Add sliding-window decoding slide

.gitmodules

@@ -1,3 +1,9 @@
 [submodule "lib/cel-slides-template-2025"]
 	path = lib/cel-slides-template-2025
-	url = git@gitlab.kit.edu:kit/cel/misc/cel-slides-template-2025.git
+	url = ssh://git@100.123.176.93:2222/an.tsouchlos/cel-slides-template-2025.git
+[submodule "lib/cel-thesis"]
+	path = lib/cel-thesis
+	url = ssh://git@100.123.176.93:2222/an.tsouchlos/cel-thesis.git
+[submodule "lib/latex-common"]
+	path = lib/latex-common
+	url = ssh://git@100.123.176.93:2222/an.tsouchlos/latex-common.git

.tmux_session.sh

@@ -0,0 +1,8 @@
+#!/bin/bash
+SESSION=$1
+
+tmux send-keys -t "$SESSION:1" "./.setup_local_env.sh" Enter
+# tmux send-keys -t "$SESSION:1" "export TEXINPUTS=./lib/cel-slides-template-2025:\$TEXINPUTS" C-m
+tmux send-keys -t "$SESSION:1" "trap './.clean_local_env.sh' EXIT" Enter
+tmux send-keys -t "$SESSION:1" "nvim" Enter
+tmux send-keys -t "$SESSION:1" "\\ll" Enter

Makefile

@@ -1,11 +1,13 @@
-DOCUMENTS := $(patsubst src/%/main.tex,build/%.pdf,$(wildcard src/*/main.tex))
+DOCUMENTS := build/midterm_presentation.pdf build/thesis.pdf
+# DOCUMENTS := build/thesis.pdf
 
 .PHONY: all
 all: $(DOCUMENTS)
 
 build/%.pdf: src/%/main.tex build/prepared
-	latexmk $<
+	TEXINPUTS=./lib/cel-slides-template-2025:$(dir $<):$$TEXINPUTS \
-	mv build/main.pdf $@
+	latexmk -outdir=build/$* $<
+	mv build/thesis/main.pdf $@
 
 build/prepared:
 	mkdir -p build

src/final_presentation/gen_sliding_window_overlap_image.py

@@ -0,0 +1,195 @@
+import warnings
+from typing import Sequence
+
+import numpy as np
+import matplotlib.pyplot as plt
+import matplotlib.patches as pt
+from scipy.sparse import csc_matrix
+
+from quits.decoder import spacetime
+from quits.decoder import detector_error_model_to_matrix
+
+from quits.qldpc_code import BbCode
+from quits import ErrorModel, CircuitBuildOptions
+
+
+def build_bb_circuit(N: int, num_rounds: int, p: float):
+    # fmt: off
+    if N == 72:
+        code = BbCode(l=6, m=6, A_x_pows=[3], A_y_pows=[1, 2], B_x_pows=[1, 2], B_y_pows=[3])
+    elif N == 90:
+        code = BbCode(l=15, m=3, A_x_pows=[9], A_y_pows=[1, 2], B_x_pows=[2, 7], B_y_pows=[0])
+    elif N == 108:
+        code = BbCode(l=9, m=6, A_x_pows=[3], A_y_pows=[1, 2], B_x_pows=[1, 2], B_y_pows=[3])
+    elif N == 144:
+        code = BbCode(l=12, m=6, A_x_pows=[3], A_y_pows=[1, 2], B_x_pows=[1, 2], B_y_pows=[3])
+    elif N == 288:
+        code = BbCode(l=12, m=12, A_x_pows=[3], A_y_pows=[2, 7], B_x_pows=[1, 2], B_y_pows=[3])
+    elif N == 360:
+        code = BbCode(l=30, m=6, A_x_pows=[9], A_y_pows=[1, 2], B_x_pows=[25, 26], B_y_pows=[3])
+    elif N == 756:
+        code = BbCode(l=21, m=18, A_x_pows=[3], A_y_pows=[10, 17], B_x_pows=[3, 19], B_y_pows=[5])
+    else:
+        assert False, "Unsupported code size"
+    # fmt: on
+
+    circuit = code.build_circuit(
+        error_model=ErrorModel(p, p, p, p),
+        num_rounds=num_rounds,
+        basis="Z",
+        circuit_build_options=CircuitBuildOptions(),
+        seed=1,
+    )
+
+    return code, circuit
+
+
+
+
+def compute_num_windows(num_rounds: int, W: int, F: int):
+    """
+    This was extracted from the function `sliding_window_circuit_mem()` of
+    `quits.decoder`.
+    """
+    if 2 + num_rounds - W >= 0:
+        # num_cor_rounds = num of windows before the last window
+        num_cor_rounds = (2 + num_rounds - W) // F
+
+        # we can slide one more window if the remaining rounds > W
+        if (2 + num_rounds - W) % F != 0:
+            num_cor_rounds += 1
+    else:
+        num_cor_rounds = 0
+        warnings.warn(
+            "Window size larger than the syndrome extraction rounds: Doing"
+            " whole history correction"
+        )
+
+    return num_cor_rounds + 1
+
+
+def get_overlap_info(
+    col_start_indices: Sequence, W: int, F: int, m: int, win_check_set: Sequence
+):
+    def i_B(k: int):
+        return col_start_indices[k]
+
+    def i_E(k: int):
+        return i_B(k) + win_check_set[k].shape[1]
+
+    def j_B(k: int):
+        return F * (k) * m
+
+    def j_E(k: int):
+        return (F * k + W) * m
+
+    num_windows = len(win_check_set)
+
+    overlap_begin_positions = []
+    for k in range(num_windows - 1):
+        overlap_begin_positions.append((j_B(k + 1) - j_B(k), i_B(k + 1) - i_B(k)))
+
+    overlap_end_positions = []
+    for k in range(1, num_windows):
+        overlap_end_positions.append((j_E(k - 1) - j_B(k), i_E(k - 1) - i_B(k)))
+
+    return overlap_begin_positions, overlap_end_positions
+
+
+def reconstruct_window_start_col_indices(win_observable_set: Sequence):
+    """
+    This function effectively just reconstructs the `col_min` values of each
+    window, from the `spacetime()` function of `quits.decoder`.
+    """
+    num_windows = len(win_observable_set)
+
+    col_mins = [0]
+
+    for k in range(num_windows - 1):
+        col_mins.append(col_mins[-1] + win_observable_set[k].shape[1])
+
+    return col_mins
+
+
+
+
+num_rounds = 12
+N = 72
+p = 0.005
+W = 5
+F = 3
+
+#
+# Get detector error matrix and split it into windows
+#
+
+code, circuit = build_bb_circuit(N, num_rounds, p)
+model = circuit.detector_error_model(decompose_errors=False)
+check_matrix, observable_matrix, priors = detector_error_model_to_matrix(model)
+
+num_windows = compute_num_windows(num_rounds, W, F)
+win_check_set, win_observable_set, win_priors_set, win_update = spacetime(
+    circuit, code.hz, W, F, num_windows - 1
+)
+
+col_start_indices = reconstruct_window_start_col_indices(win_observable_set)
+
+#
+# Paint rectangles
+#
+
+custom_colors = [
+    (162/255, 34/255, 35/255),
+    (223/255, 155/255, 27/255),
+    (70/255, 100/255, 170/255),
+    (163/255, 16/255, 124/255),
+]
+
+fig, ax = plt.subplots(1, 1, figsize=(10, 10))
+ax.spy(check_matrix.toarray())
+
+colors = [custom_colors[i % len(custom_colors)] for i in range(num_windows)]
+
+m = code.hz.shape[0]
+# for win_idx in range(num_windows):
+#     col_start_idx = col_start_indices[win_idx]
+#     row_start_idx = win_idx * F * m
+#
+#     ax.add_patch(
+#         pt.Rectangle(
+#             (col_start_idx, row_start_idx),
+#             win_check_set[win_idx].shape[1],
+#             win_check_set[win_idx].shape[0],
+#             fc="none",
+#             ec=colors[win_idx],
+#         )
+#     )
+
+# overlap_begin_positions, overlap_end_positions = get_overlap_info(
+#     col_start_indices, W, F, m, win_check_set
+# )
+
+# for k in range(len(win_check_set) - 1):
+#     ax.add_patch(
+#         pt.Rectangle(
+#             (
+#                 overlap_begin_positions[k][1] + col_start_indices[k],
+#                 overlap_begin_positions[k][0] + F * k * m,
+#             ),
+#             win_check_set[k].shape[1] - overlap_begin_positions[k][1],
+#             win_check_set[k].shape[0] - overlap_begin_positions[k][0],
+#             fc=colors[k],
+#             ec=colors[k],
+#             alpha=0.3,
+#         )
+#     )
+
+
+ax.set_xticks([])
+ax.set_yticks([])
+
+fig.savefig('72_bb_dem_no_windows.pdf', bbox_inches='tight')
+
+plt.show()
+
+

src/fundamentals/main.tex

@@ -64,8 +64,7 @@
 \Ac{qec} is a field of research combining quantum mechanics and
 ``classical'' communications engineering.
 This chapter provides the relevant theoretical background on both of
-these topics and subsequently, building on top of this, introduces the
+these topics and subsequently introduces the the fundamentals of \ac{qec}.
-the fundamentals of \ac{qec}.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % TODO: Is Quantum Information Theory the correct title here? Would someth

src/midterm_presentation/main.tex

@@ -53,7 +53,7 @@
 
 \title{Fault Tolerant Quantum Error Correction}
 \subtitle{Master's Thesis Midterm Presentation}
-\author[Andreas]{Andreas Tsouchlos}
+\author[ Andreas]{Andreas Tsouchlos}
 \date[]{}
 
 \DeclareFieldFormat{note}{}
@@ -62,7 +62,7 @@
 \DeclareFieldFormat{doi}{}
 \DeclareFieldFormat[article,book,inproceedings]{urldate}{}
 
-\addbibresource{MA.bib}
+\addbibresource{src/midterm_presentation/MA.bib}
 
 %
 %
@@ -148,7 +148,7 @@
 
 \DeclareAcronym{qldpc}{
     short=QLDPC,
-    long=quantum low - density parity - check,
+    long=quantum low-density parity-check,
 }
 
 \DeclareAcronym{scldpc}{
@@ -156,6 +156,11 @@
     long=spatially-coupled low-density parity-check
 }
 
+\DeclareAcronym{ler}{
+    short=LER,
+    long=logical error rate,
+}
+
 %
 %
 % Document body
@@ -401,7 +406,7 @@
                 \citereferencemanual{NC10}
             }
             \visible<3>{
-            \item Superposition $\rightarrow$ multiple solutions to the
+            \item Superposition $\rightarrow$ Multiple solutions to the
                 decoding problem
                 (\schlagwort{quantum degeneracy})
             \citereferencemanual{RWB$^+$20}}
@@ -520,7 +525,7 @@
         \begin{subfigure}{0.15\textwidth}
             \centering
             \begin{align*}
-                \bm{H} \bm{y}^\text{T} = \bm{s}
+                \bm{H} \bm{y}^\text{T} = \bm{H} \bm{e}^\text{T} = \bm{s}
             \end{align*}
             \vspace*{-5mm}
         \end{subfigure}
@@ -1568,18 +1573,20 @@
                     \citereferencemanual{HP23}
                     \citereferencemanual{GCR24}
             \end{itemize}
-        \item Degraded \ac{bp} performance addressed with
+            \visible<2>{
-            modification or extension
+            \item Degraded \ac{bp} performance addressed with
-            \begin{itemize}
+                modification or extension
-                \item \Ac{osd} post-processing
+                \begin{itemize}
-                    \citereferencemanual{RWB$^+$20}
+                    \item \Ac{osd} post-processing
-                \item Guided decimation \citereferencemanual{GCR24}
+                        \citereferencemanual{RWB$^+$20}
-                \item Neural approaches
+                    \item Guided decimation \citereferencemanual{GCR24}
-                    \citereferencemanual{KL22}
+                    \item Neural approaches
-                    \citereferencemanual{MSL$^+$25}
+                        \citereferencemanual{KL22}
-                \item Ensemble decoding
+                        \citereferencemanual{MSL$^+$25}
-                    \citereferencemanual{KSW$^+$25}
+                    \item Ensemble decoding
-            \end{itemize}
+                        \citereferencemanual{KSW$^+$25}
+                \end{itemize}
+            }
     \end{itemize}
 
     \vspace*{15mm}
@@ -1737,7 +1744,7 @@
 \begin{frame}
     \frametitle{Future Work}
 
-    \vspace*{-12mm}
+    \vspace*{-15mm}
 
     \begin{itemize}
         \item Completed work
@@ -1745,6 +1752,7 @@
                 \item Review literature on fault-tolerant \ac{qec}
                     using \acp{dem}
                 \item Identify research gap
+                \item Familiarize with software toolboxes
             \end{itemize}
             \vspace*{7mm}
         \item Research gap
@@ -1787,32 +1795,166 @@
 
 % TODO: Organize sections properly
 %%%%%%%%%%%%%%%%
-\section{Simulation Methodology}
+\section{Remarks on Evaluation}
-\label{sec:Simulation Methodology}
+\label{sec:Remarks on Evaluation}
 
 %%%%%%%%%%%%%%%%
-\subsection{The Code and Other Parameters}
+\subsection{Figures of Merit}
-\label{subsec:The Code and Other Parameters}
+\label{subsec:Figures of Merit}
 
 \begin{frame}
-    \frametitle{Noise Models and Figures of Merit}
+    \frametitle{Performance Evaluation}
 
-    \begin{itemize}
+    % - Gong et al. don't actually analyze the latency ->
-        \item \red{For circuit-level noise, often, all error probabilities
+    %   Benchmarking against other methods would be interesting
-                are set to the same value for simulations
-            \citereferencemanual{FSG09}}
-        \item \red{There are other approaches (e.g., SDMB noise, SI noise)
-            \citereferencemanual{DTB$^+$25}}
-    \end{itemize}
 
-    \vspace*{10mm}
+    % \item For circuit-level noise, use same
+    % \schlagwort{physical error rate} for all error
+    % locations \citereferencemanual{FSG09}
 
-    \begin{itemize}
+    \vspace*{-15mm}
-        \item \red{Footprint plots}
-        \item \red{Other figure of merit (Look into ECCentric?)}
-    \end{itemize}
 
-    \vspace*{15mm}
+    \begin{minipage}{0.35\textwidth}
+        \only<1>{
+            \begin{itemize}
+                \item Independent variables
+                    \begin{itemize}
+                        \item Physical error rate
+                        \item CNOT infidelity
+                        \item Total qubit count
+                        \item \ldots
+                    \end{itemize}
+            \end{itemize}
+        }
+        \only<2->{
+            \begin{itemize}
+                \item Independent variables
+                    \begin{itemize}
+                        \item \textbf{Physical error rate}
+                        \item CNOT infidelity
+                        \item \textbf{Total qubit count}
+                        \item \ldots
+                    \end{itemize}
+            \end{itemize}
+        }
+    \end{minipage}%
+    \begin{minipage}{0.65\textwidth}
+        \begin{itemize}
+                \only<1>{
+                \item Noise models
+                    \begin{itemize}
+                        \item Standard circuit-based depolarizing noise
+                            \citereferencemanual{FSG09}
+                        \item Superconductor inspired (SI1000)
+                            \citereferencemanual{GNF$^+$21}
+                        \item Entangling Measurements (EM3)
+                            \citereferencemanual{GNF$^+$21}
+                        \item \ldots
+                    \end{itemize}
+                }
+                \only<2->{
+                \item Noise models
+                    \begin{itemize}
+                        \item \textbf{Standard circuit-based depolarizing noise}
+                            \citereferencemanual{FSG09}
+                        \item Superconductor inspired (SI1000)
+                            \citereferencemanual{GNF$^+$21}
+                        \item Entangling Measurements (EM3)
+                            \citereferencemanual{GNF$^+$21}
+                        \item \ldots
+                    \end{itemize}
+                }
+        \end{itemize}
+    \end{minipage}
+
+    \vspace{5mm}
+
+    \visible<3->{
+        \begin{itemize}
+            \item Degeneracy, information stored in correlations
+                $\rightarrow$ Consider \schlagwort{\acl{ler}} (LER)
+        \end{itemize}
+    }
+    \visible<4->{
+        \begin{itemize}
+            \item Types of benchmarking plots
+        \end{itemize}
+
+        \vspace*{5mm}
+
+        \begin{figure}[H]
+            \centering
+            \begin{subfigure}{0.35\textwidth}
+                \centering
+                \begin{tikzpicture}
+                    \begin{axis}[
+                            domain=-5:5,
+                            width=7cm,
+                            height=5.5cm,
+                            xticklabels=\empty,
+                            yticklabels=\empty,
+                            xlabel={Physical error rate},
+                            xlabel style={yshift=5mm},
+                            ylabel={LER},
+                            ylabel style={yshift=-5mm},
+                            grid,
+                        ]
+
+                        \addplot+[
+                            mark=none,
+                            kit-red,
+                            line width=2pt,
+                        ]
+                        table[row sep=crcr] {
+                            x y \\
+                            1.134800559068837 0.5575221183357257 \\
+                            2.0632737437615223 0.9764009116710485 \\
+                            2.861072612292603 1.7787608707489788 \\
+                            3.7551580964997053 2.8407080379684153 \\
+                            4.264098875196703 3.513274267363004 \\
+                            4.573589936760932 3.9911505302955272 \\
+                            4.903713970055305 4.268436552233389 \\
+                        };
+                    \end{axis}
+                \end{tikzpicture}
+            \end{subfigure}%
+            \begin{subfigure}{0.35\textwidth}
+                \centering
+                \begin{tikzpicture}
+                    \begin{axis}[
+                            domain=-5:5,
+                            width=7cm,
+                            height=5.5cm,
+                            xticklabels=\empty,
+                            yticklabels=\empty,
+                            xlabel={Total qubit count},
+                            xlabel style={yshift=5mm},
+                            ylabel={LER},
+                            ylabel style={yshift=-5mm},
+                            grid,
+                        ]
+
+                        \addplot+[
+                            mark=none,
+                            kit-blue,
+                            line width=2pt,
+                        ]
+                        table[row sep=crcr] {
+                            x y \\
+                            1.147643096789246 3.8430493581808607 \\
+                            1.7245658892318043 2.762331811591747 \\
+                            2.3573205843145306 2.3587443650766753 \\
+                            2.9156332708646624 1.560537992857378 \\
+                            3.6352360073136527 1.0403588210329737 \\
+                            4.392060012189421 0.7130042787942606 \\
+                        };
+                    \end{axis}
+                \end{tikzpicture}
+            \end{subfigure}
+        \end{figure}
+    }
+
+    \vspace*{4mm}
 
     \addreferencesmanual
     {FSG09}{
@@ -1820,30 +1962,44 @@
         ``High-threshold universal quantum computation on the surface
         code,'' \emph{Physical Review}, 2009.
     }
-    {DTB$^+$25}{
+    {GNF$^+$21}{
-        P.- J. H. S. Derks et al., ``Designing fault-tolerant
+        C. Gidney et al., ``A Fault-Tolerant Honeycomb Memory'',
-        circuits using detector error models,'' \emph{Quantum}, 2025.
+        \emph{Quantum}, 2021.
     }
     \stopreferencesmanual
 \end{frame}
 
 %%%%%%%%%%%%%%%%
-\subsection{Proposed Methodology}
+\subsection{Conclusion and Outlook}
-\label{subsec:Proposed Methodology}
+\label{subsec:Conclusion and Outlook}
 
 \begin{frame}
-    \frametitle{Conclusion}
+    \frametitle{Conclusion and Outlook}
 
-    \vspace*{-15mm}
+    \vspace*{-10mm}
 
     \begin{minipage}[c]{0.65\textwidth}
         \begin{itemize}
-            \item \red{Noise model}
+            \item Problem setting
-            \item \red{Memory or stability experiment}
+                \begin{itemize}
-            \item \red{Figure of merit: Footprint plot}
+                    \item Research area: Decoder design for \acp{dem}
-            \item \red{Comparison with BB code also simulated by
+                        under circuit-level noise
-                \citereferencemanual{GCR24}}
+                    \item Research gap: Consideration of \acp{dem} as
-            \item \red{Comparison with surface code}
+                        \ac{scldpc} codes
+                \end{itemize}
+                \vspace*{5mm}
+            \item Future work
+                \begin{itemize}
+                    \item Modify existing decoder to pass soft information
+                    \item Test different \ac{bp} variations
+                    \item \ldots
+                \end{itemize}
+                \vspace*{5mm}
+            \item Parameters
+                \begin{itemize}
+                    \item Use standard depolarizing noise for comparability
+                    \item Compare performance with other \ac{bb} code decoders
+                \end{itemize}
         \end{itemize}
     \end{minipage}%
     \begin{minipage}[c]{0.35\textwidth}
@@ -1852,9 +2008,13 @@
         \begin{figure}[H]
             \centering
 
-            \begin{tikzpicture}[every node/.style={scale=10}]
+            \vspace*{-25mm}
-                \node at (0, 0)
+            \begin{tikzpicture}
+                \node[scale=10] at (0, 0)
                 {\textcolor{kit-blue}{{\fontfamily{phv}\selectfont ?}}};
+
+                \node[align=center] at (0,-5) {Thank you for your
+                attention! \\ Any questions?};
             \end{tikzpicture}
         \end{figure}
     \end{minipage}
@@ -1868,7 +2028,7 @@
 \begin{frame}
     \frametitle{System Level Overview}
 
-    \vspace*{-15mm}
+    \vspace*{-10mm}
 
     \begin{figure}[H]
         \centering
@@ -1933,7 +2093,7 @@
         \end{subfigure}
     \end{figure}
 
-    % \vspace*{-2mm}
+    \vspace*{5mm}
 
     \addreferencesmanual
     {ZZC$^+$23}{
@@ -1952,11 +2112,30 @@
 \begin{frame}
     \frametitle{Guided Decimation Guessing Decoding}
 
-    \begin{itemize}
+    \begin{minipage}{0.57\textwidth}
-        \item \red{Explain paper}
+        \begin{itemize}
-    \end{itemize}
+            \item BP guided decimation (BPGD) \\
+                $\rightarrow$ Iteratively fix most reliable variable node (VN)
+                \vspace*{10mm}
+            \item \schlagwort{Guided decimation guessing} (GDG)
+                \citereferencemanual{GCR24}
+                \begin{itemize}
+                    \item Choose VN with \schlagwort{lowest}
+                        log-likelihood ratio
+                    \item Choose VN to fix based on \schlagwort{LLR history}
+                    \item Explore both VN values in parallel
+                        (\schlagwort{guessing})
+                \end{itemize}
+        \end{itemize}
+    \end{minipage}%
+    \begin{minipage}{0.43\textwidth}
+        \begin{figure}[H]
+            \centering
+            \includegraphics[scale=1.3]{res/gdg.pdf}
+        \end{figure}
+    \end{minipage}%
 
-    \vspace*{25mm}
+    \vspace*{30mm}
 
     \addreferencesmanual
     {GCR24}{
@@ -1967,6 +2146,52 @@
     \stopreferencesmanual
 \end{frame}
 
+\begin{frame}
+    \frametitle{The Quantum Error Correcting Landscape}
+
+    \vspace*{-10mm}
+
+    \begin{itemize}
+        \item Taxonomy of main QEC code families \citereferencemanual{SPG$^+$25}
+    \end{itemize}
+
+    \vspace*{2mm}
+
+    \begin{figure}[H]
+        \centering
+        \includegraphics[scale=2.5]{res/taxonomy.pdf}
+    \end{figure}
+
+    \begin{itemize}
+        \item Surface code is the industry standard for
+            experimental implementations, but has poor encoding
+            efficiency \citereferencemanual{BCG$^+$24}
+        \item \Ac{qldpc} codes particularly interesting because of
+            \begin{itemize}
+                \item Constant overhead scaling \citereferencemanual{Got14}
+                \item Linear distance scaling \citereferencemanual{BCG$^+$24}
+            \end{itemize}
+    \end{itemize}
+
+    \vspace*{15mm}
+
+    \addreferencesmanual
+    {SPG$^+$25}{
+        A. Swierkowska et al., ``ECCentric: An Empirical
+        Analysis of Quantum Error Correction Codes'',
+        \emph{arXiv:2511.01062v1}, 2025.
+    }
+    {BCG$^+$24}{
+        S. Bravyi et al., ``High-threshold and low-overhead
+        fault-tolerant quantum memory,'' \emph{Nature}, 2024.
+    }
+    {Got14}{
+        D. Gottesman, ``Fault-Tolerant Quantum Computation with
+        Constant Overhead'', \emph{arXiv:1310.2984}, 2014.
+    }
+    \stopreferencesendmanual
+\end{frame}
+
 % TODO: Is this really necessary?
 % \begin{frame}
 %     \frametitle{The Quantum Error Correction Landscape}

src/qec-meeting-presentation/main.tex

@@ -0,0 +1,417 @@
+\documentclass[xcolor=dvipsnames,t,aspectratio=169]{beamer}
+\setbeamersize{text margin left=1cm,text margin right=1cm,}
+
+\usepackage{subcaption}
+\usepackage{minted}
+\usepackage[minted, most]{tcolorbox}
+\usepackage{amsmath}
+\usepackage{bm}
+\usepackage[dvipsnames]{xcolor}
+\usepackage{siunitx}
+\usepackage{graphics}
+\usepackage{xfp}
+\usepackage{quantikz}
+
+\usepackage[T1]{fontenc}
+\usepackage[scaled=.92]{helvet}
+
+\usepackage{tikz}
+\usetikzlibrary{positioning, arrows.meta, shapes.misc,
+decorations.pathreplacing, calc}
+
+\usepackage{pgfplots}
+\usepackage{pgffor}
+\usepgfplotslibrary{groupplots}
+
+% \usetikzlibrary{external}
+% \tikzexternalize
+
+%
+%
+% Set up the theme
+%
+%
+
+% General and beamer options
+
+\makeatletter
+\addtobeamertemplate{author}{\centering}{}
+\addtobeamertemplate{date}{\vspace*{-0.85cm}\centering}{}
+\makeatother
+
+\setbeamertemplate{enumerate items}[circle]
+\setbeamertemplate{itemize items}[circle]
+\setbeamercolor{title}{fg=black}
+\setbeamercolor{frametitle}{fg=black}
+
+\setbeamertemplate{footline}[frame number]
+\setbeamertemplate{navigation symbols}{}
+
+% \AtBeginSection[]{
+%     \begin{frame}
+%         \frametitle{Overview}
+%         \tableofcontents[currentsection]
+%     \end{frame}
+% }
+%
+% \AtBeginSubsection[]{
+%     \begin{frame}
+%         \frametitle{Overview}
+%         \tableofcontents[currentsubsection]
+%     \end{frame}
+% }
+
+\fontfamily{cmss}\selectfont
+
+\newcommand\wider[2][3em]{%
+    \makebox[\linewidth][c]{%
+        \begin{minipage}{\dimexpr\textwidth+#1\relax}
+            \raggedright#2
+        \end{minipage}%
+    }%
+}
+
+% Code listings
+
+\usemintedstyle{gruvbox-dark}
+\definecolor{gruvbox-bg}{HTML}{282828}
+
+% PGF color scheme
+
+\input{lib/latex-common/common.tex}
+\pgfplotsset{colorscheme/rocket}
+
+%
+%
+% Configure document
+%
+%
+
+\title{Soft-Information Aware Sliding-Window Decoding}
+\author{Andreas Tsouchlos}
+\date{March 26, 2026}
+
+%
+%
+% Custom commands
+%
+%
+
+\newcommand{\red}[1]{\textcolor{red}{#1}}
+
+\newcommand{\linkpython}[1]{
+    \scalebox{0.7}{
+        \begingroup
+        \setbox0=\hbox{\includegraphics[scale=0.18]{\res/python-logo.pdf}}%
+        \parbox{\wd0}{\box0}\endgroup
+        \hspace{1mm}
+        \colorbox{gruvbox-bg}{%
+            \textcolor{white}{%
+                \texttt{#1}%
+            }%
+        }%
+    }%
+}
+
+%
+%
+% Content
+%
+%
+
+\begin{document}
+
+\frame[plain]{\titlepage}
+\setcounter{framenumber}{0}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Motivation and Core Idea}
+\label{sec:Motivation and Core Idea}
+
+\newsavebox{\innercircuit}
+\begin{frame}
+    \frametitle{Sliding-Window Decoding for Fault-Tolerant QEC}
+
+    % \vspace*{-5mm}
+
+    \begin{figure}
+        \scalebox{0.8}{
+            \savebox{\innercircuit}{
+                \begin{tikzpicture}[
+                        block/.style={draw, rectangle, minimum
+                            height=1cm, minimum width=1.2cm, align=center,
+                        fill=white, font=\small},
+                        container/.style={draw, thick, inner sep=0.6cm}
+                    ]
+                    % Internal blocks
+                    \node (se1) [block]
+                    {$\text{SE}_1$ \\ + noise};
+                    \node (se2) [block, right=0.5cm of se1]
+                    {$\text{SE}_2$ \\ + noise};
+                    \node (dots) [right=0.3cm of se2] {\dots};
+
+                    % Wires
+                    \draw (se1.west) -- ++(-0.4,0);
+                    \draw (se1.east) -- (se2.west);
+                    \draw (se2.east) -- (dots.west);
+                    \draw (dots.east) -- ++(0.4,0);
+
+                    % The "Circuit" container
+                    \node (box) [
+                        container, fit=(se1) (se2) (dots),
+                        label={[anchor=north west, xshift=2pt]
+                        north west:Complete circuit}
+                    ] {};
+                \end{tikzpicture}
+            }
+
+            \begin{tikzpicture}[
+                    every edge/.style = {draw, -{latex}}
+                ]
+                \node (pcm) {Code~/~$\bm{H}_\text{PCM}$};
+
+                \node[
+                    rectangle,
+                    draw=black,
+                    minimum width=2cm, minimum height=1cm,
+                    right=of pcm,
+                    align=center
+                ] (sec) {Single\\ SE Circ.};
+
+                \node[right=of sec] (qc) {\usebox{\innercircuit}};
+
+                \node[right=of qc] (dem) {$\bm{H}_\text{DEM}$};
+
+                \draw (pcm.east) edge[bend left] (sec.west);
+                \draw (sec.east) edge[bend left] ($(qc.west) + (0.22,0)$);
+                \draw ($(qc.east) + (-0.22,0)$) edge[bend left]
+                node[midway, above] {Stim} (dem.west);
+            \end{tikzpicture}
+        }
+    \end{figure}
+
+    \only<1>{
+        \begin{figure}[H]
+            \centering
+            \includegraphics[scale=0.55]{./res/pcm.png}
+        \end{figure}
+    }
+    \only<2->{
+        \begin{figure}[H]
+            \centering
+            \includegraphics[scale=0.55]{./res/windowing.png}
+        \end{figure}
+    }
+
+    \vspace*{-5mm}
+
+    \visible<2->{
+        \begin{align*}
+            \colorbox{red!20}{$\bm{H}_1 \bm{e}_1 = \bm{s}_1$}
+            \hspace{2mm}\rightarrow\hspace{2mm} \hat{\bm{e_1}} = \cdots
+            \hspace{2mm}\rightarrow\hspace{2mm} \bm{s}_2' = \cdots \\
+            \colorbox{ForestGreen!20}{$\bm{H}_2 \bm{e}_2 = \bm{s}_2'$}
+            \hspace{2mm}\rightarrow\hspace{2mm} \hat{\bm{e_2}} = \cdots
+            \hspace{2mm}\rightarrow\hspace{2mm} \bm{s}_3' = \cdots \\
+            \colorbox{blue!20}{$\bm{H}_3 \bm{e}_3 = \bm{s}_3'$}
+            \hspace{2mm}\rightarrow\hspace{2mm} \hat{\bm{e_3}} = \cdots
+            \hspace{2mm}\rightarrow\hspace{2mm} \bm{s}_4' = \cdots
+            % &\hspace{2.25mm}\vdots
+        \end{align*}
+    }
+\end{frame}
+
+\begin{frame}
+    \frametitle{Similarities to Spatially-Coupled LDPC Codes}
+
+    \begin{figure}[H]
+        \centering
+        \includegraphics[scale=0.1]{./res/sc-tanner-graph.png}
+        \includegraphics[scale=0.1]{./res/sc-pcm.png}
+    \end{figure}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Current Results}
+\label{sec:Current Results}
+
+\begin{frame}
+    \frametitle{An ``Upper Bound'' on the Performance Gain}
+
+    \begin{minipage}{0.65\textwidth}
+        \begin{figure}[H]
+            % \centering
+            \hspace*{-8mm}
+            \begin{tikzpicture}
+                \begin{axis}[
+                        width=10cm,
+                        height=8cm,
+                        ymode=log,
+                        ylabel={LER},
+                        xlabel={Physical error rate},
+                        legend pos=south east,
+                        legend style={
+                            cells={anchor=west},
+                            cells={align=left},
+                        },
+                        xticklabel style={/pgf/number format/fixed},
+                        xticklabel style={/pgf/number format/precision=4},
+                        scaled x ticks=false,
+                        grid=both,
+                        xtick={0.001,0.0015,...,0.004},
+                        xmin=0.001, xmax=0.004,
+                        ymin=1e-2, ymax=1,
+                    ]
+
+                    \addplot+[NavyBlue, mark=none, line width=1pt]
+                    table[col sep=comma, x=p, y=LER_sw]
+                    {res/whole_vs_windowed_spa.csv};
+                    \addlegendentry{Windowed - SPA}
+
+                    \addplot+[BurntOrange, mark=none, line width=1pt]
+                    table[col sep=comma, x=p, y=LER_sw]
+                    {res/whole_vs_windowed_min_sum.csv};
+                    \addlegendentry{Windowed - Min-Sum}
+
+                    \addplot+[NavyBlue, mark=none, line width=1pt,
+                    densely dashed]
+                    table[col sep=comma, x=p, y=LER_whole]
+                    {res/whole_vs_windowed_spa.csv};
+                    \addlegendentry{Whole - SPA}
+
+                    \addplot+[BurntOrange, mark=none, line width=1pt,
+                    densely dashed]
+                    table[col sep=comma, x=p, y=LER_whole]
+                    {res/whole_vs_windowed_min_sum.csv};
+                    \addlegendentry{Whole - Min-Sum}
+
+                \end{axis}
+            \end{tikzpicture}
+        \end{figure}
+    \end{minipage}%
+    \begin{minipage}{0.35\textwidth}
+        \begin{itemize}
+            \item $[[144,12,12]]$ BB code
+            \item $n_\text{rounds} = 12$
+            \item $W=3$, $F=1$
+        \end{itemize}
+    \end{minipage}
+\end{frame}
+
+\begin{frame}
+    \frametitle{Soft- vs. Hard-Information Decoding Behavior I}
+
+    \begin{minipage}{0.65\textwidth}
+        \begin{figure}[H]
+            % \centering
+            \hspace*{-8mm}
+            \begin{tikzpicture}
+                \begin{axis}[
+                        width=10cm,
+                        height=8cm,
+                        ymode=log,
+                        ylabel={LER},
+                        xlabel={Physical error rate},
+                        legend pos=south east,
+                        legend style={
+                            cells={anchor=west},
+                            cells={align=left},
+                        },
+                        xticklabel style={/pgf/number format/fixed},
+                        xticklabel style={/pgf/number format/precision=4},
+                        scaled x ticks=false,
+                        grid=both,
+                        xtick={0.001,0.0015,...,0.004},
+                        xmin=0.001, xmax=0.004,
+                        ymin=1e-2, ymax=1,
+                    ]
+
+                    \addplot+[BurntOrange, mark=none,
+                    line width=1pt]
+                    table[col sep=comma, x=p, y=LER_soft]
+                    {res/hard_vs_soft_spa.csv};
+                    \addlegendentry{Soft info - SPA}
+
+                    \addplot+[BurntOrange, densely dashed, mark=none,
+                    line width=1pt]
+                    table[col sep=comma, x=p, y=LER_hard]
+                    {res/hard_vs_soft_spa.csv};
+                    \addlegendentry{Hard info - SPA}
+
+                    \addplot+[NavyBlue, mark=none, line width=1pt]
+                    table[col sep=comma, x=p, y=LER_soft]
+                    {res/hard_vs_soft_min_sum.csv};
+                    \addlegendentry{Soft info - Min-Sum}
+
+                    \addplot+[NavyBlue, densely dashed, mark=none,
+                    line width=1pt]
+                    table[col sep=comma, x=p, y=LER_hard]
+                    {res/hard_vs_soft_min_sum.csv};
+                    \addlegendentry{Hard info - Min-Sum}
+                \end{axis}
+            \end{tikzpicture}
+        \end{figure}
+    \end{minipage}%
+    \begin{minipage}{0.35\textwidth}
+        \begin{itemize}
+            \item $[[144,12,12]]$ BB code
+            \item $n_\text{rounds} = 12$
+            \item $W=3$, $F=1$
+        \end{itemize}
+    \end{minipage}
+
+    % \begin{itemize}
+    %     \item \red{Soft vs hard for min sum and spa (4 plots)}
+    %     \item \red{Different window sizes}
+    % \end{itemize}
+\end{frame}
+
+% \begin{frame}
+%     \frametitle{Soft- vs. Hard-Information Decoding Behavior II}
+%
+%     \begin{itemize}
+%         \item \red{Comparison of numbers of iterations for soft vs hard}
+%         \item \red{Convergence analysis}
+%     \end{itemize}
+% \end{frame}
+%
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Future Work and Open Questions}
+% \label{sec:Future Work and Open Questions}
+%
+% \begin{frame}
+%     \frametitle{Future Work}
+%
+%     \vspace*{15mm}
+%
+%     \begin{itemize}
+%         \item Look at behavior during iterations
+%         \item Move from BP to BPGD
+%
+%             \vspace{15mm}
+%
+%         \item Q: Theoretically rigorous proof for the
+%             ``window-friendly'' structure of the detector error matrix?
+%         \item Q: Min-sum seems to perform better than SPA? \red{Make
+%             sure this question even makes sense}
+%     \end{itemize}
+% \end{frame}
+%
+% % TODOs
+% % - Whole vs windowed for min sum and spa
+% %    - Basic implementation
+% %    - Choose parameters (e.g., window sizes)
+% % - soft vs hard for min sum and spa (4 plots)
+% %    - Basic implementation
+% %    - Choose parameters (e.g., window sizes)
+% % - Comparison of numbers of iterations for soft vs hard
+% %    - Basic implementation (more or less done)
+% %    - Choose parameters
+% % - Convergence analysis
+% %    - Basic implementation (more or less done)
+% %    - Choose parameters
+% % - Make sure min sum vs spa question makes sense
+% % - Think of a few words for each slide (and take notes)
+
+\end{document}
+

src/qec-meeting-presentation/res/hard_vs_soft_min_sum.csv

@@ -0,0 +1,8 @@
+p,LER_hard,LER_soft
+0.001,0.08,0.028
+0.0015,0.174,0.076
+0.002,0.348,0.218
+0.0025,0.54,0.38
+0.003,0.716,0.564
+0.0035,0.862,0.75
+0.004,0.948,0.906

src/qec-meeting-presentation/res/hard_vs_soft_spa.csv

@@ -0,0 +1,7 @@
+p,LER_hard,LER_soft
+0.001,0.098,0.038
+0.0015,0.174,0.11
+0.002,0.394,0.268
+0.0025,0.536,0.376
+0.003,0.688,0.574
+0.0035,0.822,0.744

src/qec-meeting-presentation/res/whole_vs_windowed_min_sum.csv

@@ -0,0 +1,8 @@
+p,LER_whole,LER_sw
+0.001,0.046,0.086
+0.0015,0.11,0.2
+0.002,0.185,0.345
+0.0025,0.346,0.546
+0.003,0.587,0.721
+0.0035,0.732,0.852
+0.004,0.871,0.941

src/qec-meeting-presentation/res/whole_vs_windowed_spa.csv

@@ -0,0 +1,8 @@
+p,LER_whole,LER_sw
+0.001,0.072,0.101
+0.0015,0.128,0.234
+0.002,0.213,0.352
+0.0025,0.374,0.517
+0.003,0.544,0.676
+0.0035,0.711,0.823
+0.004,0.83,0.905

src/results/2026-03-29/lib

@@ -0,0 +1 @@
+/home/andreas/workspace/private/ma-thesis/lib

src/results/2026-03-29/main.tex

@@ -0,0 +1,641 @@
+\documentclass{article}
+
+\usepackage[a4paper,left=2cm,right=2cm,top=2.5cm,bottom=2cm]{geometry}
+\usepackage{float}
+\usepackage{amsmath}
+\usepackage{hyperref}
+\usepackage{amsfonts}
+\usepackage{mleftright}
+\usepackage{bm}
+\usepackage{tikz}
+\usepackage{subcaption}
+\usepackage{xcolor}
+\usepackage{pgfplots}
+\usepackage{pgfplotstable}
+\pgfplotsset{compat=newest}
+\usepackage{acro}
+\usepackage{braket}
+\usepackage[
+    backend=biber,
+    style=ieee,
+    sorting=nty,
+]{biblatex}
+\usepackage{todonotes}
+\usepackage{lipsum}
+
+\usetikzlibrary{calc, positioning}
+
+\input{lib/latex-common/common.tex}
+\pgfplotsset{colorscheme/matplotlib}
+
+%
+%
+% Custom commands
+%
+%
+
+\newcommand{\red}[1]{\textcolor{red}{#1}}
+\newcommand{\figwidth}{8cm}
+\newcommand{\figheight}{6.5cm}
+
+%
+%
+% Acronyms
+%
+%
+
+\DeclareAcronym{qec}{
+    short=QEC,
+    long=quantum error correction
+}
+
+\addbibresource{src/proposal/MA.bib}
+
+%
+%
+% Content
+%
+%
+
+\title{Results: 2026-03-29}
+\author{Andreas Tsouchlos}
+\date{}
+
+\begin{document}
+
+\maketitle
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Whole vs Windowed decoding}
+\label{sec:Whole vs Windowed decoding}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=1e-3, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[
+                            mark=o, line width=1pt, densely dashed, \col,
+                        ]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_whole})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=1,+system.W=\W/2026-03-29_23-39-18/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~whole}
+                }
+
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_sw})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=1,+system.W=\W/2026-03-29_23-39-18/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~windowed}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{Min-Sum}
+    \end{subfigure}%
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=1e-3, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, densely dashed, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_whole})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=1,+system.W=\W/2026-03-29_23-39-18/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~whole}
+                }
+
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_sw})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=1,+system.W=\W/2026-03-29_23-39-18/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~windowed}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{SPA}
+    \end{subfigure}
+
+    \caption{BP simulations for the [[144,12,12]]-BB code with $F=1,
+    n_\text{iter,BP}=32,n_\text{rounds}=12$}
+\end{figure}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=1e-3, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, densely dashed, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_whole})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=\F,+system.W=5/2026-03-29_23-39-18/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~whole}
+                }
+
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_sw})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=\F,+system.W=5/2026-03-29_23-39-18/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~windowed}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{Min-Sum}
+    \end{subfigure}%
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=1e-3, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, densely dashed, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_whole})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=\F,+system.W=5/2026-03-29_23-39-18/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~whole}
+                }
+
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_sw})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=\F,+system.W=5/2026-03-29_23-39-18/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~windowed}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{SPA}
+    \end{subfigure}
+
+    \caption{BP simulations for the [[144,12,12]]-BB code with $W=5,
+    n_\text{iter,BP}=32,n_\text{rounds}=12$}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Soft- vs Hard- Information Decoding}
+\label{sec:Soft- vs Hard- Information Decoding}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=8e-4, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, densely dashed, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_hard})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=\W/2026-03-30_00-06-26/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~hard}
+                }
+
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=*, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_soft})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=\W/2026-03-30_00-06-26/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~soft}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{Min-Sum}
+    \end{subfigure}%
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=8e-4, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, densely dashed, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_hard})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=\W/2026-03-30_00-06-26/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~hard}
+                }
+
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=*, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_soft})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=\W/2026-03-30_00-06-26/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~soft}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{SPA}
+    \end{subfigure}
+
+    \caption{BP simulations for the [[144,12,12]]-BB code with $F=1,
+    n_\text{iter,BP}=32,n_\text{rounds}=12$}
+\end{figure}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=8e-4, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, densely dashed, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_hard})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=\F,+system.W=5/2026-03-30_00-06-26/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~hard}
+                }
+
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=*, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_soft})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=\F,+system.W=5/2026-03-30_00-06-26/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~soft}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{Min-Sum}
+    \end{subfigure}%
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=8e-4, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, densely dashed, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_hard})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=\F,+system.W=5/2026-03-30_00-06-26/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~hard}
+                }
+
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=*, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_soft})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=\F,+system.W=5/2026-03-30_00-06-26/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~soft}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{SPA}
+    \end{subfigure}
+
+    \caption{BP simulations for the [[144,12,12]]-BB code with $W=5,
+    n_\text{iter,BP}=32,n_\text{rounds}=12$}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Soft-Information Decoding: Best Performance}
+\label{sec:Soft-Information Decoding: Best Performance}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=8e-4, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \addplot+[mark=o, line width=1pt, densely dashed, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_hard})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_00-06-26/LERs.csv};
+
+                \addlegendentryexpanded{Windowed, hard}
+
+                \addplot+[mark=*, line width=1pt, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_soft})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_00-06-26/LERs.csv};
+
+                \addlegendentryexpanded{Windowed, soft}
+
+                \addplot+[mark=o, line width=1pt, densely dashed, scol1]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_whole})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=1,+system.W=5/2026-03-29_23-39-18/LERs.csv};
+
+                \addlegendentryexpanded{Whole window}
+            \end{axis}
+        \end{tikzpicture}
+        \caption{Min-Sum}
+    \end{subfigure}%
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=8e-4, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \addplot+[mark=o, line width=1pt, densely dashed, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_hard})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_00-06-26/LERs.csv};
+
+                \addlegendentryexpanded{Windowed, hard}
+
+                \addplot+[mark=*, line width=1pt, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_soft})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_00-06-26/LERs.csv};
+
+                \addlegendentryexpanded{Windowed, soft}
+
+                \addplot+[mark=o, line width=1pt, densely dashed, scol1]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_whole})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=1,+system.W=5/2026-03-29_23-39-18/LERs.csv};
+
+                \addlegendentryexpanded{Whole window}
+            \end{axis}
+        \end{tikzpicture}
+        \caption{SPA}
+    \end{subfigure}
+
+    \caption{BP simulations for the [[144,12,12]]-BB code,
+    $W=5, F=1, n_\text{iter,BP} = 32$}
+\end{figure}
+
+\end{document}
+

src/results/2026-03-29/src

@@ -0,0 +1 @@
+/home/andreas/workspace/private/ma-thesis/src

src/results/2026-03-30/main.tex

@@ -0,0 +1,753 @@
+\documentclass{article}
+
+\usepackage[a4paper,left=2cm,right=2cm,top=2.5cm,bottom=2cm]{geometry}
+\usepackage{float}
+\usepackage{amsmath}
+\usepackage{hyperref}
+\usepackage{amsfonts}
+\usepackage{mleftright}
+\usepackage{bm}
+\usepackage{tikz}
+\usepackage{subcaption}
+\usepackage{xcolor}
+\usepackage{pgfplots}
+\usepackage{pgfplotstable}
+\pgfplotsset{compat=newest}
+\usepackage{acro}
+\usepackage{braket}
+\usepackage[
+    backend=biber,
+    style=ieee,
+    sorting=nty,
+]{biblatex}
+\usepackage{todonotes}
+\usepackage{lipsum}
+
+\usetikzlibrary{calc, positioning}
+
+\input{lib/latex-common/common.tex}
+\pgfplotsset{colorscheme/matplotlib}
+
+%
+%
+% Custom commands
+%
+%
+
+\newcommand{\red}[1]{\textcolor{red}{#1}}
+\newcommand{\figwidth}{8cm}
+\newcommand{\figheight}{6.5cm}
+
+%
+%
+% Acronyms
+%
+%
+
+\DeclareAcronym{qec}{
+    short=QEC,
+    long=quantum error correction
+}
+
+\addbibresource{src/proposal/MA.bib}
+
+%
+%
+% Content
+%
+%
+
+\title{Results: 2026-03-30}
+\author{Andreas Tsouchlos}
+\date{}
+
+\begin{document}
+
+\maketitle
+
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Soft vs Hard Information Decoding}
+% \label{sec:Soft vs Hard Information Decoding}
+%
+% \begin{figure}[H]
+%     \centering
+%
+%     \begin{subfigure}{0.5\textwidth}
+%         \begin{tikzpicture}
+%             \begin{axis}[
+%                     width=\figwidth,
+%                     height=\figheight,
+%                     ymode=log,
+%                     legend style={
+%                         cells={anchor=west},
+%                         cells={align=left},
+%                     },
+%                     enlargelimits=false,
+%                     ymin=1e-3, ymax=2.5e-1,
+%                     grid=both,
+%                     legend pos = south west,
+%                     xtick={8, 32, 64, 96, 128, 160, 192},
+%                     xlabel={Max BP iterations},
+%                     ylabel={Per-round-LER},
+%                 ]
+%                 \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+%                     \edef\temp{\noexpand
+%                         \addplot+[
+%                             mark=o, line width=1pt, densely dashed, \col,
+%                         ]
+%                         table[
+%                             col sep=comma, x=max_bp_iter,
+%                             y expr={1 - (1-\noexpand\thisrow{LER_hard})^(1/12)}
+%                         ]
+%                         {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=1,+system.W=\W/2026-03-30_13-49-44/LERs.csv};
+%                     }
+%                     \temp
+%
+%                     \addlegendentryexpanded{$W = \W$~hard}
+%                 }
+%
+%                 \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+%                     \edef\temp{\noexpand
+%                         \addplot+[mark=o, line width=1pt, \col]
+%                         table[
+%                             col sep=comma, x=max_bp_iter,
+%                             y expr={1 - (1-\noexpand\thisrow{LER_soft})^(1/12)}
+%                         ]
+%                         {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=1,+system.W=\W/2026-03-30_13-49-44/LERs.csv};
+%                     }
+%                     \temp
+%
+%                     \addlegendentryexpanded{$W = \W$~soft}
+%                 }
+%             \end{axis}
+%         \end{tikzpicture}
+%         \caption{Min-Sum}
+%     \end{subfigure}%
+%     \begin{subfigure}{0.5\textwidth}
+%         \begin{tikzpicture}
+%             \begin{axis}[
+%                     width=\figwidth,
+%                     height=\figheight,
+%                     ymode=log,
+%                     legend style={
+%                         cells={anchor=west},
+%                         cells={align=left},
+%                     },
+%                     enlargelimits=false,
+%                     ymin=1e-3, ymax=2.5e-1,
+%                     grid=both,
+%                     legend pos = south west,
+%                     xtick={8, 32, 64, 96, 128, 160, 192},
+%                     xlabel={Max BP iterations},
+%                     ylabel={Per-round-LER},
+%                 ]
+%                 \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+%                     \edef\temp{\noexpand
+%                         \addplot+[mark=o, line width=1pt, densely dashed, \col]
+%                         table[
+%                             col sep=comma, x=max_bp_iter,
+%                             y expr={1 - (1-\noexpand\thisrow{LER_hard})^(1/12)}
+%                         ]
+%                         {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=1,+system.W=\W/2026-03-30_13-49-44/LERs.csv};
+%                     }
+%                     \temp
+%
+%                     \addlegendentryexpanded{$W = \W$~hard}
+%                 }
+%
+%                 \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+%                     \edef\temp{\noexpand
+%                         \addplot+[mark=o, line width=1pt, \col]
+%                         table[
+%                             col sep=comma, x=max_bp_iter,
+%                             y expr={1 - (1-\noexpand\thisrow{LER_soft})^(1/12)}
+%                         ]
+%                         {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=1,+system.W=\W/2026-03-30_13-49-44/LERs.csv};
+%                     }
+%                     \temp
+%
+%                     \addlegendentryexpanded{$W = \W$~windowed}
+%                 }
+%             \end{axis}
+%         \end{tikzpicture}
+%         \caption{SPA}
+%     \end{subfigure}
+%
+%     \caption{BP simulations for the [[144,12,12]]-BB code with $F=1,
+%     n_\text{iter,BP}=32,n_\text{rounds}=12$}
+% \end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Whole vs Windowed decoding}
+\label{sec:Whole vs Windowed decoding}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=4e-5, ymax=2e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[
+                            mark=o, line width=1pt, densely dashed, \col,
+                        ]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_whole})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=1,+system.W=\W/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~whole}
+                }
+
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_sw})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=1,+system.W=\W/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~windowed}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{Min-Sum}
+    \end{subfigure}%
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=4e-5, ymax=2e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, densely dashed, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_whole})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=1,+system.W=\W/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~whole}
+                }
+
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_sw})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=1,+system.W=\W/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~windowed}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{SPA}
+    \end{subfigure}
+
+    \caption{BP simulations for the [[144,12,12]]-BB code with $F=1,
+    n_\text{iter,BP}=200,n_\text{rounds}=12$}
+\end{figure}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=4e-5, ymax=2e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, densely dashed, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_whole})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=\F,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~whole}
+                }
+
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_sw})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=\F,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~windowed}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{Min-Sum}
+    \end{subfigure}%
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=4e-5, ymax=2e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, densely dashed, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_whole})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=\F,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~whole}
+                }
+
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_sw})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=\F,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~windowed}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{SPA}
+    \end{subfigure}
+
+    \caption{BP simulations for the [[144,12,12]]-BB code with $W=5,
+    n_\text{iter,BP}=200,n_\text{rounds}=12$}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Soft- vs Hard- Information Decoding}
+\label{sec:Soft- vs Hard- Information Decoding}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=4e-5, ymax=2e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, densely dashed, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_hard})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=\W/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~hard}
+                }
+
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=*, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_soft})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=\W/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~soft}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{Min-Sum}
+    \end{subfigure}%
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=4e-5, ymax=2e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, densely dashed, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_hard})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=\W/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~hard}
+                }
+
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=*, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_soft})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=\W/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~soft}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{SPA}
+    \end{subfigure}
+
+    \caption{BP simulations for the [[144,12,12]]-BB code with $F=1,
+    n_\text{iter,BP}=32,n_\text{rounds}=12$}
+\end{figure}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=8e-4, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, densely dashed, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_hard})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=\F,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~hard}
+                }
+
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=*, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_soft})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=\F,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~soft}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{Min-Sum}
+    \end{subfigure}%
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=8e-4, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, densely dashed, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_hard})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=\F,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~hard}
+                }
+
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=*, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=physical_p,
+                            y expr={1 - (1-\noexpand\thisrow{LER_soft})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=\F,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~soft}
+                }
+            \end{axis}
+        \end{tikzpicture}
+        \caption{SPA}
+    \end{subfigure}
+
+    \caption{BP simulations for the [[144,12,12]]-BB code with $W=5,
+    n_\text{iter,BP}=200,n_\text{rounds}=12$}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Soft-Information Decoding: Best Performance}
+\label{sec:Soft-Information Decoding: Best Performance}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=8e-4, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \addplot+[mark=o, line width=1pt, densely dashed, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_hard})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+
+                \addlegendentryexpanded{Windowed, hard}
+
+                \addplot+[mark=*, line width=1pt, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_soft})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+
+                \addlegendentryexpanded{Windowed, soft}
+
+                \addplot+[mark=o, line width=1pt, densely dashed, scol1]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_whole})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=1,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+
+                \addlegendentryexpanded{Whole window}
+            \end{axis}
+        \end{tikzpicture}
+        \caption{Min-Sum}
+    \end{subfigure}%
+    \begin{subfigure}{0.5\textwidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=8e-4, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = south east,
+                    xtick={0.001,0.0015,...,0.004},
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    xlabel={Physical error rate},
+                    ylabel={Per-round-LER},
+                ]
+                \addplot+[mark=o, line width=1pt, densely dashed, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_hard})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+
+                \addlegendentryexpanded{Windowed, hard}
+
+                \addplot+[mark=*, line width=1pt, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_soft})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+
+                \addlegendentryexpanded{Windowed, soft}
+
+                \addplot+[mark=o, line width=1pt, densely dashed, scol1]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_whole})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=windowed_vs_whole_param_exploration,+system.F=1,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+
+                \addlegendentryexpanded{Whole window}
+            \end{axis}
+        \end{tikzpicture}
+        \caption{SPA}
+    \end{subfigure}
+
+    \caption{BP simulations for the [[144,12,12]]-BB code,
+    $W=5, F=1, n_\text{iter,BP} = 200, n_\text{rounds}=12$}
+\end{figure}
+
+\end{document}
+

src/results/2026-04-01/main.tex

@@ -0,0 +1,587 @@
+\documentclass{article}
+
+\usepackage[a4paper,left=2cm,right=2cm,top=2.5cm,bottom=2cm]{geometry}
+\usepackage{float}
+\usepackage{amsmath}
+\usepackage{hyperref}
+\usepackage{amsfonts}
+\usepackage{mleftright}
+\usepackage{bm}
+\usepackage{tikz}
+\usepackage{subcaption}
+\usepackage{xcolor}
+\usepackage{pgfplots}
+\usepackage{pgfplotstable}
+\pgfplotsset{compat=newest}
+\usepackage{acro}
+\usepackage{braket}
+\usepackage[
+    backend=biber,
+    style=ieee,
+    sorting=nty,
+]{biblatex}
+\usepackage{todonotes}
+\usepackage{lipsum}
+
+\usetikzlibrary{calc, positioning}
+
+\input{lib/latex-common/common.tex}
+\pgfplotsset{colorscheme/matplotlib}
+
+%
+%
+% Custom commands
+%
+%
+
+\newcommand{\red}[1]{\textcolor{red}{#1}}
+\newcommand{\figwidth}{7.5cm}
+\newcommand{\figheight}{6.5cm}
+
+%
+%
+% Acronyms
+%
+%
+
+\DeclareAcronym{qec}{
+    short=QEC,
+    long=quantum error correction
+}
+
+\addbibresource{src/proposal/MA.bib}
+
+%
+%
+% Content
+%
+%
+
+\title{Results: 2026-04-01}
+\author{Andreas Tsouchlos}
+\date{}
+
+\begin{document}
+
+\maketitle
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Soft vs Hard Information Decoding}
+\label{sec:Soft vs Hard Information Decoding}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{subfigure}{0.4\textwidth}
+        \centering
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    enlargelimits=false,
+                    ymin=5e-3, ymax=2.5e-1,
+                    grid=both,
+                    xtick={8, 32, 64, 96, 128, 160, 192},
+                    xlabel={Max BP iterations},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[
+                            mark=o, line width=1pt, densely dashed, \col,
+                        ]
+                        table[
+                            col sep=comma, x=max_bp_iter,
+                            y expr={1 - (1-\noexpand\thisrow{LER_hard})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=1,+system.W=\W/2026-03-30_23-30-38/LERs.csv};
+                    }
+                    \temp
+                }
+
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=max_bp_iter,
+                            y expr={1 - (1-\noexpand\thisrow{LER_soft})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=1,+system.W=\W/2026-03-30_23-30-38/LERs.csv};
+                    }
+                    \temp
+                }
+
+                \addplot+[mark=o, line width=1pt, black]
+                table[
+                    col sep=comma, x=max_bp_iter,
+                    y expr={1 - (1-\noexpand\thisrow{LER_whole})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=1,+system.W=5/2026-03-30_23-30-38/LERs.csv};
+            \end{axis}
+        \end{tikzpicture}
+        \caption{Min-Sum}
+    \end{subfigure}%
+    \begin{subfigure}{0.6\textwidth}
+        \centering
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=5e-3, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = outer north east,
+                    xtick={8, 32, 64, 96, 128, 160, 192},
+                    xlabel={Max BP iterations},
+                    ylabel={},
+                    yticklabels = {},
+                ]
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[
+                            mark=o, line width=1pt, densely dashed, \col,
+                        ]
+                        table[
+                            col sep=comma, x=max_bp_iter,
+                            y expr={1 - (1-\noexpand\thisrow{LER_hard})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=1,+system.W=\W/2026-03-30_23-30-38/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~hard}
+                }
+
+                \foreach \W/\col in {3/scol2,4/scol1,5/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=max_bp_iter,
+                            y expr={1 - (1-\noexpand\thisrow{LER_soft})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=1,+system.W=\W/2026-03-30_23-30-38/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$W = \W$~soft}
+                }
+
+                \addplot+[mark=o, line width=1pt, black]
+                table[
+                    col sep=comma, x=max_bp_iter,
+                    y expr={1 - (1-\noexpand\thisrow{LER_whole})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=1,+system.W=5/2026-03-30_23-30-38/LERs.csv};
+
+                \addlegendentryexpanded{Whole}
+            \end{axis}
+        \end{tikzpicture}
+        \caption{SPA}
+    \end{subfigure}%
+
+    \caption{BP simulations for the [[144,12,12]]-BB code with
+    $n_\text{rounds}=12,p_\text{phys} = 0.0025,F=1$}
+\end{figure}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{subfigure}{0.4\textwidth}
+        \centering
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    enlargelimits=false,
+                    ymin=5e-3, ymax=2.5e-1,
+                    grid=both,
+                    xtick={8, 32, 64, 96, 128, 160, 192},
+                    xlabel={Max BP iterations},
+                    ylabel={Per-round-LER},
+                ]
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[
+                            mark=o, line width=1pt, densely dashed, \col,
+                        ]
+                        table[
+                            col sep=comma, x=max_bp_iter,
+                            y expr={1 - (1-\noexpand\thisrow{LER_hard})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=\F,+system.W=5/2026-03-30_23-30-38/LERs.csv};
+                    }
+                    \temp
+                }
+
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=max_bp_iter,
+                            y expr={1 - (1-\noexpand\thisrow{LER_soft})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=\F,+system.W=5/2026-03-30_23-30-38/LERs.csv};
+                    }
+                    \temp
+                }
+
+                \addplot+[mark=o, line width=1pt, black]
+                table[
+                    col sep=comma, x=max_bp_iter,
+                    y expr={1 - (1-\noexpand\thisrow{LER_whole})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=1,+system.W=5/2026-03-30_23-30-38/LERs.csv};
+            \end{axis}
+        \end{tikzpicture}
+        \caption{Min-Sum}
+    \end{subfigure}%
+    \begin{subfigure}{0.6\textwidth}
+        \centering
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=5e-3, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = outer north east,
+                    xtick={8, 32, 64, 96, 128, 160, 192},
+                    xlabel={Max BP iterations},
+                    ylabel={},
+                    yticklabels = {},
+                ]
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[
+                            mark=o, line width=1pt, densely dashed, \col,
+                        ]
+                        table[
+                            col sep=comma, x=max_bp_iter,
+                            y expr={1 - (1-\noexpand\thisrow{LER_hard})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=\F,+system.W=5/2026-03-30_23-30-38/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~hard}
+                }
+
+                \foreach \F/\col in {1/scol2,2/scol1,3/scol0} {
+                    \edef\temp{\noexpand
+                        \addplot+[mark=o, line width=1pt, \col]
+                        table[
+                            col sep=comma, x=max_bp_iter,
+                            y expr={1 - (1-\noexpand\thisrow{LER_soft})^(1/12)}
+                        ]
+                        {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=\F,+system.W=5/2026-03-30_23-30-38/LERs.csv};
+                    }
+                    \temp
+
+                    \addlegendentryexpanded{$F = \F$~soft}
+                }
+
+                \addplot+[mark=o, line width=1pt, black]
+                table[
+                    col sep=comma, x=max_bp_iter,
+                    y expr={1 - (1-\noexpand\thisrow{LER_whole})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_over_iter_param_exploration,+system.F=1,+system.W=5/2026-03-30_23-30-38/LERs.csv};
+
+                \addlegendentryexpanded{Whole}
+            \end{axis}
+        \end{tikzpicture}
+        \caption{SPA}
+    \end{subfigure}%
+
+    \caption{BP simulations for the [[144,12,12]]-BB code with
+    $n_\text{rounds}=12,p_\text{phys} = 0.0025,W=5$}
+\end{figure}
+
+\newpage
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Whole vs Windowed Decoding with Constant Total Number of Iterations}
+\label{sec:Whole vs Windowed Decoding with Constant Total Number of Iterations}
+
+%%%%%%%%%%%%%%%%
+\subsection{32 Sliding-Window Iterations}
+\label{subsec:32 Sliding-Window Iterations}
+
+\begin{minipage}[t]{0.5\textwidth}
+    \begin{itemize}
+        \item Sliding-window decoding
+            \begin{itemize}
+                \item $n_\text{iter,BP} = 32$
+                \item $W=5, F=1 \Rightarrow n_\text{windows} = 8$
+                \item $n_\text{iter,BP}^\text{total} = n_\text{windows}
+                    \cdot n_\text{iter,BP} = 256$
+            \end{itemize}
+    \end{itemize}
+\end{minipage}%
+\begin{minipage}[t]{0.5\textwidth}
+    \begin{itemize}
+        \item Whole decoding
+            \begin{itemize}
+                \item $n_\text{iter,BP} = 256$
+            \end{itemize}
+    \end{itemize}
+\end{minipage}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{subfigure}{0.4\textwidth}
+        \centering
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    enlargelimits=false,
+                    ymin=5e-5, ymax=2.5e-1,
+                    grid=both,
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    grid=both,
+                    xtick={0.001,0.0015,...,0.004},
+                    xlabel={Physical Error Rate},
+                    ylabel={Per-round-LER},
+                ]
+
+                \addplot+[
+                    mark=o, line width=1pt, densely dashed, black,
+                ]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_whole})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=whole_more_iterations,+system.F=1,+system.W=3/2026-04-01_15-41-23/LERs.csv};
+
+                \addplot+[mark=o, line width=1pt, densely dashed, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_hard})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_00-06-26/LERs.csv};
+
+                \addplot+[mark=*, line width=1pt, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_soft})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_00-06-26/LERs.csv};
+            \end{axis}
+        \end{tikzpicture}
+        \caption{Min-Sum}
+    \end{subfigure}%
+    \begin{subfigure}{0.6\textwidth}
+        \centering
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=5e-5, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = outer north east,
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    grid=both,
+                    xtick={0.001,0.0015,...,0.004},
+                    xlabel={Physical Error Rate},
+                    ylabel={},
+                    yticklabels = {},
+                ]
+
+                \addplot+[mark=o, line width=1pt, densely dashed, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_hard})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_00-06-26/LERs.csv};
+
+                \addlegendentry{Hard}
+
+                \addplot+[mark=*, line width=1pt, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_soft})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_00-06-26/LERs.csv};
+
+                \addlegendentry{Soft}
+
+                \addplot+[
+                    mark=o, line width=1pt, densely dashed, black,
+                ]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_whole})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=whole_more_iterations,+system.F=1,+system.W=3/2026-04-01_15-41-24/LERs.csv};
+
+                \addlegendentryexpanded{Whole}
+            \end{axis}
+        \end{tikzpicture}
+        \caption{SPA}
+    \end{subfigure}%
+
+    \caption{BP simulations for the [[144,12,12]]-BB code with
+    $n_\text{rounds}=12, W=5, F=1$}
+\end{figure}
+
+%%%%%%%%%%%%%%%%
+\subsection{200 Sliding-Window Iterations}
+\label{subsec:200 Sliding-Window Iterations}
+
+\begin{minipage}[t]{0.5\textwidth}
+    \begin{itemize}
+        \item Sliding-window decoding
+            \begin{itemize}
+                \item $n_\text{iter,BP} = 200$
+                \item $W=5, F=1 \Rightarrow n_\text{windows} = 8$
+                \item $n_\text{iter,BP}^\text{total} = n_\text{windows}
+                    \cdot n_\text{iter,BP} = 1600$
+            \end{itemize}
+    \end{itemize}
+\end{minipage}%
+\begin{minipage}[t]{0.5\textwidth}
+    \begin{itemize}
+        \item Whole decoding
+            \begin{itemize}
+                \item $n_\text{iter,BP} = 1536$
+            \end{itemize}
+    \end{itemize}
+\end{minipage}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{subfigure}{0.4\textwidth}
+        \centering
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    enlargelimits=false,
+                    ymin=5e-5, ymax=2.5e-1,
+                    grid=both,
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    grid=both,
+                    xtick={0.001,0.0015,...,0.004},
+                    xlabel={Physical Error Rate},
+                    ylabel={Per-round-LER},
+                ]
+
+                \addplot+[
+                    mark=o, line width=1pt, densely dashed, black,
+                ]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_whole})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=whole_more_iterations,+system.F=1,+system.W=5/2026-04-01_18-31-29/LERs.csv};
+
+                \addplot+[mark=o, line width=1pt, densely dashed, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_hard})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+
+                \addplot+[mark=*, line width=1pt, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_soft})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=minimum_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+            \end{axis}
+        \end{tikzpicture}
+        \caption{Min-Sum}
+    \end{subfigure}%
+    \begin{subfigure}{0.6\textwidth}
+        \centering
+        \begin{tikzpicture}
+            \begin{axis}[
+                    width=\figwidth,
+                    height=\figheight,
+                    ymode=log,
+                    legend style={
+                        cells={anchor=west},
+                        cells={align=left},
+                    },
+                    enlargelimits=false,
+                    ymin=5e-5, ymax=2.5e-1,
+                    grid=both,
+                    legend pos = outer north east,
+                    xticklabel style={/pgf/number format/fixed},
+                    xticklabel style={/pgf/number format/precision=4},
+                    scaled x ticks=false,
+                    grid=both,
+                    xtick={0.001,0.0015,...,0.004},
+                    xlabel={Physical Error Rate},
+                    ylabel={},
+                    yticklabels = {},
+                ]
+
+                \addplot+[mark=o, line width=1pt, densely dashed, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_hard})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+
+                \addlegendentry{Hard}
+
+                \addplot+[mark=*, line width=1pt, scol0]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_soft})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=soft_vs_hard_param_exploration,+system.F=1,+system.W=5/2026-03-30_08-57-21/LERs.csv};
+
+                \addlegendentry{Soft}
+
+                \addplot+[
+                    mark=o, line width=1pt, densely dashed, black,
+                ]
+                table[
+                    col sep=comma, x=physical_p,
+                    y expr={1 - (1-\thisrow{LER_whole})^(1/12)}
+                ]
+                {/home/andreas/workspace/private/ma-sw-results/outputs/+decoder.bp_method=product_sum,+experiment=whole_more_iterations,+system.F=1,+system.W=3/2026-04-01_18-31-29/LERs.csv};
+
+                \addlegendentryexpanded{Whole}
+            \end{axis}
+        \end{tikzpicture}
+        \caption{SPA}
+    \end{subfigure}%
+
+    \caption{BP simulations for the [[144,12,12]]-BB code with
+    $n_\text{rounds}=12, W=5, F=1$}
+\end{figure}
+
+\end{document}
+

src/thesis/acronyms.tex

@@ -0,0 +1,74 @@
+\DeclareAcronym{qec}{
+    short=QEC,
+    long=quantum error correction
+}
+
+\DeclareAcronym{bp}{
+    short=BP,
+    long=belief propagation
+}
+
+\DeclareAcronym{nms}{
+    short=NMS,
+    long=normalized min-sum
+}
+
+\DeclareAcronym{spa}{
+    short=SPA,
+    long=sum-product algorithm
+}
+
+\DeclareAcronym{llr}{
+    short=LLR,
+    long=log-likelihood ratio
+}
+
+\DeclareAcronym{sc}{
+    short=SC,
+    long=spatially coupled
+}
+
+\DeclareAcronym{ldpc}{
+    short=LDPC,
+    long=low-density parity-check
+}
+
+\DeclareAcronym{ml}{
+    short=ML,
+    long=maximum likelihood
+}
+
+\DeclareAcronym{map}{
+    short=MAP,
+    long=maximum a posteriori
+}
+
+\DeclareAcronym{pcm}{
+    short=PCM,
+    long=parity-check matrix
+}
+
+\DeclareAcronym{vn}{
+    short=VN,
+    long=variable node
+}
+
+\DeclareAcronym{cn}{
+    short=CN,
+    long=check node
+}
+
+\DeclareAcronym{ber}{
+    short=BER,
+    long=bit error rate
+}
+
+\DeclareAcronym{fer}{
+    short=FER,
+    long=frame error rate
+}
+
+\DeclareAcronym{awgn}{
+    short=AWGN,
+    long=additive white Gaussian noise
+}

src/thesis/chapters/1_introduction.tex

@@ -0,0 +1 @@
+\chapter{Introduction}

src/thesis/chapters/2_fundamentals.tex

@@ -0,0 +1,762 @@
+\chapter{Fundamentals}
+\label{ch:Fundamentals}
+
+\Ac{qec} is a field of research combining ``classical''
+communications engineering and quantum information science.
+This chapter provides the relevant theoretical background on both of
+these topics and subsequently introduces the fundamentals of \ac{qec}.
+
+% TODO: Is an explanation of BP with guided decimation needed in this chapter?
+% TODO: Is an explanation of OSD needed chapter?
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Classical Error Correction}
+\label{sec:Classical Error Correction}
+
+The core concept underpinning error correcting codes is the
+realization that introducing a finite amount of redundancy to
+information before transmission can considerably reduce the error rate.
+Specifically, Shannon proved in 1948 that for any channel, a block
+code can be found that achieves arbitrarily small probability of
+error at any communication rate up to the capacity of the channel
+when the block length approaches infinity
+\cite[Sec.~13]{shannon_mathematical_1948}.
+
+In this section, we explore the concepts of ``classical'' (as in non-quantum)
+error correction that are central to this work.
+We start by looking at different ways of encoding information,
+first considering binary linear block codes in general and then \ac{ldpc} and
+\ac{sc}-\ac{ldpc} codes.
+Finally, we pivot to the decoding process, specifically the \ac{bp}
+algorithm.
+
+\subsection{Binary Linear Block Codes}
+
+%
+% Codewords, n, k, rate
+%
+
+One particularly important class of coding schemes is that of binary
+linear block codes.
+The information to be protected takes the form of a sequence of
+binary symbols, which is split into separate blocks.
+Each block is encoded, transmitted, and decoded separately.
+The encoding step introduces redundancy by mapping input messages
+$\bm{u} \in \mathbb{F}_2^k$ of length $k \in \mathbb{N}$ (called the
+\textit{information length}) onto \textit{codewords} $\bm{x} \in
+\mathbb{F}_2^n$ of length $n \in \mathbb{N}$ (called the
+\textit{block length}) with $n > k$.
+A measure of the amount of introduced redundancy is the \textit{code
+rate} $R = k/n$.
+We call the set of all codewords $\mathcal{C}$ the \textit{code}
+\cite[Sec.~3.1.1]{ryan_channel_2009}.
+
+%
+% d_min and the [] Notation
+%
+
+During the encoding process, a mapping from $\mathbb{F}_2^k$
+onto $\mathcal{C} \subset \mathbb{F}_2^n$ takes place.
+The input messages are mapped onto an expanded vector space, where
+they are ``further apart'', giving rise to the error correcting
+properties of the code.
+This notion of the distance between two codewords $\bm{x}_1$ and
+$\bm{x}_2$ can be expressed using the \textit{Hamming distance} $d(\bm{x}_1,
+\bm{x}_2)$, which is defined as the number of positions in which they differ.
+We define the \textit{minimum distance} of a code $\mathcal{C}$ as
+%
+\begin{align*}
+    d_\text{min} := \min \left\{ d(\bm{x}_1, \bm{x}_2) : \bm{x}_1,
+    \bm{x}_2 \in \mathcal{C}, \bm{x}_1 \neq \bm{x}_2 \right\}
+    .
+\end{align*}
+%
+We can signify that a binary linear block code has information length
+$k$, block length $n$ and minimum distance $d_\text{min}$ using the
+notation $[n,k,d_\text{min}]$ \cite[Sec.~1.3]{macwilliams_theory_1977}.
+
+%
+% Parity checks, H, and the syndrome
+%
+
+A particularly elegant way of describing the subspace $C$ of
+$\mathbb{F}_2^n$ that the codewords make up is the notion of
+\textit{parity checks}.
+Since $\lvert \mathcal{C} \rvert = 2^k$ and $\lvert \mathbb{F}_2^n
+\rvert = 2^n$, we could introduce $n-k$ conditions to constrain the
+additional degrees of freedom.
+These conditions, called parity checks, take the form of equations
+over $\mathbb{F}_2^n$, linking the individual positions of each codeword.
+We can arrange the coefficients of these equations in a
+\textit{parity-check matrix} (\acs{pcm}) $\bm{H} \in
+\mathbb{F}_2^{(n-k) \times n}$ and equivalently define the code as
+\cite[Sec.~3.1.1]{ryan_channel_2009}
+%
+\begin{align*}
+    \mathcal{C} = \left\{ \bm{x} \in \mathbb{F}_2^n :
+    \bm{H}\bm{x}^\text{T} = \bm{0} \right\}
+    .%
+\end{align*}
+Note that in general we may have linearly dependent parity checks,
+prompting us to define the \ac{pcm} as $\bm{H} \in
+\mathbb{F}_2^{m\times n}$ with $\hspace{2mm} m \ge n-k$ instead.
+The \textit{syndrome} $\bm{s} = \bm{H} \bm{v}^\text{T}$ describes
+which parity checks a candidate codeword $\bm{v} \in \mathbb{F}_2^n$ violates.
+The representation using the \ac{pcm} has the benefit of providing a
+description of the code, the memory complexity of which doesn't grow
+exponentially with $n$, in contrast to keeping track of all codewords directly.
+
+%
+% The decoding problem
+%
+
+Figure \ref{fig:Diagram of a transmission system} visualizes the
+communication process \cite[Sec.~1.1]{ryan_channel_2009}.
+An input message $\bm{u}\in \mathbb{F}_2^k$ is mapped onto a codeword $\bm{x}
+\in \mathbb{F}_2^n$. This is passed on to a modulator, which
+interacts with the physical channel.
+A demodulator processes the channel output and forwards the result
+$\bm{y}$ to a decoder.
+We differentiate between \textit{soft-decision} decoding, where
+$\bm{y} \in \mathbb{R}^n$, and \textit{hard-decision} decoding, where
+$\bm{y} \in \mathbb{F}_2^n$ \cite[Sec.~1.5.1.3]{ryan_channel_2009}.
+Finally, the decoder is responsible for obtaining an estimate
+$\hat{\bm{u}} \in \mathbb{F}_2^k$ of the original input message.
+This is done by first finding an estimate $\hat{\bm{x}}$ of the sent
+codeword and undoing the encoding.
+The decoding problem that we generally attempt to solve thus consists
+in finding the best estimate $\hat{\bm{x}}$ given $\bm{y}$.
+
+\begin{figure}[t]
+    \centering
+
+    \tikzset{
+        box/.style={
+            rectangle, draw=black, minimum width=17mm, minimum height=8mm,
+        },
+    }
+
+    \begin{tikzpicture}
+        [
+            node distance = 2mm and 7mm,
+        ]
+        \node                                              (in)  {};
+        \node[box, right=of in]                            (enc) {Encoder};
+        \node[box, minimum width=25mm, right=of enc]       (mod) {Modulator};
+        \node[box, below right=of mod]                     (cha) {Channel};
+        \node[box, minimum width=25mm, below left=of cha]  (dem) {Demodulator};
+        \node[box, left=of dem]                            (dec) {Decoder};
+        \node[left=of dec]                                 (out) {};
+
+        \draw[-{latex}] (in)  -- (enc) node[midway, above] {$\bm{u}$};
+        \draw[-{latex}] (enc) -- (mod) node[midway, above] {$\bm{x}$};
+        \draw[-{latex}] (mod) -| (cha);
+        \draw[-{latex}] (cha) |- (dem);
+        \draw[-{latex}] (dem) -- (dec) node[midway, above] {$\bm{y}$};
+        \draw[-{latex}] (dec) -- (out) node[midway, above] {$\hat{\bm{u}}$};
+    \end{tikzpicture}
+
+    \caption{Overview of a transmission system.}
+    \label{fig:Diagram of a transmission system}
+\end{figure}
+%
+
+%
+% Hard vs. soft information
+%
+
+\subsection{Low-Density Parity-Check Codes}
+
+%
+% Core concept
+%
+
+Shannon's noisy-channel coding theorem is stated for codes whose block
+length approaches infinity. This suggests that as the block length
+becomes larger, the performance of the considered codes should
+generally improve.
+However, the size of the \ac{pcm}, and thus in general the decoding complexity,
+of a linear block code grows quadratically with $n$.
+This would quickly render decoding intractable as we increase the block length.
+We can get around this problem by constructing $\bm{H}$ in such a
+manner that the number of nonzero entries grows less than quadratically, e.g.,
+only linearly.
+This is exactly the motivation behind \ac{ldpc} codes
+\cite[Ch.~1]{gallager_low_1960}.
+
+%
+% Tanner Graph, VNs and CNs
+%
+
+\ac{ldpc} codes belong to a class sometimes referred to as ``modern codes''.
+These differ from ``classical codes'' in their decoding algorithms:
+Classical codes are usually decoded using one-step hard-decision decoding,
+whereas modern codes are suitable for iterative soft-decision
+decoding \cite[Preface]{ryan_channel_2009}. The iterative decoding algorithms
+in question are generally defined in terms of message passing on the
+\textit{Tanner graph} of the code. The Tanner graph is a bipartite
+graph that constitutes an alternative representation of the \ac{pcm}.
+We define two types of nodes: \acp{vn}, corresponding to codeword
+bits, and \acp{cn}, corresponding to individual parity checks.
+We then construct the Tanner graph by connecting each \ac{cn} to
+the \acp{vn} that make up the corresponding parity check
+\cite[Sec.~5.1.2]{ryan_channel_2009}.
+Figure \ref{PCM and Tanner graph of the Hamming code} shows this
+construction for the [7,4,3]-Hamming code.
+%
+\begin{figure}[t]
+    \centering
+
+    \begin{align*}
+        \bm{H} =
+        \begin{pmatrix}
+            0 & 1 & 1 & 1 & 1 & 0 & 0 \\
+            1 & 0 & 1 & 1 & 0 & 1 & 0 \\
+            1 & 1 & 0 & 1 & 0 & 0 & 1 \\
+        \end{pmatrix}
+    \end{align*}
+
+    \vspace*{2mm}
+
+    \tikzset{
+        VN/.style={
+            circle, fill=KITgreen, minimum width=1mm, minimum height=1mm,
+        },
+        CN/.style={
+            rectangle, fill=KITblue, minimum width=1mm, minimum height=1mm,
+        },
+    }
+
+    \begin{tikzpicture}
+        \node[VN, label=above:$x_1$]                    (vn1) {};
+        \node[VN, right=12mm of vn1, label=above:$x_2$] (vn2) {};
+        \node[VN, right=12mm of vn2, label=above:$x_3$] (vn3) {};
+        \node[VN, right=12mm of vn3, label=above:$x_4$] (vn4) {};
+        \node[VN, right=12mm of vn4, label=above:$x_5$] (vn5) {};
+        \node[VN, right=12mm of vn5, label=above:$x_6$] (vn6) {};
+        \node[VN, right=12mm of vn6, label=above:$x_7$] (vn7) {};
+
+        \node[
+            CN, below=25mm of vn4,
+            label={below:$x_1 + x_3 + x_4 + x_6 = 0$}
+        ] (cn2) {};
+        \node[
+            CN, left=40mm of cn2,
+            label={below:$x_2 + x_3 + x_4 + x_5 = 0$}
+        ] (cn1) {};
+        \node[
+            CN, right=40mm of cn2,
+            label={below:$x_1 + x_2 + x_4 + x_7 = 0$}
+        ] (cn3) {};
+
+        \foreach \n in {2,3,4,5} {
+            \draw (cn1) -- (vn\n);
+        }
+
+        \foreach \n in {1,3,4,6} {
+            \draw (cn2) -- (vn\n);
+        }
+
+        \foreach \n in {1,2,4,7} {
+            \draw (cn3) -- (vn\n);
+        }
+    \end{tikzpicture}
+
+    \caption{The \ac{pcm} and corresponding Tanner graph of the
+    [7,4,3]-Hamming code.}
+    \label{PCM and Tanner graph of the Hamming code}
+\end{figure}
+
+%
+% N_V(j), N_C(i)
+%
+
+Mathematically, we represent a \ac{vn} using the index $i \in
+\mathcal{I} := \left[
+1 : n \right]$ and a \ac{cn} using the index $j \in \mathcal{J}
+:= \left[ 1 : m \right]$.
+We can then encode the information contained in the graph by defining
+the neighborhood of a variable node $i$ as
+$\mathcal{N}_\text{V} (i) = \left\{ j \in \mathcal{J} : \bm{H}_{j,i}
+= 1 \right\}$
+and that of a check node $j$ as
+$\mathcal{N}_\text{C} (j) = \left\{ i \in \mathcal{I} : \bm{H}_{j,i}
+= 1 \right\}$.
+
+%
+% Error floor and waterfall regions
+%
+
+We typically evaluate the performance of LDPC codes using the
+\ac{ber} or the \ac{fer} (a \textit{frame} referes to one whole
+transmitted block in this context).
+Considering an \ac{awgn} channel, \autoref{fig:ldpc-perf} shows a
+qualitative performance characteristic of an \ac{ldpc} code
+\cite[Fig.~1]{costello_spatially_2014}. We talk of the
+\textit{waterfall} and the \textit{error floor} regions.
+
+\begin{figure}[t]
+    \centering
+
+    \begin{tikzpicture}
+        \begin{axis}[
+                width=12cm,
+                height=9cm,
+                xlabel={Signal-to-noise ratio},
+                ylabel={Error rate},
+                % xmin=0, xmax=6,
+                enlarge x limits=false,
+                ymin=1e-9, ymax=1,
+                ticks=none,
+                % y tick label={},
+                ymode=log,
+                grid=both,
+                grid style={line width=0.2pt, draw=gray!30},
+                major grid style={line width=0.4pt, draw=gray!50},
+                legend pos=north east,
+                legend cell align={left},
+            ]
+
+            \addplot+[mark=none, solid, smooth, KITblue] coordinates {
+                (4.5789E-01, 1.1821E-01)
+                (6.6842E-01, 9.4575E-02)
+                (8.6316E-01, 5.2657E-02)
+                (1.0421E+00, 2.2183E-02)
+                (1.1789E+00, 8.3588E-03)
+                (1.3368E+00, 1.4835E-03)
+                (1.4895E+00, 1.6852E-04)
+                (1.5842E+00, 2.8285E-05)
+                (1.6737E+00, 4.2465E-06)
+                (1.7684E+00, 3.4519E-07)
+                (1.8316E+00, 3.9213E-08)
+                (1.8684E+00, 6.2247E-09)
+                (1.9053E+00, 1E-09)
+            };
+            \addlegendentry{Regular}
+
+            \addplot+[mark=none, solid, smooth, KITorange] coordinates {
+                (4.5789E-01, 1.1821E-01)
+                (6.4211E-01, 4.9800E-02)
+                (7.5263E-01, 1.2700E-02)
+                (8.1579E-01, 2.3177E-03)
+                (8.6842E-01, 3.5779E-04)
+                (9.1053E-01, 5.3716E-05)
+                (9.4737E-01, 4.8818E-06)
+                (9.8947E-01, 6.5555E-07)
+                (1.0421E+00, 9.5713E-08)
+                % (1.0684E+00, 2.9670E-08)
+                (1.1474E+00, 1.2499E-08)
+                (1.3000E+00, 7.1560E-09)
+                (1.4579E+00, 6.0535E-09)
+                % (1.6105E+00, 5E-09)
+                (1.9579E+00, 4E-09)
+                (2.2947E+00, 3.1876E-09)
+                % (2.8842E+00, 2.0403E-09)
+            };
+            \addlegendentry{Irregular}
+
+            \draw[gray, densely dashed]
+            (axis cs:0.65, 2e-3) rectangle (axis cs:1.65, 5e-5);
+            \node[below] at (axis cs:1.15, 6e-5) {Waterfall};
+
+            \draw[gray, densely dashed]
+            (axis cs:1, 6e-8) rectangle (axis cs:2, 2e-9);
+            \node[above] at (axis cs:1.5, 7e-8) {Error floor};
+        \end{axis}
+    \end{tikzpicture}
+
+    \caption{
+        Qualitative performance characteristic of \ac{ldpc} code
+        in an \ac{awgn} channel. Adapted from
+        \cite[Fig.~1]{costello_spatially_2014}.
+    }
+    \label{fig:ldpc-perf}
+\end{figure}
+
+Broadly, there are two kinds of \ac{ldpc} codes, \textit{regular} and
+\textit{irregular}.
+Regular codes are characterized by the fact that the weights, i.e.,
+the numbers of ones, of their rows and columns are constant
+\cite[Sec.~5.1.1]{ryan_channel_2009}.
+Already during their introduction, regular \ac{ldpc} codes were shown to have
+a minimum distance scaling linearly with the block length $n$ for
+large values \cite[Ch.~2,~Theorem~1]{gallager_low_1960},
+which leads to them not exhibiting an error floor under \ac{ml} decoding.
+Irregular codes, on the other hand, generally do exhibit an error floor,
+their redeeming quality being the ability to reach near-capacity
+performance in the waterfall region \cite[Intro.]{costello_spatially_2014}.
+
+\subsection{Spatially-Coupled LDPC Codes}
+
+A relatively recent development in the world of \ac{ldpc} codes is
+that of \ac{sc}-\ac{ldpc} codes.
+Their key feature is that they combine the best properties of regular
+and irregular codes.
+They have a minimum distance that grows linearly with $n$, promising
+good error floor behavior, and capacity approaching
+iterative decoding behavior, promising good performance in the
+waterfall region \cite[Intro.]{costello_spatially_2014}.
+
+The essential property of \ac{sc}-\ac{ldpc} codes is that codewords
+from different \textit{spatial positions}, that would ordinarily be sent
+one after the other independently, are coupled.
+This is achieved by connecting some \acp{vn} of one spatial position to
+\acp{cn} of another, resulting in a \ac{pcm} of the form
+\cite[Eq.~1]{hassan_fully_2016}
+%
+\begin{align*}
+    \bm{H} =
+    \begin{pmatrix}
+        \bm{H}_0(1) &        &             \\
+        \vdots      & \ddots &             \\
+        \bm{H}_K(1) &        & \bm{H}_0(L) \\
+        & \ddots &             \\
+        &        & \bm{H}_K(L) \\
+    \end{pmatrix}
+    ,
+\end{align*}
+%
+where $K \in \mathbb{N}$ is the \textit{coupling width} and $L \in
+\mathbb{N}$ is the number of spatial positions.
+This construction results in a Tanner graph as depicted in
+\autoref{fig:sc-ldpc-tanner}.
+
+\begin{figure}[t]
+    \centering
+
+    \tikzset{
+        VN/.style={
+            circle, fill=KITgreen, minimum width=1mm, minimum height=1mm,
+        },
+        CN/.style={
+            rectangle, fill=KITblue, minimum width=1mm, minimum height=1mm,
+        },
+    }
+
+    \begin{tikzpicture}[node distance=7mm and 1cm]
+        \node[VN]                  (vn00) {};
+        \node[VN, below = of vn00] (vn01) {};
+        \node[VN, below = of vn01] (vn02) {};
+        \node[VN, below = of vn02] (vn03) {};
+        \node[VN, below = of vn03] (vn04) {};
+
+        \coordinate (temp) at ($(vn01)!0.5!(vn02)$);
+
+        \node[CN, right = of temp] (cn00) {};
+        \node[CN, below = of cn00] (cn01) {};
+
+        \draw (vn00) -- (cn00);
+        \draw (vn01) -- (cn00);
+        \draw (vn03) -- (cn00);
+        \draw (vn01) -- (cn01);
+        \draw (vn02) -- (cn01);
+        \draw (vn04) -- (cn01);
+
+        \foreach \i in {1,2,3} {
+            \pgfmathtruncatemacro{\previ}{\i-1}
+            \node[VN, right = 25mm of vn\previ 0] (vn\i0) {};
+
+            \foreach \j in {1,...,4} {
+                \pgfmathtruncatemacro{\prevj}{\j-1}
+                \node[VN, below = of vn\i\prevj] (vn\i\j) {};
+            }
+
+            \coordinate (temp) at ($(vn\i1)!0.5!(vn\i2)$);
+
+            \node[CN, right = of temp]  (cn\i0) {};
+            \node[CN, below = of cn\i0] (cn\i1) {};
+
+            \draw (vn\i0) -- (cn\i0);
+            \draw (vn\i1) -- (cn\i0);
+            \draw (vn\i3) -- (cn\i0);
+            \draw (vn\i1) -- (cn\i1);
+            \draw (vn\i2) -- (cn\i1);
+            \draw (vn\i4) -- (cn\i1);
+        }
+
+        \node[right = 25mm of vn30] (vn40) {};
+        \node[below = of vn40]      (vn41) {};
+        \node[below = of vn41]      (vn42) {};
+        \node[below = of vn42]      (vn43) {};
+        \node[below = of vn43]      (vn44) {};
+
+        \coordinate (temp) at ($(vn41)!0.5!(vn42)$);
+
+        \node[right = of temp] (cn40) {};
+        \node[below = of cn40] (cn41) {};
+
+        \foreach \i in {0,1,2} {
+            \pgfmathtruncatemacro{\next}{\i+1}
+            \pgfmathtruncatemacro{\nextnext}{\i+2}
+
+            \draw (vn\i 3) to[bend right] (cn\next 1);
+            \draw (vn\i 1) to[bend left]  (cn\nextnext 0);
+        }
+
+        \draw (vn33) to[bend right] (cn41);
+
+        \node at ($(cn40)!0.5!(cn41)$) {\dots};
+
+        \draw[decorate, decoration={brace, amplitude=10pt}]
+        ([xshift=-5mm,yshift=2mm]vn00.north) --
+        ([xshift=5mm,yshift=2mm]vn00.north -| cn20.north)
+        node[midway, above=4mm] {K};
+    \end{tikzpicture}
+
+    \caption{
+        Visualization of the coupling between the Tanner graphs
+        of individual spatial positions.
+    }
+    \label{fig:sc-ldpc-tanner}
+\end{figure}
+
+Note that at the first and last few spatial positions, some \acp{cn}
+have lower degrees.
+This leads to more reliable information about the
+\acp{vn} that, as we will see, is
+later passed to subsequent spatial positions during decoding.
+This is precisely the effect that leads to the good performance of
+\ac{sc}-\ac{ldpc} codes in the waterfall region \cite{costello_spatially_2014}.
+
+\subsection{Iterative Decoding}
+
+% Introduction
+
+\ac{ldpc} codes are generally decoded using efficient iterative
+algorithms, something that is possible due to their sparsity
+\cite[Sec.~5.3]{ryan_channel_2009}.
+The algorithm originally proposed alongside LDPC codes for this
+purpose by Gallager in 1960 is now known as the \ac{spa}
+\cite[5.4.1]{ryan_channel_2009}, also called \ac{bp}.
+
+The optimality criterion the \ac{spa} is built around is a
+symbol-wise \ac{map} decision \cite[Sec.~5.4.1]{ryan_channel_2009}.
+The core idea of the resulting algorithm is to view \acp{cn} as
+representing single-parity check codes and \acp{vn} as representing
+repetition codes.
+The algorithm alternates between consolidating soft information about
+the \acp{vn} in the \acp{cn}, and consolidating soft information about
+the \acp{cn} in the \acp{vn}.
+To this end, messages are passed back and forth along the edges of
+the Tanner graph.
+$L_{i\rightarrow j}$ represents a message passed from \ac{vn} $i$ to
+\ac{cn} j, $L_{i\leftarrow j}$ represents a message passed from
+\ac{cn} j to \ac{vn} i.
+The \acp{vn} additionally receive messages \cite[5.4.2]{ryan_channel_2009}
+\begin{align*}
+    \tilde{L}_i = \log \frac{P(X=0 \vert Y=y)}{P(X=1 \vert Y=y)},
+\end{align*}
+computed from the channel outputs.
+The consolidation of the information occurs in the \ac{vn} update
+\begin{align*}
+    L_{i\rightarrow j} = \tilde{L}_i + \sum_{j'\in \mathcal{N}(i)\setminus
+    j} L_{i\leftarrow j'}
+\end{align*}
+and the \ac{cn} update
+\begin{align*}
+    L_{i\leftarrow j} = 2\cdot \tanh^{-1} \left( \prod_{i'\in
+    \mathcal{N}(j)\setminus i} \tanh \frac{L_{i'\rightarrow j}}{2} \right)
+    .
+\end{align*}
+
+A basic assumption for the derivation of the \ac{spa} is that the
+messages are statistically independent.
+If the Tanner graph has cycles, however, this
+condition is not met.
+The shorter the cycles, the sooner this condition is violated and the
+worse the approximation becomes \cite[Sec.~5.4.4]{ryan_channel_2009}.
+Cycles of length four (so-called \emph{$4$-cycles}) are the shortest
+possible cycles and are thus especially problematic.
+
+% Min-sum algorithm
+
+A simplification of the \ac{spa} is the min-sum decoder. Here, the
+\ac{cn} update is approximated as \cite[Sec.~5.5.1]{ryan_channel_2009}
+\begin{align*}
+    L_{i \leftarrow j} = \prod_{i' \in \mathcal{N}(j)\setminus i}
+    \sign \left( L_{i' \rightarrow j} \right)
+    \cdot \min_{i' \in \mathcal{N}(j)\setminus i} \lvert
+    L_{i'\rightarrow j} \rvert
+    .
+\end{align*}
+
+% Sliding-window decoding
+
+For \ac{sc}-\ac{ldpc} codes, the iterative decoding process is wrapped by a
+windowing step. This is done to reduce the latency and memory requirements and
+also the overall computational complexity \cite{costello_spatially_2014}.
+To this end, the Tanner graph is split into several overlapping windows.
+During decoding, the messages that are passed along the edges of the
+graph in the overlapping regions are kept in memory and used for the
+decoding of subsequent blocks \cite[Sec.~III.~C.]{hassan_fully_2016}.
+
+\section{Quantum Mechanics and Quantum Information Science}
+\label{sec:Quantum Mechanics and Quantum Information Science}
+
+% TODO: Should the brief intro to QC be made later on or here?
+%%%%%%%%%%%%%%%%
+\subsection{Core Concepts and Notation}
+\label{subsec:Notation}
+
+\ldots can be very elegantly expressed using the language of
+linear algebra.
+\todo{Mention that we model the state of a quantum mechanical system
+as a vector}
+The so called Bra-ket or Dirac notation is especially appropriate,
+having been proposed by Paul Dirac in 1939 for the express purpose
+of simplifying quantum mechanical notation \cite{dirac_new_1939}.
+Two new symbols are defined, \emph{bra}s $\bra{\cdot}$ and
+\emph{ket}s $\ket{\cdot}$.
+Kets denote ordinary vectors, while bras denote their Hermitian conjugates.
+For example, two vectors specified by the labels $a$ and $b$
+respectively are written as $\ket{a}$ and $\ket{b}$.
+Their inner product is $\braket{a\vert b}$.
+
+\red{\textbf{Tensor product}}
+\red{\ldots
+    \todo{Introduce determinate state or use a different word?}
+    Take for example two systems with the determinate states $\ket{0}$
+    and $\ket{1}$. In general, the state of each can be written as the
+    superposition%
+    %
+    \begin{align*}
+        \alpha \ket{0} + \beta \ket{1}
+        .%
+    \end{align*}
+    %
+    Combining these two sytems into one, the overall state becomes%
+    %
+    \begin{align*}
+        &\mleft( \alpha_1 \ket{0} + \beta_1 \ket{1} \mright) \otimes
+        \mleft( \alpha_2 \ket{0} + \beta_2 \ket{1} \mright) \\
+        = &\alpha_1 \alpha_2 \ket{0} \ket{0}
+        + \alpha_1 \alpha_2 \ket{0} \ket{1}
+        + \beta_1 \alpha_2 \ket{1} \ket{0}
+        + \beta_1 \beta_2 \ket{1} \ket{1}
+        % =: &\alpha_{00} \ket{00}
+        % + \alpha_{01} \ket{01}
+        % + \alpha_{10} \ket{10}
+        % + \alpha_{11} \ket{11}
+        .%
+    \end{align*}%
+    %
+    \ldots When not ambiguous in the context, the tensor product
+    symbol may be omitted, e.g.,
+    \begin{align*}
+        \ket{0} \otimes \ket{0} = \ket{0}\ket{0}
+        .%
+    \end{align*}
+}
+
+As we will see, the core concept that gives quantum computing its
+power is entanglement. When two quantum mechanical systems are
+entangled, measuring the state of one will collapse that of the other.
+Take for example two subsystems with the overall state
+%
+\begin{align*}
+    \ket{\psi} = \frac{1}{\sqrt{2}} \mleft( \ket{0}\ket{0} +
+    \ket{1}\ket{1} \mright)
+    .%
+\end{align*}
+%
+If we measure the first subsystem as being in $\ket{0}$, we can
+be certain that a measurement of the second subsystem will also yield $\ket{0}$.
+Introducing a new notation for entangled states, we can write%
+%
+\begin{align*}
+    \ket{\psi} = \frac{1}{\sqrt{2}} \left( \ket{00} + \ket{11} \right)
+    .%
+\end{align*}
+%
+
+\subsection{Projective Measurements}
+\label{subsec:Projective Measurements}
+
+% TODO: Write
+
+%%%%%%%%%%%%%%%%
+\subsection{Quantum Gates}
+\label{subsec:Quantum Gates}
+
+\red{
+    \textbf{Content:}
+    \begin{itemize}
+        \item Bra-ket notation
+        \item The tensor product
+        \item Projective measurements (the related operators,
+            eigenvalues/eigenspaces, etc.)
+            \begin{itemize}
+                \item First explain what an operator is
+            \end{itemize}
+        \item Abstract intro to QC: Use gates to process qubit
+            states, similar to classical case
+        \item X, Z, Y operators/gates
+        \item Hadamard gate (+ X and Z are the same thing in differt bases)
+        \item Notation of operators on multi-qubit states
+        \item The Pauli, Clifford and Magic groups
+    \end{itemize}
+}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Quantum Error Correction}
+\label{sec:Quantum Error Correction}
+
+\red{
+    \textbf{Content:}
+    \begin{itemize}
+        \item  General context
+            \begin{itemize}
+                \item Why we want QC
+                \item Why we need QEC (correcting errors due to noisy gates)
+                \item Main challenges of QEC compared to classical
+                    error correction
+            \end{itemize}
+        \item Stabilizer codes
+            \begin{itemize}
+                \item Definition of a stabilizer code
+                \item The stabilizer its generators (note somewhere
+                        that the generators have to commute to be able to
+                    be measured without disturbing each other)
+                \item syndrome extraction circuit
+                \item Stabilizer codes are effectively the QM
+                    % TODO: Actually binary linear codes or just linear codes?
+                    equivalent of binary linear codes (e.g.,
+                    expressible via check matrix)
+            \end{itemize}
+        \item Digitization of errors
+        \item CSS codes
+        \item Color codes?
+        \item Surface codes?
+        \item Fault tolerant error correction (gates with which we do
+            error correction are also noisy)
+            \begin{itemize}
+                \item Transversal operations
+                \item \dots
+            \end{itemize}
+        \item Circuit level noise
+        \item Detector error model
+            \begin{itemize}
+                \item Columns of the check matrix represent different
+                    possible error patterns $\rightarrow$ Check matrix
+                    doesn't quite correspond to the codewords we used
+                    initially anymore, but some similar structure ist
+                    still there (compare with syndrome)
+            \end{itemize}
+    \end{itemize}
+    \textbf{General Notes:}
+    \begin{itemize}
+        \item Give a brief overview of the history of QEC
+        \item Note (and research if this is actually correct) that QC
+            was developed on an abstract level before thinking of
+            what hardware to use
+        \item Note that there are other codes than stabilizer codes
+            (and research and give some examples), but only
+            stabilizer codes are considered in this work
+        \item Degeneracy
+        \item The QEC decoding problem (considering degeneracy)
+    \end{itemize}
+}
+
+\subsection{Stabilizer Codes}
+\subsection{CSS Codes}
+\subsection{Quantum Low-Density Parity-Check Codes}
+

src/thesis/chapters/3_fault_tolerant_qec.tex

@@ -0,0 +1,13 @@
+\chapter{Fault Tolerant QEC}
+\section{Fault Tolerance}
+\section{Noise Models}
+\subsection{Depolarizing Channel}
+\subsection{Phenomenological Noise}
+\subsection{Circuit-Level Noise}
+\section{Detector Error Models}
+\subsection{Measurement Syndrome Matrix}
+\subsection{Detector Error Matrix}
+\subsection{Detector Error Models}
+\section{Practical Considerations}
+\subsection{Practical Methodology}
+\subsection{Stim}

src/thesis/chapters/4_decoding_under_dems.tex

@@ -0,0 +1,5 @@
+\chapter{Decoding under Detector Error Models}
+\section{Sliding-Window Decoding}
+\section{Treating Detector Error Matrices like SC-LDPC Codes}
+\section{Soft-Information Aware Sliding-Window Decoding}
+\section{Numerical Results and Analysis}

src/thesis/chapters/5_conclusion_and_outlook.tex

@@ -0,0 +1 @@
+\chapter{Conclusion and Outlook}

src/thesis/main.tex

@@ -0,0 +1,107 @@
+\documentclass{lib/cel-thesis/cel-thesis}
+
+\usepackage[a4paper,left=3cm,right=3cm,top=2.5cm,bottom=2.5cm]{geometry}
+\usepackage{float}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{mleftright}
+\usepackage{bm}
+\usepackage{tikz}
+\usepackage{xcolor}
+\usepackage{pgfplots}
+\pgfplotsset{compat=newest}
+\usepackage{acro}
+\usepackage{braket}
+% \usepackage[
+%     backend=biber,
+%     style=ieee,
+%     sorting=nty,
+% ]{biblatex}
+\usepackage{todonotes}
+
+\usetikzlibrary{calc, positioning, arrows, fit}
+
+\usetikzlibrary{external}
+\tikzexternalize
+
+\makeatletter
+\renewcommand{\todo}[2][]{\tikzexternaldisable\@todo[#1]{#2}\tikzexternalenable}
+\makeatother
+
+%
+%
+% Custom commands
+%
+%
+
+\newcommand{\red}[1]{\textcolor{red}{#1}}
+\newcommand{\figwidth}{10cm}
+\newcommand{\figheight}{7.5cm}
+
+%
+%
+% Acronyms
+%
+%
+
+\input{acronyms.tex}
+
+\usepackage{babelbib}
+\setlanguage
+
+\usepackage{caption}
+\usepackage{bm}
+\usepackage{subcaption}
+\usepackage{todonotes} % great for draft annotations
+\DeclareCaptionLabelFormat{bf-nodot}{\textbf{#1}~\textbf{#2}}
+\captionsetup{labelformat=bf-nodot,labelsep=colon}
+
+%
+%
+% Content
+%
+%
+
+\thesisTitle{Fault Tolerant Quantum Error Correction}
+\thesisType{Master's Thesis}
+\thesisAuthor{Andreas Tsouchlos}
+\thesisAdvisor{Prof. Dr.-Ing. Laurent Schmalen}
+\thesisHeadOfInstitute{Prof. Dr.-Ing. Laurent Schmalen}
+% \thesisHeadOfInstitute{Prof. Dr.-Ing. Peter Rost}
+%\thesisHeadOfInstitute{Prof. Dr.-Ing. Peter Rost\\Prof. Dr.-Ing.
+% Laurent Schmalen}
+\thesisSupervisor{Jonathan Mandelbaum}
+\thesisStartDate{01.11.2025}
+\thesisEndDate{04.05.2026}
+\thesisSignatureDate{Signature date}
+\thesisLanguage{english}
+
+\begin{document}
+\pagenumbering{roman}  % all the preliminaries should be counted roman style
+
+\maketitle
+\newpage
+
+% \include{chapters/abstract}
+
+\cleardoublepage
+\pagenumbering{arabic}
+
+\tableofcontents
+\cleardoublepage
+
+\input{chapters/1_introduction.tex}
+\input{chapters/2_fundamentals.tex}
+\input{chapters/3_fault_tolerant_qec.tex}
+\input{chapters/4_decoding_under_dems.tex}
+\input{chapters/5_conclusion_and_outlook.tex}
+
+% \appendix
+% \listoffigures
+% \listoftables
+% \include{abbreviations}
+
+\bibliography{lib/cel-thesis/IEEEabrv,src/thesis/bibliography}
+
+\end{document}
+