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Fix typos

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Andreas Tsouchlos
2026-04-29 21:03:26 +02:00
parent 94e4c9f8c9
commit 8071c9f485
1 changed files with 10 additions and 9 deletions
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src/thesis/chapters/3_fault_tolerant_qec.tex
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@@ -20,7 +20,7 @@ introduces two new challenges \cite[Sec.~4]{gottesman_introduction_2009}:
\item \ac{qec} systems are themselves partially implemented in \item \ac{qec} systems are themselves partially implemented in
quantum hardware. In addition to the errors we have quantum hardware. In addition to the errors we have
originally introduced them for, these systems must originally introduced them for, these systems must
be able to acount for the fact they are implemented on noisy be able to account for the fact they are implemented on noisy
hardware themselves. hardware themselves.
\end{itemize} \end{itemize}
In the literature, both of these points are viewed under the umbrella In the literature, both of these points are viewed under the umbrella
@@ -190,7 +190,7 @@ data qubits are possible \cite[Appendix~A]{gidney_new_2023}.
This type of noise model is shown in \Cref{subfig:bit_flip}. This type of noise model is shown in \Cref{subfig:bit_flip}.
Note that we cannot use bit-flip noise to develop fault-tolerant Note that we cannot use bit-flip noise to develop fault-tolerant
systems, as it doesnt't account for errors during the syndrome extraction. systems, as it does not account for errors during the syndrome extraction.
%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%
\subsection{Depolarizing Channel} \subsection{Depolarizing Channel}
@@ -230,7 +230,7 @@ phenomenological noise is already a significant step beyond the code
capacity noise models. capacity noise models.
Additionally, there are applications where the Additionally, there are applications where the
consideration of phenomenological noise is enough. consideration of phenomenological noise is enough.
It can, for example, be used for guiding the design of fault-tolerant It can, for example, be used to guide the design of fault-tolerant
circuitry [DTTBE25, Sec. 4.2]. circuitry [DTTBE25, Sec. 4.2].
%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%
@@ -238,7 +238,7 @@ circuitry [DTTBE25, Sec. 4.2].
\label{subsec:Circuit-Level Noise} \label{subsec:Circuit-Level Noise}
The most general type of noise model is \emph{circuit-level noise}. The most general type of noise model is \emph{circuit-level noise}.
Here we not only consider noise inbetween syndrome extraction rounds Here we not only consider noise between syndrome extraction rounds
and at the measurements, but at each gate. and at the measurements, but at each gate.
Specifically, we allow arbitrary $n$-qubit Pauli errors after each Specifically, we allow arbitrary $n$-qubit Pauli errors after each
$n$-qubit gate \cite[Def.~2.5]{derks_designing_2025}. $n$-qubit gate \cite[Def.~2.5]{derks_designing_2025}.
@@ -277,7 +277,8 @@ error locations.
\section{Detector Error Models} \section{Detector Error Models}
\label{sec:Detector Error Models} \label{sec:Detector Error Models}
\emph{Detector error models} (\acsp{dem}) constitue a standardized framework for \emph{Detector error models} (\acsp{dem}) constitute a standardized
framework for
passing information about a circuit used for \ac{qec} to a decoder. passing information about a circuit used for \ac{qec} to a decoder.
They are also useful as a theoretical tool to aid in the design of They are also useful as a theoretical tool to aid in the design of
fault-tolerant \ac{qec} schemes. fault-tolerant \ac{qec} schemes.
@@ -439,7 +440,7 @@ circuit and each \ac{cn} corresponds to a syndrome measurement.
% Mathematical definition % Mathematical definition
We describe the circuit code using the \emph{measurement syndrome We describe the circuit code using the \emph{measurement syndrome
matrix} matrix $\bm{\Omega} \in \mathbb{F}_2^{M\times N}$, with matrix} $\bm{\Omega} \in \mathbb{F}_2^{M\times N}$, with
\begin{align*} \begin{align*}
\Omega_{\ell,i} = \Omega_{\ell,i} =
\begin{cases} \begin{cases}
@@ -824,7 +825,7 @@ For two detector matrices $\bm{D}_1$ and $\bm{D}_2$, as long as
they describe the same set of possible measurement outcomes (under they describe the same set of possible measurement outcomes (under
the absence of noise) and thus the same circuit. the absence of noise) and thus the same circuit.
In fact, as long as \Cref{eq:kern_condition} holds, the detector In fact, as long as \Cref{eq:kern_condition} holds, the detector
error matrices we construct from them can distinguish between the error matrices constructed from them can distinguish between the
same pairs of error sets \cite[Lemma~6]{derks_designing_2025}. same pairs of error sets \cite[Lemma~6]{derks_designing_2025}.
To see this, we note that we can distinguish between two circuit To see this, we note that we can distinguish between two circuit
error vectors $\bm{e}_1$ and $\bm{e}_2$ as long as they do not error vectors $\bm{e}_1$ and $\bm{e}_2$ as long as they do not
@@ -1121,6 +1122,6 @@ include many utilities for building syndrome extraction circuitry automatically.
The user has to define most, if not all, of the circuit manually, The user has to define most, if not all, of the circuit manually,
depending on the code in question. depending on the code in question.
This is somewhat natural, as stim is meant first and foremost as a This is somewhat natural, as stim is meant first and foremost as a
simulator, and circuit generation is contigent upon the \ac{qec} simulator, and circuit generation is contingent upon the \ac{qec}
scheme in question. scheme in question.