an.tsouchlos/ma-thesis
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases 12 Wiki Activity

Fix typos

Browse Source
This commit is contained in:
Andreas Tsouchlos
2026-04-29 21:03:26 +02:00
parent 94e4c9f8c9
commit 8071c9f485
1 changed files with 10 additions and 9 deletions
Download Patch File Download Diff File Expand all files Collapse all files

19
src/thesis/chapters/3_fault_tolerant_qec.tex
View File

@@ -20,7 +20,7 @@ introduces two new challenges \cite[Sec.~4]{gottesman_introduction_2009}:
\item \ac{qec} systems are themselves partially implemented in
quantum hardware. In addition to the errors we have
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.
\end{itemize}
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}.
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}
@@ -230,7 +230,7 @@ phenomenological noise is already a significant step beyond the code
capacity noise models.
Additionally, there are applications where the
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].
%%%%%%%%%%%%%%%%
@@ -238,7 +238,7 @@ circuitry [DTTBE25, Sec. 4.2].
\label{subsec: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.
Specifically, we allow arbitrary $n$-qubit Pauli errors after each
$n$-qubit gate \cite[Def.~2.5]{derks_designing_2025}.
@@ -277,7 +277,8 @@ error locations.
\section{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.
They are also useful as a theoretical tool to aid in the design of
fault-tolerant \ac{qec} schemes.
@@ -439,7 +440,7 @@ circuit and each \ac{cn} corresponds to a syndrome measurement.
% Mathematical definition
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*}
\Omega_{\ell,i} =
\begin{cases}
@@ -759,7 +760,7 @@ Instead of using stabilizer measurement results directly, we
generalize the notion of what constitutes a parity check slightly.
We formally define a \emph{detector} as a deterministic parity constraint on
a set of measurement outcomes \cite[Def.~2.1]{derks_designing_2025}.
In the most straight forward case, we may simply use the stabilizer
In the most straightforward case, we may simply use the stabilizer
measurements as detectors.
We immediately recognize that we will have as many linearly
independent detectors as there are separate deterministic measurements.
@@ -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
the absence of noise) and thus the same circuit.
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}.
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
@@ -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,
depending on the code in question.
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.