Fix bibliography

src/thesis/bibliography.bib

@@ -69,12 +69,13 @@
 	howpublished = {arXiv:2508.15743},
 }
 
-@article{koutsioumpas_automorphism_2025,
+@misc{koutsioumpas_automorphism_2025,
 	title = {Automorphism Ensemble Decoding of Quantum {LDPC} Codes},
 	language = {en},
 	author = {Koutsioumpas, Stergios and Sayginel, Hasan and Webster, Mark and Browne, Dan E},
 	month = mar,
 	year = {2025},
+        howpublished = {arXiv:2503.01738},
 }
 
 @misc{gottesman_heisenberg_1998,
@@ -291,7 +292,7 @@
 	author = {Gottesman, Daniel},
 	month = may,
 	year = {1997},
-	howpublished = {arXiv:quant-ph/9705052},
+	howpublished = {Ph.D. {Thesis}, arXiv:quant-ph/9705052},
 }
 
 @misc{shor_fault-tolerant_1997,
@@ -554,6 +555,7 @@
 	author = {Wang, Ming and Li, Ang and Mueller, Frank},
 	month = jun,
 	year = {2025},
+        howpublished = {arXiv:2507.00254},
 }
 
 @misc{ye_beam_2025,
@@ -793,8 +795,9 @@
 	title = {Algorithms for quantum computation: discrete logarithms and factoring},
 	shorttitle = {Algorithms for quantum computation},
 	doi = {10.1109/SFCS.1994.365700},
-	booktitle = {Proceedings 35th Annual Symposium on Foundations of Computer Science},
+	booktitle = {Proc. Annual Symposium on Foundations of Computer Science},
 	author = {Shor, P.W.},
+        address = {Santa Fe},
 	month = nov,
 	year = {1994},
 	pages = {124--134},
@@ -928,6 +931,7 @@
 	doi = {10.1109/ACCESS.2015.2503267},
 	journal = {IEEE Access},
 	author = {Babar, Zunaira and Botsinis, Panagiotis and Alanis, Dimitrios and Ng, Soon Xin and Hanzo, Lajos},
+        month = nov,
 	year = {2015},
 	pages = {2492--2519},
 }
@@ -1012,11 +1016,11 @@
 	title = {Fully parallel window decoder architecture for spatially-coupled {LDPC} codes},
 	issn = {1938-1883},
 	doi = {10.1109/ICC.2016.7511553},
-	booktitle = {2016 {IEEE} International Conference on Communications ({ICC})},
+	booktitle = {Proc. {IEEE} International Conference on Communications ({ICC})},
 	author = {Hassan, Najeeb Ul and Schlüter, Martin and Fettweis, Gerhard P.},
+        address = {Kuala Lumpur},
 	month = may,
 	year = {2016},
-	pages = {1--6},
 }
 
 @article{costello_spatially_2014,

src/thesis/chapters/3_fault_tolerant_qec.tex

@@ -25,7 +25,7 @@ introduces two new challenges \cite[Sec.~4]{gottesman_introduction_2009}:
         hardware themselves.
 \end{itemize}
 In the literature, both of these points are viewed under the umbrella
-of \emph{fault tolerance}.
+of \emph{fault-tolerant} quantum computing.
 We focus only on the second aspect in this work.
 
 It was recognized early on as a challenge of \ac{qec} that the correction
@@ -188,9 +188,11 @@ We visualize the different types of noise models in
 The simplest type of noise model is \emph{bit-flip} noise.
 This corresponds to the classical \ac{bsc}, i.e., only $X$ errors on the
 data qubits are possible \cite[Appendix~A]{gidney_new_2023}.
+The occurrence of bit-flip errors is modeled as a Bernoulli process
+$\text{Bern}(p)$.
 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 bit-flip noise is not suitable for developing fault-tolerant
 systems, as it does not account for errors during the syndrome extraction.
 
 %%%%%%%%%%%%%%%%
@@ -243,12 +245,12 @@ 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}.
-An $n$-qubit Pauli error is simply a series of correlated Pauli
+An $n$-qubit Pauli error can be written as a series of correlated Pauli
 errors on each related individual qubit.
 This type of noise model is shown in \Cref{subfig:circuit_level}.
 
 While phenomenological noise is useful for some design aspects of
-fault tolerant circuitry, for simulations, circuit-level noise should
+fault-tolerant circuitry, for simulations, circuit-level noise should
 always be used \cite[Sec.~4.2]{derks_designing_2025}.
 Note that this introduces new challenges during the decoding process,
 as the decoding complexity is increased considerably due to the many
@@ -286,7 +288,7 @@ fault-tolerant \ac{qec} schemes.
 E.g., they can be used to easily determine whether a measurement
 schedule is fault-tolerant \cite[Example~12]{derks_designing_2025}.
 
-Other approaches of implementing fault tolerance exist, such as
+Other approaches of implementing fault-tolerance circuits exist, such as
 flag error correction, which uses additional ancilla qubits to detect
 potentially damaging high-weight errors \cite[Sec.~1]{chamberland_flag_2018}.
 However, \acp{dem} offer some unique advantages
@@ -300,8 +302,7 @@ However, \acp{dem} offer some unique advantages
         treated in a unified manner. This leads to a more powerful
         description of the overall circuit.
 \end{itemize}
-In this work, we only consider the process of decoding under the
+In this work, we consider the process of decoding under the \ac{dem} framework.
-\ac{dem} framework.
 
 % Core idea
 
@@ -459,15 +460,22 @@ circuit, tracking which measurements they affect
 
 We turn to our example of the three-qubit repetition code to
 illustrate the construction of the syndrome measurement matrix.
-We begin by extending our check matrix in \Cref{eq:rep_code_H}
+We begin by extending our check matrix $\bm{H}_Z$ in
-to represent three rounds of syndrome extraction.
+\Cref{eq:rep_code_H} to represent three rounds of syndrome extraction.
 Each round yields an additional set of syndrome bits,
 and we combine them by stacking them in a new vector
-$\bm{s} \in \mathbb{F}_2^{R(n-k)}$.
+$\bm{s} \in \mathbb{F}_2^{R(n-k)}$, where $R \in \mathbb{N}$ is the
-We thus have to replicate the rows of $\bm{\Omega}$, once for each
+number of syndrome measurement rounds.
+We thus have to replicate the rows of $\bm{H}_Z$, once for each
 additional syndrome measurement, to obtain
 \begin{align*}
-    \bm{\Omega} =
+    \bm{\Omega}_0 =
+    \begin{pmatrix}
+        \bm{H}_Z \\
+        \bm{H}_Z \\
+        \bm{H}_Z
+    \end{pmatrix}
+    =
     \begin{pmatrix}
         1 & 1 & 0 \\
         0 & 1 & 1 \\
@@ -482,7 +490,7 @@ additional syndrome measurement, to obtain
 depicts the corresponding circuit.
 Note that we have not yet introduced error locations in the syndrome
 extraction circuitry, so we still consider only bit flip noise at this stage.
-Recall that $\bm{\Omega}$ describes which \ac{vn} is connected to
+Recall that $\bm{\Omega}_0$ describes which \ac{vn} is connected to
 which parity check and the syndrome indicates which parity checks
 are violated.
 This means that if an error exists at only a single \ac{vn}, we can
@@ -491,7 +499,7 @@ If errors occur at multiple locations, the resulting syndrome will be
 the linear combination of the respective columns.
 We thus have
 \begin{align*}
-    \bm{s} \in \text{span} \{\bm{\Omega}\}
+    \bm{s} \in \text{span} \{\bm{\Omega}_0\}
     .%
 \end{align*}
 
@@ -502,11 +510,11 @@ only considering $X$ errors in this case.
 We introduce new error locations at the appropriate positions,
 arriving at the circuit depicted in
 \Cref{fig:rep_code_multiple_rounds_phenomenological}.
-For each additional error location, we extend $\bm{\Omega}$ by
+For each additional error location, we extend $\bm{\Omega}_0$ by
 appending the corresponding syndrome vector as a column.
 \begin{gather}
     \label{eq:syndrome_matrix_ex}
-    \bm{\Omega} =
+    \bm{\Omega}_1 =
     \left(
         \begin{array}{ccccccccccccccc}
             1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0
@@ -523,24 +531,25 @@ appending the corresponding syndrome vector as a column.
             & 0 & 1 & 1 & 0 & 1
         \end{array}
     \right) . \\[-6mm]
-    \hspace*{-58.7mm}
+    \hspace*{-56.7mm}
     \underbrace{
         \phantom{
             \begin{array}{ccc}
                 0 & 0 & 0
             \end{array}
         }
-    }_\text{Original matrix}
+    }_{\bm{\Omega}_0} \nonumber
 \end{gather}
 Notice that the first three columns correspond to the original
-measurement syndrome matrix, as these columns correspond to the error
+measurement syndrome matrix $\bm{\Omega}_0$, as these columns
-locations on the data qubits.
+correspond to the error locations on the data qubits.
 
 In this example, all measurements we considered were syndrome measurements.
-Assuming no errors, the results of those measurements were
+Assuming no errors, the results of those measurements are
-deterministic, irrespective of the actual logical state
+deterministic: They are not subject to any probabilistic behavior
-$\ket{\psi}_\text{L}$, as they only depend on whether
+despite the quantum mechanical nature of the underlying system.
-$\ket{\psi}_\text{L} \in \mathcal{C}$, not on the concrete state.
+They only depend on whether $\ket{\psi}_\text{L} \in \mathcal{C}$,
+not on the concrete state.
 It is, in general, possible to also consider non-deterministic measurements.
 As an example, it is usual to consider a round of noiseless
 measurements of the actual data qubit states after the last syndrome
@@ -557,7 +566,7 @@ extraction round.
         \centering
         \begin{tikzpicture}
             \node{$%
-                \bm{\Omega} =
+                \bm{\Omega}_0 =
                 \begin{pmatrix}
                     1 & 1 & 0 \\
                     0 & 1 & 1 \\
@@ -667,7 +676,7 @@ extraction round.
     \end{gather*}
     \vspace*{-8mm}
     \begin{gather*}
-        \bm{\Omega} =
+        \bm{\Omega}_1 =
         \left(
             \begin{array}{
                     cccccc%
@@ -761,10 +770,10 @@ 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}.
+It can be seen that we will have as many linearly
+independent detectors as there are separate deterministic measurements.
 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.
 We generally aim to utilize the maximum number of linearly
 independent detectors \cite[Sec.~2.2]{derks_designing_2025}.
 
@@ -775,8 +784,8 @@ the \emph{detector matrix} $\bm{D} \in \mathbb{F}_2^{D\times M}$
 \cite[Def.~2.2]{derks_designing_2025}, with $~D\in \mathbb{N}$
 denoting the number of detectors.
 Similar to the way a \ac{pcm} associates bits with parity checks, the
-detector matrix links measurements and detectors.
+detector matrix links measurement outcomes and detectors.
-Each column corresponds to a measurement, while each rows corresponds
+Each column corresponds to a measurement, while each row corresponds
 to a detector.
 We should note at this point that the combination of measurements
 into detectors has no bearing on the actual construction of the
@@ -786,12 +795,12 @@ affects the decoder.
 
 Note that we can use the detector matrix $\bm{D}$ to describe the set
 of possible measurement outcomes under the absence of noise.
-The same way we use a \ac{pcm} to describe the code space as
+Similar to the we use a \ac{pcm} to describe the code space as
-\begin{align*}
+\begin{equation*}
     \mathcal{C}
     = \{ \bm{x} \in \mathbb{F}_2^{n} : \bm{H}\bm{x}^\text{T} = \bm{0} \}
     ,%
-\end{align*}
+\end{equation*}
 the set of possible measurement outcomes is simply $\text{kern}\{\bm{D}\}$
 \cite[Sec.~2.2]{derks_designing_2025}.
 
@@ -915,7 +924,8 @@ with $\bm{m}^{(0)} = \bm{0}$.
 
 We again turn our attention to the three-qubit repetition code.
 In \Cref{fig:rep_code_multiple_rounds_phenomenological} we can see
-that $E_6$ has occurred and has subsequently tripped the last four measurements.
+that $E_6$ has occurred and has subsequently triggered the last four
+measurements.
 We now take those measurements and combine them according to
 \Cref{eq:measurement_combination}.
 We can see this process graphically in
@@ -923,13 +933,13 @@ We can see this process graphically in
 To understand why this way of defining the detectors is useful, we
 note that the error $E_6$ in
 \Cref{fig:rep_code_multiple_rounds_phenomenological} has not only
-tripped the measurements in the syndrome extraction round immediately
+triggered the measurements in the syndrome extraction round immediately
 afterwards, but all subsequent ones as well.
 To only see errors in the rounds immediately following them, we
 consider our newly defined detectors instead of the measurements,
 that effectively compute the difference between the measurements.
 
-Each error can only trip syndrome bits that follow it.
+Each error can only trigger syndrome bits that follow it.
 This is reflected in the triangular structure of $\bm{\Omega}$ in
 \Cref{eq:syndrome_matrix_ex}.
 Combining the measurements into detectors according to
@@ -949,7 +959,7 @@ The detector error matrix
         \end{array}
     \right)
 \end{align*}
-we obtain this way has a block-diagonal structure.
+obtained this way has a block-diagonal structure.
 Note that we exploit the fact that each syndrome measurement round is
 identical to obtain this structure.
 
@@ -1030,11 +1040,11 @@ measurements) models \cite[Sec.~2.1]{gidney_fault-tolerant_2021}.
 These differ in the way they compute individual error probabilities
 from the physical error rate.
 
-In this work we only consider \emph{standard circuit-based depolarizing
+In this work we consider the \emph{standard circuit-based depolarizing
-noise}, as this is the standard approach in the literature.
+noise} variant of circuit-level noise, as this is the standard
-We thus set the error probabilities of all error locations in the
+approach in the literature:
-circuit-level noise model to the same value, the physical error rate
+We set the error probabilities of all error locations to the same
-$p_\text{phys}$.
+value, the physical error rate $p_\text{phys}$.
 
 %%%%%%%%%%%%%%%%
 \subsection{Per-Round Logical Error Rate}
@@ -1065,13 +1075,12 @@ The overall probability of error is then
 \end{align}
 We approximate $p_\text{e,total}$ using a Monte Carlo simulation and
 compute the per-round-\ac{ler} using \Cref{eq:per_round_ler}.
-This is a common approach taken in the literature
+This is the approach taken in \cite{gong_toward_2024}\cite{wang_fully_2025}.
-\cite{gong_toward_2024}\cite{wang_fully_2025}.
 
-Another common approach \cite{chen_exponential_2021}%
+Another approach \cite{chen_exponential_2021}%
 \cite{bausch_learning_2024}\cite{beni_tesseract_2025} is to assume an
 exponential decay for the decoder's \emph{logical fidelity}
-\cite[Eq.~2]{bausch_learning_2024}
+\cite[Eq.~(2)]{bausch_learning_2024}
 \begin{align*}
     F_\text{total} = (F_\text{round})^{R}
     .%
@@ -1079,7 +1088,7 @@ exponential decay for the decoder's \emph{logical fidelity}
 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}
+\cite[Eq.~(4)]{bausch_learning_2024}
 \begin{align}
     (1 - 2p_\text{e,total}) &= (1 - 2p_\text{e,round})^{R} \nonumber\\
     \implies \hspace{15mm} p_\text{e,round} &= \frac{1}{2}

src/thesis/main.tex

@@ -42,7 +42,7 @@
 
 \Crefname{equation}{}{}
 \Crefname{section}{Section}{Sections}
-\Crefname{subsection}{Subsection}{Subsections}
+\Crefname{subsection}{Section}{Sections}
 \Crefname{figure}{Figure}{Figures}
 
 %
@@ -89,7 +89,7 @@
 % \thesisHeadOfInstitute{Prof. Dr.-Ing. Peter Rost}
 %\thesisHeadOfInstitute{Prof. Dr.-Ing. Peter Rost\\Prof. Dr.-Ing.
 % Laurent Schmalen}
-\thesisSupervisor{Jonathan Mandelbaum}
+\thesisSupervisor{M.Sc. Jonathan Mandelbaum}
 \thesisStartDate{01.11.2025}
 \thesisEndDate{04.05.2026}
 \thesisSignatureDate{04.05.2026}