a softer treatment of the decimation state Future work could include a softer treatment of the decimation state

10
src/thesis/chapters/5_conclusion_and_outlook.tex
View File

@@ -7,7 +7,7 @@ This thesis investigates decoding under \acp{dem} for fault-tolerant
\ac{qec}, with a focus on low-latency decoding methods for \ac{qldpc} codes. \ac{qec}, with a focus on low-latency decoding methods for \ac{qldpc} codes.
The repetition of the syndrome measurements, especially under The repetition of the syndrome measurements, especially under
consideration of circuit-level noise, leads to a significant increase consideration of circuit-level noise, leads to a significant increase
in decoding complexity: in our experiments on the $\llbracket in decoding complexity: In our experiments on the $\llbracket
144,12,12 \rrbracket$ \ac{bb} code with $12$ syndrome extraction 144,12,12 \rrbracket$ \ac{bb} code with $12$ syndrome extraction
rounds, the check matrix grows from 144 \acp{vn} and 72 rounds, the check matrix grows from 144 \acp{vn} and 72
\acp{cn} to 9504 \acp{vn} and 1008 \acp{cn}. \acp{cn} to 9504 \acp{vn} and 1008 \acp{cn}.
@@ -46,18 +46,18 @@ min-sum algorithm.
For standard min-sum \ac{bp}, the warm start is consistently For standard min-sum \ac{bp}, the warm start is consistently
beneficial to the cold start, across the considered parameter ranges. beneficial to the cold start, across the considered parameter ranges.
The size of the gain depends on the overlap between consecutive The size of the gain depends on the overlap between consecutive
windows: enlarging $W$ or shrinking $F$, both of which enlarge the windows: Enlarging $W$ or shrinking $F$, both of which enlarge the
overlap, result in larger gains of the warm-start. overlap, result in larger gains of the warm-start.
We observe that the underlying mechanism is an effective increase in We observe that the underlying mechanism is an effective increase in
the number of \ac{bp} iterations spent on the \acp{vn} in the overlap 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 region: Each such \ac{vn} is processed by multiple consecutive window
invocations, and the warm start lets these invocations accumulate invocations, and the warm start lets these invocations accumulate
iterations on the same \acp{vn} rather than restarting from scratch. iterations on the same \acp{vn} rather than restarting from scratch.
The gain was most pronounced at low numbers of maximum iterations, where The gain was most pronounced at low numbers of maximum iterations, where
every additional iteration carries proportionally more information. every additional iteration carries proportionally more information.
For \ac{bpgd}, we note that more information is available in the For \ac{bpgd}, we note that more information is available in the
overlap region of a window: in addition to the \ac{bp} messages, 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. there is information about which \acp{vn} were decimated and to what value.
Passing this decimation information to the next window in addition to Passing this decimation information to the next window in addition to
the messages turned out to worsen the performance considerably, which the messages turned out to worsen the performance considerably, which
@@ -66,7 +66,7 @@ overlap region.
Restricting the warm start to the \ac{bp} messages alone, removed this effect. Restricting the warm start to the \ac{bp} messages alone, removed this effect.
The resulting message-only warm start recovered a consistent The resulting message-only warm start recovered a consistent
improvement over cold-start that followed the same qualitative improvement over cold-start that followed the same qualitative
behaviour as for standard \ac{bp}: larger overlap, achieved by larger behaviour as for standard \ac{bp}: Larger overlap, achieved by larger
$W$ or smaller $F$, yielded a larger gain, and the $W$ or smaller $F$, yielded a larger gain, and the
performance difference is most pronounced at low numbers of maximum iterations. performance difference is most pronounced at low numbers of maximum iterations.

2
src/thesis/chapters/abstract.tex
View File

@@ -49,7 +49,7 @@ For both standard \ac{bp} and \ac{bpgd} decoding, the warm-start
initialization provides a consistent improvement across all examined initialization provides a consistent improvement across all examined
parameter settings. parameter settings.
We attribute this to an effective increase in \ac{bp} iterations on We attribute this to an effective increase in \ac{bp} iterations on
variable nodes in the overlap regions: each such VN is processed by variable nodes in the overlap regions: Each such VN is processed by
multiple consecutive windows, and warm-starting lets these multiple consecutive windows, and warm-starting lets these
invocations accumulate iterations rather than restart from scratch. invocations accumulate iterations rather than restart from scratch.
Crucially, the warm-start modification incurs no additional Crucially, the warm-start modification incurs no additional