Finish first draft of text for fundamentals

src/thesis/acronyms.tex

@@ -8,6 +8,11 @@
     long=belief propagation
 }
 
+\DeclareAcronym{bpgd}{
+    short=BPGD,
+    long=belief propagation with guided decimation
+}
+
 \DeclareAcronym{nms}{
     short=NMS,
     long=normalized min-sum

src/thesis/chapters/2_fundamentals.tex

@@ -517,6 +517,7 @@ 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}
+\label{subsec:Iterative Decoding}
 
 % Introduction
 
@@ -1373,12 +1374,10 @@ We can describe it using the check matrix
     \right]
     .%
 \end{align}
-We can understand each row as defining a stabilizer.
+% TODO: Check X vs. Z
 The first $n$ columns correspond to $X$ operators acting on the
 corresponding physical qubit, the rest to the $Z$ operators.
 
-% TODO: Write
-
 \begin{figure}[t]
     \centering
 
@@ -1400,26 +1399,6 @@ corresponding physical qubit, the rest to the $Z$ operators.
     \label{fig:sec}
 \end{figure}
 
-% %%%%%%%%%%%%%%%%
-% \subsection{Decoding Stabilizer Codes}
-% \label{subsec:Decoding Stabilizer Codes}
-%
-%
-%
-% \noindent\indent\red{[The QEC decoding problem
-% \cite[Sec.~2.3]{yao_belief_2024}]} \\
-% \indent\red{[+ Degeneracy]} \\
-% \indent\red{[``The task of decoding is therefore to infer, from a
-%         measured syndrome, the most likely error coset rather than the exact
-%         physical error.''
-% % tex-fmt: off
-% \cite[Sec.~II~B)]{koutsioumpas_colour_2025}%
-% % tex-fmt: on
-% ]} \\
-% \indent\red{[Fixing the error after finding it
-%         \cite[Sec.~10.5.5]{nielsen_quantum_2010} -> This
-% may require introducing the gates as unitary]}
-
 %%%%%%%%%%%%%%%%
 \subsection{Calderbank-Shor-Steane Codes}
 \label{subsec:Calderbank-Shor-Steane Codes}
@@ -1477,14 +1456,17 @@ $\mathcal{C}_2 \subset \mathcal{C}_1$.
 
 Various methods of constructing \ac{qec} codes exist
 \cite{swierkowska_eccentric_2025}.
-\red{[topological codes]}
+Topological codes, for example, encode information in the features of
-\red{[(?) Mention color and surface codes]}.
+a lattice and are intrinsically robust against local errors.
-A more recent development is that of quantum \ac{ldpc} (\acs{qldpc}) codes.
+Among these, the \emph{surface code} is the most widely studied.
-
+Another example are concatenated codes, which nest one code within
-% Why QLDPC codes are interesting
+another, allowing for especially simple and flexible constructions
-
+\cite[Sec.~3.2]{swierkowska_eccentric_2025}.
-\indent\red{[Constant overhead scaling]} \\
+An area of research that has recently seen more attention is that of
-\indent\red{[Scaling of minimum distance with code length]} \\
+quantum \ac{ldpc} (\acs{qldpc}) codes.
+They have much better encoding efficiency than, e.g., the surface
+code, scaling up of which would be prohibitively expensive
+\cite[Sec.~I]{bravyi_high-threshold_2024}.
 
 % Bivariate Bicycle codes
 
@@ -1527,9 +1509,168 @@ Additionally, they posess short-depth syndrome measurement circuits,
 leading to lower time requirements for the syndrome extraction
 and thus lower error rates \cite[Sec.~1]{bravyi_high-threshold_2024}.
 
-% Decoding QLDPC codes
+% Syndrome-based BP
 
-\indent\red{[Decoding QLDPC codes (syndrome-based BP)]} \\
+As we saw in \autoref{subsec:Stabilizer Measurements}, we work only
-\indent\red{[Short cycles]} \\
+with the parity information contained in the syndrome, to avoid
-\indent\red{[Degeneracy + short cycles -> BP+OSD, BPGD]} \\
+disturbing the quantum states of indivudual qubits.
+This necessitates a modification of the standard \ac{bp} algorithm
+introduced in \autoref{subsec:Iterative Decoding}
+\cite[Sec.~3.1]{yao_belief_2024}.
+Instead of attempting to find the most likely codeword directly, the
+algorithm will now try to find an error pattern $\hat{\bm{e}} \in
+\mathbb{F}_2^n$ that satisfies
+\begin{align*}
+    \bm{H} \hat{\bm{e}}^\text{T} = \bm{s}
+    .%
+\end{align*}
+To this end, we initialize the channel \acp{llr} as
+\begin{align*}
+    \tilde{L}_i = \log{\frac{P(X_i = 0)}{P(X_i = 1)}} = \frac{1 - p_i}{p_i}
+    ,%
+\end{align*}
+where $p_i$ is the prior probability of error of \ac{vn} $i$.
+Additionally, we amend the \ac{cn} update to consider the parity
+indicated by the syndrome, calculating
+\begin{align*}
+    L_{i\leftarrow j} = 2\cdot (-1)^{s_j} \cdot \tanh^{-1} \left( \prod_{i'\in
+    \mathcal{N}(j)\setminus \{i\}} \tanh \frac{L_{i'\rightarrow j}}{2} \right)
+    .
+\end{align*}
+The resulting syndrome-based \ac{bp} algorithm is shown in
+algorithm \ref{alg:syndome_bp}.
+
+% tex-fmt: off
+\tikzexternaldisable
+\begin{algorithm}[t]
+    \caption{Binary syndrome-based belief propagation (BP) algorithm.}
+    \label{alg:syndome_bp}
+    \begin{algorithmic}[1]
+        \State \textbf{Initialize:} $\tilde{L}_i \leftarrow
+               \log \frac{1-p_i}{p_i}$ for all $i \in \mathcal{I}$
+        \State \textbf{Initialize:} $L_{i \rightarrow j} \leftarrow
+               \tilde{L}_i$ for all $i \in \mathcal{I},\, j \in \mathcal{N}_\text{V}(i)$
+        \State \textbf{Initialize:} $\hat{e} \leftarrow \bm{0}$
+
+        \For{$\ell = 1, \ldots, n_\text{iter}$}
+
+            \For{$j \in \mathcal{J}$}
+                \For{$i \in \mathcal{N}_\text{C}(j)$}
+                    \State $\displaystyle L_{i \leftarrow j} \leftarrow
+                           2\cdot(-1)^{s_j}\cdot\tanh^{-1}
+                           \!\left(
+                               \prod_{i' \in \mathcal{N}_\text{C}(j)\setminus\{i\}}
+                               \tanh\frac{L_{i'\rightarrow j}}{2}
+                           \right)$
+                \EndFor
+            \EndFor
+
+            \For{$i \in \mathcal{I}$}
+                \For{$j \in \mathcal{N}_\text{V}(i)$}
+                    \State $\displaystyle L_{i \rightarrow j} \leftarrow
+                           \tilde{L}_i +
+                           \sum_{j' \in \mathcal{N}_\text{V}(i)\setminus\{j\}}
+                           L_{i \leftarrow j'}$
+                \EndFor
+            \EndFor
+
+            \For{$i \in \mathcal{I}$}
+                \State $\displaystyle \hat{e}_i \leftarrow
+                       \mathbbm{1}\left\{
+                           \tilde{L}_i +
+                           \sum_{j \in \mathcal{N}_\text{V}(i)} L_{i \leftarrow j} < 0
+                       \right\}$
+            \EndFor
+
+            \If{$\bm{H}\hat{\bm{e}}^\text{T} = \bm{s}$}
+            \State \textbf{break}
+            \EndIf
+
+        \EndFor
+        \State \textbf{return} $\hat{\bm{e}}$
+    \end{algorithmic}
+\end{algorithm}
+\tikzexternalenable
+% tex-fmt: on
+
+% Degeneracy and short cycles
+
+Decoding \ac{qldpc} codes brings with it some unique challenges.
+One issue is that of \emph{quantum degeneracy}.
+Because errors that differ by a stabilizer have the same impact on
+all codewords, there can be multiple minimum-weight solutions to the
+quantum decoding problem \cite[Sec.~II.C.]{babar_fifteen_2015}
+\cite[Sec.~V]{roffe_decoding_2020}.
+This leads to the decoding algorithm getting confused about the
+direction to proceed in \cite[Sec.~5]{yao_belief_2024}.
+Another problem is that due to the commutativity property of the stabilizers,
+quantum codes inherently contain short cycles
+\cite[Sec.~IV.C]{babar_fifteen_2015}.
+As discussed in \autoref{subsec:Iterative Decoding}, these lead to
+the violation of the independence assumption of the messages passed
+during decoding, impeding performance.
+
+% BPGD
+
+The aforementioned issues both manifest themselves as convergence problems
+of the \ac{bp} algorithm, and different ways of modifying the algorithm
+to aide with convergence exist.
+One approach is to use \ac{bp} with guided decimation (\acs{bpgd})
+\cite[Alg.~1]{yao_belief_2024}.
+Here, a number $T\in \mathbb{N}$ of \ac{bp} iterations are performed,
+before \emph{decimating} the most reliable \ac{vn}, i.e., performing
+a hard decision and excluding it from further decoding.
+This constrains the solution space more and more as the decoding
+progresses, encouraging the algorithm to converge to one of the
+solutions \cite[Sec.~5]{yao_belief_2024}.
+Algorithm \ref{alg:bpgd} shows this process.
+Note that as the Tanner graph only has $n$ \acp{vn}, this is a
+natural constraint on the maximum number of iterations.
+
+% TODO: Explain that setting the channel LLR to infinity is the same
+% as a hard decision and ignoring the VN in the further decoding
+% tex-fmt: off
+\tikzexternaldisable
+\begin{algorithm}[t]
+    \caption{Belief propagation with guided decimation (BPGD) algorithm.}
+    \label{alg:bpgd}
+    \begin{algorithmic}[1]
+        \State \textbf{Initialize:} $\tilde{L}_i \leftarrow
+               \log \frac{1-p_i}{p_i}$ for all $i \in \mathcal{I}$
+        \State \textbf{Initialize:} $L_{i \rightarrow j} \leftarrow
+               \tilde{L}_i$ for all $i \in \mathcal{I},\, j \in \mathcal{N}_\text{V}(i)$
+        \State \textbf{Initialize:} $\hat{e} \leftarrow \bm{0}$
+        \State \textbf{Initialize:} $\mathcal{I}' \leftarrow \mathcal{I}$
+
+        \For{$r = 1, \ldots, n$}
+
+            \For{$\ell = 1, \ldots, T$}
+                \State Perform \ac{cn} update
+                \State Perform \ac{vn} update
+                \State  $L^\text{total}_i \leftarrow \tilde{L}_i + \sum_{j \in \mathcal{N}_\text{V}(i)} L_{i \leftarrow j}$
+            \EndFor
+
+            \For{$i \in \mathcal{I}$}
+                \State $\displaystyle \hat{e}_i \leftarrow
+                       \mathbbm{1}\left\{ L^\text{total}_i \right\}$
+            \EndFor
+
+            \If{$\bm{H}\hat{\bm{e}}^\text{T} = \bm{s}$}
+                \State \textbf{break}
+            \Else
+                \State $i_\text{max} \leftarrow \argmax_{i \in \mathcal{I}'} \lvert L^\text{total}_i \rvert $
+                \If{$L^\text{total}_{i_\text{max}} < 0$}
+                    \State $\tilde{L}_{i_\text{max}} \leftarrow -\infty$
+                \Else
+                    \State $\tilde{L}_{i_\text{max}} \leftarrow +\infty$
+                \EndIf
+                \State $\mathcal{I}' \leftarrow \mathcal{I}'\setminus\{i_\text{max}\}$
+            \EndIf
+
+        \EndFor
+        \State \textbf{return} $\hat{\bm{e}}$
+    \end{algorithmic}
+\end{algorithm}
+\tikzexternalenable
+% tex-fmt: on
 

src/thesis/main.tex

@@ -6,12 +6,15 @@
 \usepackage{amsfonts}
 \usepackage{mleftright}
 \usepackage{bm}
+\usepackage{bbm}
 \usepackage{tikz}
 \usepackage{xcolor}
 \usepackage{pgfplots}
 \pgfplotsset{compat=newest}
 \usepackage{acro}
 \usepackage{braket}
+\usepackage{listings}
+\usepackage{caption}
 % \usepackage[
 %     backend=biber,
 %     style=ieee,
@@ -20,9 +23,10 @@
 \usepackage{todonotes}
 \usepackage{quantikz}
 \usepackage{stmaryrd}
+\usepackage{algorithm}
+\usepackage[noEnd=false]{algpseudocodex}
 
 \usetikzlibrary{calc, positioning, arrows, fit}
-
 \usetikzlibrary{external}
 \tikzexternalize