Fix notation
This commit is contained in:
@@ -1,4 +1,3 @@
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% TODO: Make all [H] -> [t]
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\chapter{Fault-Tolerant Quantum Error Correction}
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% Intro
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@@ -40,11 +39,10 @@ address both.
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% Definition of fault tolerance
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% TODO: Different variable name for N?
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We model the possible occurrence of errors during any processing
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stage as different \emph{error locations} $E_i,~i\in \{1,\ldots,N\}$
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in the circuit.
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$N \in \mathbb{N}$ is the total number of error locations.
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$N \in \mathbb{N}$ is the total number of considered error locations.
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The \emph{circuit error vector} $\bm{e} \in \{0,1\}^N$ is a vector
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indicating which errors occurred, with
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\begin{align*}
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@@ -151,8 +149,7 @@ Typically, the number of syndrome extraction rounds is chosen as $d_\text{min}$.
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% Intro
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We collect the probabilities of error at each location in the
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\emph{noise model}, a vector $\bm{p} \in [0,1]^N$, where $N \in
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\mathbb{N}$ is the number of possible error locations.
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\emph{noise model}, a vector $\bm{p} \in [0,1]^N$.
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There are different types of noise models, each allowing for
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different error locations in the circuit.
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@@ -442,32 +439,29 @@ circuit and each \ac{cn} corresponds to a syndrome measurement.
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% Mathematical definition
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We describe the circuit code using the \emph{measurement syndrome
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matrix} matrix $\bm{\Omega} \in \mathbb{F}_2^{m\times N}$, with
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matrix} matrix $\bm{\Omega} \in \mathbb{F}_2^{M\times N}$, with
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\begin{align*}
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\Omega_{j,i} =
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\Omega_{\ell,i} =
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\begin{cases}
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1, & \text{Error $i$ flips measurement $j$}\\
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1, & \text{Error $i$ flips measurement $\ell$}\\
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0, & \text{otherwise}
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\end{cases}
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.%
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,%
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\end{align*}
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This matrix thus defines this new code based on which error mechanism
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flips which measurement, rather than the Pauli type and location of
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each error \cite[Sec.~1.4.3]{higgott_practical_2024}.
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where $M \in \mathbb{N}$ is the number of measurements.
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To obtain $\bm{\Omega}$, we must propagate Pauli errors through the
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circuit, tracking which measurements they affect
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\cite[Sec.~2.4]{derks_designing_2025}.
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% Example
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% TODO: Fix syndrome dimension notation
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We turn to our example of the three-qubit repetition code to
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illustrate the construction of the syndrome measurement matrix.
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We begin by extending our check matrix in \autoref{eq:rep_code_H}
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to represent three rounds of syndrome extraction.
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Each round yields an additional set of syndrome bits,
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and we combine them by stacking them in a new vector
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$\bm{s} \in \mathbb{F}_2^{n_\text{rounds}\cdot(n-k)}$.
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$\bm{s} \in \mathbb{F}_2^{R(n-k)}$.
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We thus have to replicate the rows of $\bm{\Omega}$, once for each
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additional syndrome measurement, to obtain
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\begin{align*}
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@@ -501,7 +495,7 @@ We thus have
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% Expand to phenomenological
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We now whish to expand the error model to phenomenological noise, though
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We now wish to expand the error model to phenomenological noise, though
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only considering $X$ errors in this case.
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We introduce new error locations at the appropriate positions,
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arriving at the circuit depicted in
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@@ -774,10 +768,10 @@ independent detectors \cite[Sec.~2.2]{derks_designing_2025}.
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% The detector matrix
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% TODO: Fix the notation mess
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We describe the relationship between measurements and detectors using
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the \emph{detector matrix} $\bm{D} \in \mathbb{F}_2^{d\times m}$
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\cite[Def.~2.2]{derks_designing_2025}.
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the \emph{detector matrix} $\bm{D} \in \mathbb{F}_2^{D\times M}$
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\cite[Def.~2.2]{derks_designing_2025}, with $~D\in \mathbb{N}$
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denoting the number of detectors.
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Similar to the way a \ac{pcm} associates bits with parity checks, the
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detector matrix links measurements and detectors.
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Each column corresponds to a measurement, while each rows corresponds
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@@ -811,7 +805,7 @@ measurements relate to the detectors (through $\bm{D}$).
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For decoding, we are interested in the effect of the errors on the
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detectors directly.
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We thus construct the \emph{detector error matrix} $\bm{H} \in
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\mathbb{F}_2^{d\times N}$ \cite[Def.~2.9]{derks_designing_2025} as
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\mathbb{F}_2^{D\times N}$ \cite[Def.~2.9]{derks_designing_2025} as
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\begin{align*}
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\bm{H} := \bm{D}\bm{\Omega}
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.%
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@@ -852,9 +846,6 @@ It may, however, change the decoding performance when using a practical decoder.
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% How to choose the detectors
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% TODO: Fix notation (n used for both number of measurements and
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% measurements themselves)
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% TODO: Properly define the ranges i and r belong to
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What constitutes a good set of detectors is difficult to assess
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without performing explicit decoding simulations, since it ultimately
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depends on the decoder employed.
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@@ -866,26 +857,29 @@ at a later stage.
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To the measurement results from each syndrome extraction round we
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can add the results from the previous round, as illustrated in
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\autoref{fig:detectors_from_measurements_general}.
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Concretely, we denote the outcome of the
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$i$-th measurement in round $r$ by $m_i^{(r)} \in \mathbb{F}_2$ and define
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We thus have $D=n-k$.
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Concretely, we denote the outcome of
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measurement $\ell \in \{1,\ldots,n-k\}$ in round $r \in \{1,\ldots,R\}$ by
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$m_\ell^{(r)} \in \mathbb{F}_2$
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and define
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\begin{gather*}
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\bm{m}^{(r)} :=
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\begin{pmatrix}
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m_1^{(r)} \\
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\vdots \\
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m_m^{(r)}
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m_{n-k}^{(r)}
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\end{pmatrix}
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.%
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\end{gather*}
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Similarly, we denote the the $i$-th detector in round $r$ by
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$d_i^{(r)} \in \mathbb{F}_2$ and define
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Similarly, we denote the outcome of detector $j\in\{1,\ldots,D\}$ in
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round $r$ by $d_j^{(r)} \in \mathbb{F}_2$ and define
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\begin{gather}
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\label{eq:measurement_combination}
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\bm{d}^{(r)} =
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\begin{pmatrix}
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d_1^{(r)} \\
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\vdots \\
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d_m^{(r)}
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d_D^{(r)}
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\end{pmatrix}
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:= \bm{m}^{(r)} + \bm{m}^{(r-1)}
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,%
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@@ -1011,10 +1005,10 @@ It contains all information necessary for the decoding process.
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\section{Practical Considerations}
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\label{sec:Practical Considerations}
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The previous sections give \red{[theoretical overview of noise models
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and DEMs]}.
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In order to apply them successfully in practice, we must consider a
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few further aspects.
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The previous sections give an overview over available noise models
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and the function of \acp{dem}.
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In order to successfully apply these concepts in practice, we must
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consider a few further aspects.
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%%%%%%%%%%%%%%%%
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\subsection{Choice of Noise Model}
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@@ -1037,7 +1031,8 @@ from the physical error rate.
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In this work we only consider \emph{standard circuit-based depolarizing
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noise}, as this is the standard approach in the literature.
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We thus set the error probabilities of all error locations in the
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circuit-level noise model to the same value, the physical error rate $p$.
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circuit-level noise model to the same value, the physical error rate
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$p_\text{phys}$.
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%%%%%%%%%%%%%%%%
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\subsection{Per-Round Logical Error Rate}
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@@ -1056,18 +1051,18 @@ use the \emph{per-round-\ac{ler}}.
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The simplest way of calculating the per-round \ac{ler} is by modeling
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each round as an independent experiment.
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For each experiment, an error might occur with a certain probability
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$p_\text{round}$.
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$p_\text{e,round}$.
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The overall probability of error is then
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\begin{align}
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\hspace{-12mm}
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p_\text{total} &= 1 - (1 - p_\text{round})^{n_\text{rounds}} \nonumber\\
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p_\text{e,total} &= 1 - (1 - p_\text{e,round})^{R} \nonumber\\
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\label{eq:per_round_ler}
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\implies \hspace{3mm} p_\text{round} &=
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1 - (1 - p_\text{total})^{1 / n_\text{rounds}}
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\implies \hspace{3mm} p_\text{e,round} &=
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1 - (1 - p_\text{e,total})^{1 / R}
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.%
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\hspace{12mm}
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\end{align}
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We approximate $p_\text{total}$ using a Monte Carlo simulation and
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We approximate $p_\text{e,total}$ using a Monte Carlo simulation and
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compute the per-round-\ac{ler} using \autoref{eq:per_round_ler}.
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This is a common approach taken in the literature
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\cite{gong_toward_2024}\cite{wang_fully_2025}.
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@@ -1077,7 +1072,7 @@ Another common approach \cite{chen_exponential_2021}%
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exponential decay for the decoder's \emph{logical fidelity}
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\cite[Eq.~2]{bausch_learning_2024}
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\begin{align*}
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F_\text{total} = (F_\text{round})^{n_\text{rounds}}
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F_\text{total} = (F_\text{round})^{R}
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.%
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\end{align*}
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The logical fidelity is a measure of the quality of a logical state
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@@ -1085,9 +1080,9 @@ The logical fidelity is a measure of the quality of a logical state
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As it is related to the error rate through $F = 1 - 2p$, we obtain
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\cite[Eq.~4]{bausch_learning_2024}
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\begin{align}
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(1 - 2p_\text{total}) &= (1 - 2p_\text{round})^{n_\text{rounds}} \nonumber\\
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\implies \hspace{15mm} p_\text{total} &= \frac{1}{2}
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\left[ 1 - (1 - 2p_\text{round})^{1/n_\text{rounds}} \right]
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(1 - 2p_\text{e,total}) &= (1 - 2p_\text{e,round})^{R} \nonumber\\
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\implies \hspace{15mm} p_\text{e,round} &= \frac{1}{2}
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\left[ 1 - (1 - 2p_\text{e,total})^{1/R} \right]
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.%
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\end{align}
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@@ -1116,18 +1111,17 @@ first introduced.
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One capability of stim, and \acp{dem} in general, that we didn't go
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into detail about in this chapter is the merging of error mechanisms.
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Since \acp{dem} differentiate errors based on their effect on the
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measurements and not on their type and location, it is natural to
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group errors that have the same effect.
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measurements and not on their Pauli type and location
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\cite[Sec.~1.4.3]{higgott_practical_2024}, it is natural to group
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errors that have the same effect.
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This slightly lowers the computational complexity of decoding, as the
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number of resulting \acp{vn} is reduced.
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While stim is a useful tool for circuit simulation, it doesn't
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include many utilities for building syndrome extraction circuitry automatically.
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The user has to define most, if not, all of the circuit manually,
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The user has to define most, if not all, of the circuit manually,
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depending on the code in question.
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This is somewhat natural, as stim is meant first and foremost as a
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simulator, and circuit generation is contigent upon the \ac{qec}
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scheme in question.
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\content{Introduce logical error rate}
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