Rewrite parts of measurement syndrome matrix subsection

src/thesis/chapters/3_fault_tolerant_qec.tex

@@ -161,18 +161,17 @@ different error locations in the circuit.
 We will illustrate the most widely used types of error models on the
 example of the three-qubit repetition code for $X$ errors.
 This is a code with check matrix
-\begin{align*}
-    \bm{H} =
+\begin{gather}
+    \label{eq:rep_code_H}
+    \bm{H}_Z =
     \left[
-        \begin{array}{ccc|ccc}
-            0 & 0 & 0 & 0 & 0 & 0 \\
-            0 & 0 & 0 & 0 & 0 & 0 \\
-            0 & 0 & 0 & 1 & 1 & 0 \\
-            0 & 0 & 0 & 0 & 1 & 1
+        \begin{array}{ccc}
+            1 & 1 & 0 \\
+            0 & 1 & 1
         \end{array}
     \right]
     .
-\end{align*}
+\end{gather}
 We can see that it has stabilizers $Z_1Z_2$ and $Z_2Z_3$.
 \autoref{fig:pure_syndrome_extraction} shows the corresponding
 syndrome extraction circuit.
@@ -452,7 +451,7 @@ matrix} matrix $\bm{\Omega} \in \mathbb{F}_2^{m\times N}$, with
     \end{cases}
     .%
 \end{align*}
-This matrix thus defines the code based on which error mechanism
+This matrix thus defines this new code based on which error mechanism
 flips which measurement, rather than the Pauli type and location of
 each error \cite[Sec.~1.4.3]{higgott_practical_2024}.
 To obtain $\bm{\Omega}$, we must propagate Pauli errors through the
@@ -464,16 +463,13 @@ circuit, tracking which measurements they affect
 % TODO: Fix syndrome dimension notation
 We turn to our example of the three-qubit repetition code to
 illustrate the construction of the syndrome measurement matrix.
-We begin by replicating the syndrome extraction circuitry, three
-times in this case, as can be seen in
-\autoref{fig:rep_code_multiple_rounds_bit_flip}.
-We consider only bit flip noise at this stage.
-For each syndrome extraction round we get an additional set of
-syndrome measurements.
-We combine these measurements by stacking them in a new vector $\bm{s}
-\in \mathbb{F}_2^{n_\text{rounds}\cdot(n-k)}$.
-To accomodate the additional syndrome bits, we extend the
-matrix $\bm{\Omega}$ representing the circuit by replicating the rows as well:
+We begin by extending our check matrix in \autoref{eq:rep_code_H}
+to represent three rounds of syndrome extraction.
+Each round yields an additional set of syndrome bits,
+and we combine them by stacking them in a new vector
+$\bm{s} \in \mathbb{F}_2^{n_\text{rounds}\cdot(n-k)}$.
+We thus have to replicate the rows of $\bm{\Omega}$, once for each
+additional syndrome measurement, to obtain
 \begin{align*}
     \bm{\Omega} =
     \begin{pmatrix}
@@ -486,21 +482,32 @@ matrix $\bm{\Omega}$ representing the circuit by replicating the rows as well:
     \end{pmatrix}
     .%
 \end{align*}
+\autoref{fig:rep_code_multiple_rounds_bit_flip}
+depicts the corresponding circuit.
+Note that we have not yet introduced error locations in the syndrome
+extraction circuitry, so we still consider only bit flip noise at this stage.
 Recall that $\bm{\Omega}$ describes which \ac{vn} is connected to
 which parity check and the syndrome indicates which parity checks
 are violated.
 This means that if an error exists at only a single \ac{vn}, we can
 read off the syndrome in the corresponding column.
+If errors occur at multiple locations, the resulting syndrome will be
+the linear combination of the respective columns.
+We thus have
+\begin{align*}
+    \bm{s} \in \text{span} \{\bm{\Omega}\}
+    .%
+\end{align*}
 
 % Expand to phenomenological
 
 We now whish to expand the error model to phenomenological noise, though
 only considering $X$ errors in this case.
-We introduce new error locations at the respective positions,
+We introduce new error locations at the appropriate positions,
 arriving at the circuit depicted in
 \autoref{fig:rep_code_multiple_rounds_phenomenological}.
 For each additional error location, we extend $\bm{\Omega}$ by
-appending the corresponding syndrome vector as a column:
+appending the corresponding syndrome vector as a column.
 \begin{gather*}
     \bm{\Omega} =
     \left(
@@ -518,8 +525,15 @@ appending the corresponding syndrome vector as a column:
             0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0
             & 0 & 1 & 1 & 0 & 1
         \end{array}
-    \right)
-    .%
+    \right) . \\[-6mm]
+    \hspace*{-58.7mm}
+    \underbrace{
+        \phantom{
+            \begin{array}{ccc}
+                0 & 0 & 0
+            \end{array}
+        }
+    }_\text{Original matrix}
 \end{gather*}
 Notice that the first three columns correspond to the original
 measurement syndrome matrix, as these columns correspond to the error
@@ -954,13 +968,12 @@ For circuit-level noise, various options exist, such as the \emph{SI1000}
 measurements) models \cite[Sec.~2.1]{gidney_fault-tolerant_2021}.
 These differ in the way they compute individual error probabilities
 from the physical error rate.
+
 In this work we only consider \emph{standard circuit-based depolarizing
 noise}, as this is the standard approach in the literature.
 We thus set the error probabilities of all error locations in the
 circuit-level noise model to the same value, the physical error rate $p$.
 
-\content{Intro}
-
 %%%%%%%%%%%%%%%%
 \subsection{Practical Methodology}
 \label{subsec:Practical Methodology}