Write CSS codes section

src/thesis/chapters/2_fundamentals.tex

@@ -1359,7 +1359,8 @@ as we have to consider both the $X$ and $Z$ type operators that make up
 the stabilizers.
 Take for example the Steane code \cite[Eq.~10.83]{nielsen_quantum_2010}.
 We can describe it using the check matrix
-\begin{align*}
+\begin{align}
+    \label{eq:steane}
     \left[
         \begin{array}{ccccccc|ccccccc}
             0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
@@ -1371,7 +1372,7 @@ We can describe it using the check matrix
         \end{array}
     \right]
     .%
-\end{align*}
+\end{align}
 We can understand each row as defining a stabilizer.
 The first $n$ columns correspond to $X$ operators acting on the
 corresponding physical qubit, the rest to the $Z$ operators.
@@ -1405,13 +1406,43 @@ corresponding physical qubit, the rest to the $Z$ operators.
 
 Stabilizer codes are especially practical to work with when they can
 handle $X$- and $Z$-type errors independently.
-We can then separate the stabilizer generators into some with only
-$Z$ operators and some with only $X$ operators.
+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
+of the Steane code.
 
-\indent\red{[Z-type operators for X type errors and vice versa ]} \\
-\indent\red{[Construction from two binary linear codes
-\cite[p.~452,469]{nielsen_quantum_2010}]} \\
+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
+$\mathcal{C}_2$, 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}
+\begin{align*}
+    \mathcal{C}_2^\perp := \left\{ \bm{x}' \in \mathbb{F}^2 :
+    \bm{x}' \bm{x}^\text{T} = 0 ~\forall \bm{x} \in \mathcal{C}_2 \right\}
+    ,%
+\end{align*}
+we can construct the check matrix as
+\begin{align*}
+    \left[
+        \begin{array}{c|c}
+            \bm{H}(\mathcal{C}_2^\perp) & \bm{0} \\
+            \bm{0} & \bm{H}(\mathcal{C}_1)
+        \end{array}
+    \right]
+    .%
+\end{align*}
+In order to yield a valid stabilizer code, $\mathcal{C}_1$ and
+$\mathcal{C}_2$ must satisfy the commutativity condition
+\begin{align*}
+    \bm{H}(\mathcal{C}_2^\perp) \bm{H}(\mathcal{C}_1)^\text{T} = \bm{0}
+    .%
+\end{align*}
+We can ensure this is the case by choosing them such that
+$\mathcal{C}_2 \subset \mathcal{C}_1$.
 
 %%%%%%%%%%%%%%%%
 \subsection{Quantum Low-Density Parity-Check Codes}