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Polish per-round logical error rate subsection

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Andreas Tsouchlos
2026-04-29 17:40:37 +02:00
parent b73a66649c
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18
src/thesis/chapters/3_fault_tolerant_qec.tex
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@@ -1059,7 +1059,7 @@ The simplest way of calculating the per-round \ac{ler} is by modeling
each round as an independent experiment. each round as an independent experiment.
For each experiment, an error might occur with a certain probability For each experiment, an error might occur with a certain probability
$p_\text{round}$. $p_\text{round}$.
The overall probability of error is thus The overall probability of error is then
\begin{align} \begin{align}
\hspace{-12mm} \hspace{-12mm}
p_\text{total} &= 1 - (1 - p_\text{round})^{n_\text{rounds}} \nonumber\\ p_\text{total} &= 1 - (1 - p_\text{round})^{n_\text{rounds}} \nonumber\\
@@ -1075,15 +1075,17 @@ This is a common approach taken in the literature
\cite{gong_toward_2024}\cite{wang_fully_2025}. \cite{gong_toward_2024}\cite{wang_fully_2025}.
Another common approach \cite{chen_exponential_2021}% Another common approach \cite{chen_exponential_2021}%
\cite{bausch_learning_2024}\cite{maan_decoding_2025}\cite{cao_exact_2025}% \cite{bausch_learning_2024}\cite{beni_tesseract_2025} is to assume an
\cite{beni_tesseract_2025} is to assume a exponential decay for the exponential decay for the decoder's \emph{logical fidelity}
decoder's \emph{fidelity} \red{explain what this is}
\cite[Eq.~2]{bausch_learning_2024} \cite[Eq.~2]{bausch_learning_2024}
\begin{align*} \begin{align*}
F_\text{total} = (F_\text{round})^{n_\text{rounds}} F_\text{total} = (F_\text{round})^{n_\text{rounds}}
.% .%
\end{align*} \end{align*}
As the fidelity is related to the error rate through $F = 1 - 2p$, we obtain The logical fidelity is a measure of the quality of a logical state
\cite[Appendix~E]{postler_demonstration_2024}.
As it is related to the error rate through $F = 1 - 2p$, we obtain
\cite[Eq.~4]{bausch_learning_2024}
\begin{align} \begin{align}
(1 - 2p_\text{total}) &= (1 - 2p_\text{round})^{n_\text{rounds}} \nonumber\\ (1 - 2p_\text{total}) &= (1 - 2p_\text{round})^{n_\text{rounds}} \nonumber\\
\implies \hspace{15mm} p_\text{total} &= \frac{1}{2} \implies \hspace{15mm} p_\text{total} &= \frac{1}{2}
@@ -1091,7 +1093,9 @@ As the fidelity is related to the error rate through $F = 1 - 2p$, we obtain
.% .%
\end{align} \end{align}
\content{We choose the first approach} We have chosen to use the first approach, i.e.,
\autoref{eq:per_round_ler}, as the related literature is closer in
topic to our own work.
%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%
\subsection{Stim} \subsection{Stim}
@@ -1108,6 +1112,8 @@ propagate through the circuit an what detectors they affect}
\content{Merging of error mechanisms} \content{Merging of error mechanisms}
\content{Stim is were DEMs were first introduced}
\content{Stim is a software package that generates DEMs from circuits} \content{Stim is a software package that generates DEMs from circuits}
\content{The user still has to define the circuit themselves, and \content{The user still has to define the circuit themselves, and
especially the detectors \cite[Sec~2.5]{derks_designing_2025}} especially the detectors \cite[Sec~2.5]{derks_designing_2025}}