From 8071c9f485ab6482451db0f7a39615f96952022e Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Wed, 29 Apr 2026 21:03:26 +0200 Subject: [PATCH] Fix typos --- src/thesis/chapters/3_fault_tolerant_qec.tex | 19 ++++++++++--------- 1 file changed, 10 insertions(+), 9 deletions(-) diff --git a/src/thesis/chapters/3_fault_tolerant_qec.tex b/src/thesis/chapters/3_fault_tolerant_qec.tex index 95de0f2..56caa74 100644 --- a/src/thesis/chapters/3_fault_tolerant_qec.tex +++ b/src/thesis/chapters/3_fault_tolerant_qec.tex @@ -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.