Refactor the intro to numerical results

src/thesis/chapters/4_decoding_under_dems.tex

@@ -1114,32 +1114,16 @@ messages, pass decimation info}
 \section{Numerical Results}
 \label{sec:Numerical Results}
 
-% Simulation setup
+% Intro
 
 In this section, we perform numerical experiments to evaluate the
 modification to sliding-window decoding we introduced in
 \Cref{sec:warm_start_bp}.
-We chose to carry out our simulations on \ac{bb} codes, as they have
+For the practical aspects of implementation, several layers of
-recently emerged as particularly promising candidates for practical
+abstraction must be considered.
-\ac{qec}, offering high encoding rates and large minimum distances
-while admitting short-depth syndrome extraction circuits
-\cite[Sec.~1]{bravyi_high-threshold_2024}.
-Specifically, we chose the $\llbracket 144, 12, 12 \rrbracket$ BB
-code, as it represents a good trade-off between code size and
-simulation tractability \cite{gong_toward_2024}.
-We employ standard circuit-based depolarizing noise as described in
-\Cref{subsec:Choice of Noise Model}, and report performance in terms
-of the per-round \ac{ler} as defined in
-\Cref{subsec:Per-Round Logical Error Rate}.
-All datapoints have been generated by simulating at least $200$
-logical error events.
-
-\content{Mention the number of syndrome extraction rounds}
 
 % Software stack: Layer 1
 
-For the practical aspects of implementation, several layers of
-abstraction must be considered.
 The lowest layer is the circuit-level simulator.
 This serves as the backbone of all further simulations, handling the
 quantum mechanical aspects of the system, including the modeling of
@@ -1183,6 +1167,29 @@ reimplementation in Rust to achieve higher simulation speeds due to
 the compiled nature of the language.
 We reimplemented both the window splitting and the decoders themselves.
 
+% Simulation setup
+
+We chose to carry out our simulations on \ac{bb} codes, as they have
+recently emerged as particularly promising candidates for practical
+\ac{qec}, offering high encoding rates and large minimum distances
+while admitting short-depth syndrome extraction circuits
+\cite[Sec.~1]{bravyi_high-threshold_2024}.
+Specifically, we chose the $\llbracket 144, 12, 12 \rrbracket$ BB
+code, as it represents a good trade-off between code size and
+simulation tractability.
+For the generation of the \ac{dem} we set the number of syndrome
+extraction rounds to $12$, similarly to \cite{gong_toward_2024}, and
+we defined our detectors as in the example in
+\Cref{subsec:Detector Error Matrix}.
+We employed circuit-lose noise as described in
+\Cref{subsec:Choice of Noise Model} as our noise model, specifically standard
+ciruit-based depolarizing noise \cite[Sec.~VIII]{fowler_high-threshold_2009},
+i.e., all error locations in the circuit get assigned the same
+physical error probability.
+We report performance in terms of the per-round \ac{ler} as defined
+in \Cref{subsec:Per-Round Logical Error Rate} and all datapoints were
+generated by simulating at least $200$ logical error events.
+
 %%%%%%%%%%%%%%%%
 \subsection{Belief Propagation}
 \label{subsec:Belief Propagation}