Write BB code paragraph

src/thesis/acronyms.tex

@@ -38,6 +38,11 @@
     long=low-density parity-check
 }
 
+\DeclareAcronym{qldpc}{
+    short=QLDPC,
+    long=quantum low-density parity-check
+}
+
 \DeclareAcronym{ml}{
     short=ML,
     long=maximum likelihood
@@ -82,3 +87,8 @@
     short=PDF,
     long=probability density function
 }
+
+\DeclareAcronym{bb}{
+    short=BB,
+    long=bivariate bicycle
+}

src/thesis/chapters/2_fundamentals.tex

@@ -1400,23 +1400,47 @@ 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}
 
+% Intro
+
 Stabilizer codes are especially practical to work with when they can
-handle $X$- and $Z$-type errors independently.
+handle $X$ and $Z$ type errors independently.
 As $Z$ errors anti-commute with $X$ operators in the stabilizers and
 vice versa, this property translates into being able to split the
 stabilizers into some being made up of only $X$
 operators and some only of $Z$ operators.
 We call such codes \ac{css} codes.
-We can see this property in \autoref{eq:steane}, in the check matrix
+We can see this property in \autoref{eq:steane} in the check matrix
 of the Steane code.
 
+% Construction
+
 We can exploit this separate consideration of $X$ and $Z$ errors in
 the construction of \ac{css} codes.
-We can combine two binary linear codes $\mathcal{C}_1$ and
+We combine two binary linear codes $\mathcal{C}_1$ and
 $\mathcal{C}_2$, each responsible for correcting one type of error
 \cite[Sec.~10.5.6]{nielsen_quantum_2010}.
 Using the dual code of $\mathcal{C}_2$ \cite[Eq.~3.4]{ryan_channel_2009}
@@ -1437,10 +1461,11 @@ we can construct the check matrix as
 \end{align*}
 In order to yield a valid stabilizer code, $\mathcal{C}_1$ and
 $\mathcal{C}_2$ must satisfy the commutativity condition
-\begin{align*}
+\begin{align}
+    \label{eq:css_condition}
     \bm{H}(\mathcal{C}_2^\perp) \bm{H}(\mathcal{C}_1)^\text{T} = \bm{0}
     .%
-\end{align*}
+\end{align}
 We can ensure this is the case by choosing them such that
 $\mathcal{C}_2 \subset \mathcal{C}_1$.
 
@@ -1448,29 +1473,63 @@ $\mathcal{C}_2 \subset \mathcal{C}_1$.
 \subsection{Quantum Low-Density Parity-Check Codes}
 \label{subsec:Quantum Low-Density Parity-Check Codes}
 
-\noindent\red{[(?) Mention color and surface codes]} \\
-\noindent\red{[Constant overhead scaling]} \\
-\noindent\red{[Scaling of minimum distance with code length]} \\
-\noindent\red{[Bivariate Bicycle codes]} \\
-\noindent\red{[Decoding QLDPC codes (syndrome-based BP)]} \\
-\noindent\red{[Degeneracy -> BP+OSD, BPGD]} \\
-\noindent\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
-]} \\
+% Intro
 
-%%%%%%%%%%%%%%%%
-\subsection{Decoding Quantum Codes}
-\label{subsec:Decoding Quantum Codes}
+Various methods of constructing \ac{qec} codes exist
+\cite{swierkowska_eccentric_2025}.
+\red{[topological codes]}
+\red{[(?) Mention color and surface codes]}.
+A more recent development is that of quantum \ac{ldpc} (\acs{qldpc}) codes.
 
-\noindent\indent\red{[The QEC decoding problem
-\cite[Sec.~2.3]{yao_belief_2024}]} \\
-\indent\red{[+ Degeneracy]} \\
-\indent\red{[+ Short cycles]} \\
-\indent\red{[Fixing the error after finding it
-        \cite[Sec.~10.5.5]{nielsen_quantum_2010} -> This
-may require introducing the gates as unitary]}
+% Why QLDPC codes are interesting
+
+\indent\red{[Constant overhead scaling]} \\
+\indent\red{[Scaling of minimum distance with code length]} \\
+
+% Bivariate Bicycle codes
+
+% TODO: Introduce H_X and H_Z above
+A recent addition to the class of \ac{qldpc} codes is that of \ac{bb}
+codes \cite[Sec.~3]{bravyi_high-threshold_2024}.
+These are a special type of \ac{css} code, where $\bm{H}_X$ and
+$\bm{H}_Z$ are constructed from two matrices $\bm{A}$ and $\bm{B}$ as
+\begin{align*}
+    \bm{H}_X = [\bm{A} \vert \bm{B}]
+    \hspace*{5mm} \text{and} \hspace*{5mm}
+    \bm{H}_Z = [\bm{B}^\text{T} \vert \bm{A}^\text{T}]
+    .%
+\end{align*}
+This way, we can guarantee the satisfaction of the commutativity
+condition (\autoref{eq:css_condition}).
+To define $\bm{A}$ and $\bm{B}$ we first introduce some additional notation.
+We denote the identity matrix as $\bm{I_l} \in \mathbb{F}^{l\times l}$ and
+the \emph{cyclic shift matrix} as $\bm{S_l} \in \mathbb{F}^{l\times
+l},~S_{l,i,j}= \delta_{i+1,j}$, with $l \in \mathbb{N}$.
+We further define
+\begin{align*}
+    x = \bm{S}_l \otimes \bm{I}_m
+    \hspace*{5mm} \text{and} \hspace*{5mm}
+    y = \bm{I}_l \otimes \bm{S}_m
+    .%
+\end{align*}
+We can then construct $\bm{A}$ and $\bm{B}$ as bivariate polynomials
+\begin{align*}
+    \bm{A} = \bm{A}_1 + \bm{A}_2 + \bm{A}_3
+    \hspace*{5mm} \text{and} \hspace*{5mm}
+    \bm{B} = \bm{B}_1 + \bm{B}_2 + \bm{B}_3
+    ,%
+\end{align*}
+where $\bm{A}_i$ and $\bm{B}_i$ are powers of $\bm{x}$ or $\bm{y}$.
+\ac{bb} codes have large minimum distance $d_\text{min}$ and high rate,
+offering a more than 10-time reduction of encoding overhead over the
+surface code.
+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
+
+\indent\red{[Decoding QLDPC codes (syndrome-based BP)]} \\
+\indent\red{[Short cycles]} \\
+\indent\red{[Degeneracy + short cycles -> BP+OSD, BPGD]} \\
 

src/thesis/chapters/3_fault_tolerant_qec.tex

@@ -10,9 +10,5 @@
 \subsection{Detector Error Models}
 \section{Practical Considerations}
 \subsection{Practical Methodology}
-
-\indent\red{[(?) Figure from presentation, showing where the LER
-calculation takes place]} \\
-
 \subsection{Stim}