Add clean_bibliography.sh; Incorporate LLM corrections

src/thesis/chapters/2_fundamentals.tex

@@ -1,4 +1,4 @@
-\chapter{Fundamentals}
+\chapter{Fundamentals of Classical and Quantum Error Correction}
 \label{ch:Fundamentals}
 
 \Ac{qec} is a field of research combining ``classical''
@@ -606,21 +606,23 @@ The purpose of this section is to convey these concepts to the reader.
 
 In quantum mechanics, the state of a particle is described by a
 \emph{wave function} $\psi(x,t)$.
-The connection between this function and the observable world
+Born's statistical interpretation provides a connection between this
-is Born's statistical interpretation:
+function and the observable world:
-$\lvert \psi (x,t) \rvert^2$ is the \ac{pdf} of finding a praticle at
+$\lvert \psi (x,t) \rvert^2$ is the \ac{pdf} of finding a particle at
 position $x$ and time $t$ \cite[Sec.~1.2]{griffiths_introduction_1995}.
+Note that this presupposes a normalization of $\psi$ such that
+$\int_{-\infty}^{\infty} \lvert \psi(x,t) \rvert^2 dx = 1$.
 
 % Dirac notation
 
-A lot of the related mathematics can be very elegantly expressed
+Much of the related mathematics can be very elegantly expressed
 using the language of linear algebra.
-The so called Bra-ket or Dirac notation is especially appropriate,
+The so-called Bra-ket or Dirac notation is especially appropriate,
 having been proposed by Paul Dirac in 1939 for the express purpose
 of simplifying quantum mechanical notation \cite{dirac_new_1939}.
 Two new symbols are defined, \emph{bra}s $\bra{\cdot}$ and
 \emph{ket}s $\ket{\cdot}$.
-Kets denote ordinary vectors, while bras denote their Hermitian conjugates.
+Kets denote column vectors, while bras denote their Hermitian conjugates.
 For example, two vectors specified by the labels $a$ and $b$
 respectively are written as $\ket{a}$ and $\ket{b}$.
 Their inner product is $\braket{a\vert b}$.
@@ -629,7 +631,7 @@ Their inner product is $\braket{a\vert b}$.
 
 The connection we will make between quantum mechanics and linear
 algebra is that we will model the state space of a system as a
-\emph{function space}.
+\emph{function space}, the Hilbert space $L_2$.
 We will represent the state of a particle with wave function
 $\psi(x,t)$ using the vector $\ket{\psi}$
 \cite[Sec.~3.3]{griffiths_introduction_1995}.
@@ -642,8 +644,17 @@ output \cite[Sec.~3.2.2]{griffiths_introduction_1995}.
 Operators are useful to describe the relations between different
 quantities relating to a particle.
 An example of this is the differential operator $\partial x$.
-Two operators $P_1$ and $P_2$ are said to \emph{commute}, if $P_1P_2
+We define the \emph{commutator} of two operators $P_1$ and $P_2$ as
-= P_2P_1$ and \emph{anti-commute} if $P_1P_2 = -P_2P_1$.
+\begin{align*}
+    [P_1,P_2] = P_1P_2 - P_2P_1
+\end{align*}
+and the \emph{anticommutator} as
+\begin{align*}
+    [P_1,P_2]_+ = P_1P_2 + P_2P_1
+    .%
+\end{align*}
+We say the two operators \emph{commute} iff $[P_1,P_2] = 0$, and they
+\emph{anti-commute} iff $[P_1,P_2]_+ = 0$.
 
 %%%%%%%%%%%%%%%%
 \subsection{Observables}
@@ -664,8 +675,8 @@ observations \cite[Sec.~3.3]{griffiths_introduction_1995}.
 
 If we know the wave function of a particle, we should be able to
 compute the expected value $\braket{Q}$ of any observable quantity we wish.
-It can be shown that for any $Q$, we can compute a
+It can be shown that for any $Q$, we can find a
-corresponding operator $\hat{Q}$ such that
+corresponding Hermitian operator $\hat{Q}$ such that
 \cite[Sec.~3.3]{griffiths_introduction_1995}
 \begin{align}
     \label{eq:gen_expr_Q_exp}
@@ -690,7 +701,7 @@ formula simplifies to the direct calculation of the expected value.
 
 Let us now examine how the observable operator $\hat{Q}$ relates to
 the determinate states of the observable quantity.
-We begin by translating \autoref{eq:gen_expr_Q_exp} into linear alebra as
+We begin by translating \autoref{eq:gen_expr_Q_exp} into linear algebra as
 \cite[Eq.~3.114]{griffiths_introduction_1995}
 \begin{align}
     \label{eq:gen_expr_Q_exp_lin}
@@ -698,7 +709,7 @@ We begin by translating \autoref{eq:gen_expr_Q_exp} into linear alebra as
     .%
 \end{align}
 \autoref{eq:gen_expr_Q_exp_lin} expresses an inherently probabilistic
-relationhip.
+relationship.
 The determinate states are inherently deterministic.
 To relate the two, we note that since determinate states should
 always yield the same measurement results, the variance of the
@@ -743,6 +754,9 @@ We can use the determinate states for this purpose, expressing the state as%
     c_n := \braket{e_n \vert \psi}
     .%
 \end{align}
+Because of the normalization of the wave function such that
+$\int_{-\infty}^{\infty} \lvert \psi(x,t) \rvert^2 dx = 1$, we have
+$\sum_{n=1}^{\infty} \lvert c_n \rvert ^2 = 1$.
 Inserting \autoref{eq:determinate_basis} into
 \autoref{eq:gen_expr_Q_exp_lin} we obtain
 % tex-fmt: off
@@ -772,7 +786,7 @@ We can decompose an arbitrary state as $\ket{\psi} = \sum_{n=1}^{\infty} c_n
 \ket{e_n}$, where $\lvert c_n \rvert ^2$ represents the probability
 of obtaining a certain measurement value.
 Note that when we speak of an \emph{observable}, we are usually
-refering to the operator $\hat{Q}$.
+referring to the operator $\hat{Q}$.
 
 %%%%%%%%%%%%%%%%
 \subsection{Projective Measurements}
@@ -838,7 +852,7 @@ These project a vector onto the subspace spanned by $\ket{e_n}$.
 \subsection{Qubits and Multi-Qubit States}
 \label{subsec:Qubits and Multi-Qubit States}
 
-% The qubit
+% Intro
 
 % TODO: Make sure `quantum gate` is proper terminology
 A central concept for quantum computing is that of the \emph{qubit}.
@@ -847,20 +861,35 @@ For classical computers, we alter bits' states using \emph{gates}.
 We can chain multiple of these gates together to build up more complex logic,
 such as half-adders or eventually a full processor.
 In principle, quantum computers work in a similar fashion, only that
-instead of bits we use qubits and instead of, e.g. {AND}, OR, and XOR
+instead of bits we use qubits and instead of, e.g., AND, OR, and XOR
 operations we use \emph{quantum gates} \cite[Sec.~1.3]{nielsen_quantum_2010}.
-We define a qubit to be a component with determinate
+
-states $\ket{0}$ and $\ket{1}$.
+% Qubits and multi-qubit states
-The general description of the state $\ket{\psi}$ of a qubit is thus
+
-\begin{align}
+We fix an orthonormal basis of $\mathbb{C}^2$ to be
-    \label{eq:gen_qubit_state}
+\begin{align*}
-    \ket{\psi} = \alpha\ket{0} + \beta\ket{1}, \hspace{5mm} \alpha,
+    \ket{0} =
-    \beta \in \mathbb{C}
+    \begin{pmatrix}
+        1 \\
+        0
+    \end{pmatrix}, \hspace{5mm}
+    \ket{1} =
+    \begin{pmatrix}
+        0 \\
+        1
+    \end{pmatrix}
     .%
-\end{align}
+\end{align*}
-
+A qubit is defined to be a system with quantum state
-% The tensor product and multi-qubit states
+\begin{align*}
-
+    \ket{\psi} =
+    \begin{pmatrix}
+        \alpha \\
+        \beta
+    \end{pmatrix}
+    = \alpha \ket{0} + \beta \ket{1}
+    .%
+\end{align*}
 The overall state of a composite quantum system is described using
 the \emph{tensor product}, denoted as $\otimes$
 \cite[Sec.~2.2.8]{nielsen_quantum_2010}.
@@ -907,9 +936,9 @@ An example of such states are the \emph{Bell states}
 \begin{align*}
     \begin{split}
         \ket{\psi_{00}} &= \frac{\ket{00} + \ket{11}}{\sqrt{2}} \hspace{15mm}
-        \ket{\psi_{01}} = \frac{\ket{01} - \ket{10}}{\sqrt{2}} \\
+        \ket{\psi_{01}} = \frac{\ket{01} + \ket{10}}{\sqrt{2}} \\
-        \ket{\psi_{10}} &= \frac{\ket{00} + \ket{11}}{\sqrt{2}} \hspace{15mm}
+        \ket{\psi_{10}} &= \frac{\ket{01} - \ket{10}}{\sqrt{2}} \hspace{15mm}
-        \ket{\psi_{11}} = \frac{\ket{01} - \ket{10}}{\sqrt{2}}
+        \ket{\psi_{11}} = \frac{\ket{00} - \ket{11}}{\sqrt{2}}
     \end{split}
     \hspace{4mm}.%
 \end{align*}
@@ -941,7 +970,7 @@ Unlike classical systems, where there are only two possible states and
 thus the only possible state change is a bit-flip, a general qubit
 state as shown in \autoref{eq:gen_qubit_state} lives on a continuum of values.
 We thus technically also have an infinite number of possible state changes.
-Luckily, we can express any operator as a linear combination of the
+Fortunately, we can express any operator as a linear combination of the
 \emph{Pauli operators} \cite[Sec.~2.2]{gottesman_stabilizer_1997}
 \cite[Sec.~2.2]{roffe_quantum_2019}
 \begin{align*}
@@ -962,14 +991,14 @@ Luckily, we can express any operator as a linear combination of the
     \begin{array}{c}
         Z\text{ Operator} \\
         \hline\\
-        \ket{0} \mapsto -\ket{0} \\
+        \ket{0} \mapsto \phantom{-}\ket{0} \\
         \ket{1} \mapsto -\ket{1}
     \end{array}%
     \hspace{10mm}%
     \begin{array}{c}
         Y\text{ Operator} \\
         \hline\\
-        \ket{0} \mapsto -j\ket{1} \\
+        \ket{0} \mapsto \phantom{-}j\ket{1} \\
         \hspace{2.75mm}\ket{1} \mapsto -j\ket{0} \hspace*{1mm}.
     \end{array}
 \end{align*}
@@ -984,7 +1013,7 @@ group} over $n$ qubits.
 
 In the context of modifying qubit states, we also call operators \emph{gates}.
 When working with multi-qubit systems, we can also apply Pauli gates
-to individual qubits independently, which we write ask e.g., $I_1 X_2
+to individual qubits independently, which we write, e.g., as $I_1 X_2
 I_3 Z_4 Y_5$.
 We often omit the identity operators, instead writing, e.g., $X_2 Z_4 Y_5$.
 Other important operators include the \emph{Hadamard} and
@@ -1082,8 +1111,8 @@ quantum computer is the inevitability of errors during quantum
 computation. These arise due to the difficulty in sufficiently isolating the
 qubits from external noise \cite[Intro.]{roffe_quantum_2019}.
 This isolation is critical for quantum systems, as the constant interactions
-with the environment act as small measurements, leading to the
+with the environment act as small measurements, an effect called
-eventual \emph{decoherence} of the quantum state
+\emph{decoherence} of the quantum state
 \cite[Intro.]{gottesman_stabilizer_1997}.
 \ac{qec} is one approach of dealing with this problem, by protecting
 the quantum state in a similar fashion to information in classical error
@@ -1096,7 +1125,7 @@ Three main restrictions apply \cite[Sec.~2.4]{roffe_quantum_2019}:
 \begin{itemize}
     \item The no-cloning theorem states that it is
         impossible to exactly copy the state of one qubit into another.
-    \item Qubit are susceptible to more types of errors than
+    \item Qubits are susceptible to more types of errors than
         just bit-flips, as we saw in
         \autoref{subsec:Qubits and Multi-Qubit States}.
     \item Directly measuring the state of a qubit collapses it onto
@@ -1128,7 +1157,7 @@ low overall computational complexity.
 This is due to the \emph{backlog problem}
 \cite[Sec.~II.G.3.]{terhal_quantum_2015}: There are certain gates
 at which the effect of existing errors on single qubits may be
-exacerbated by transforming them to mutli-qubit errors.
+exacerbated by transforming them to multi-qubit errors.
 We wish to correct the errors before passing qubits through such gates.
 If the \ac{qec} system is not fast enough, there will be an increasing
 backlog of information at this point in the circuit, leading to an
@@ -1194,7 +1223,7 @@ To do this without directly observing (and thus potentially
 collapsing) the logical state $\ket{\psi}_\text{L}$, we prepare an
 ancilla qubit with state $\ket{0}_\text{A}$ and entangle it with
 $\ket{\psi}_\text{L}$ in such a way that the eigenvalue is indicated
-by measuring that instead.
+by measuring the ancilla qubit instead.
 More specifically, using a stabilizer measurement circuit as shown in
 \autoref{fig:stabilizer_measurement}, we transform the state of the
 three-qubit system as
@@ -1248,7 +1277,8 @@ E.g., $P_\mathcal{C}$ will eliminate all components of $E
 \ket{\psi}_\text{L}$ that lie in $\mathcal{F}$.
 This process, together with the fact that any coherent error can be
 decomposed into a linear combination of $X$ and $Z$ errors, means
-that it is enough for a \ac{qec} to be able to correct only $X$ and $Z$ errors.
+that it is sufficient for \ac{qec} to be able to correct only these
+types of errors.
 This effect is referred to as error \emph{digitization}
 \cite[Sec.~2.2]{roffe_quantum_2019}.
 
@@ -1261,10 +1291,11 @@ $[[n, k, d_\text{min}]]$ code $\mathcal{C}$, if
     \item It stabilizes all logical states, i.e.,
         $P_i\ket{\psi}_\text{L} = (+1)\ket{\psi}_\text{L} ~\forall~
         \ket{\psi}_\text{L} \in \mathcal{C}$.
-    \item It commutes with all other stabilizers of the code. This
+    \item It commutes with all other stabilizers $P_j$ of the code,
-        property is important to be able to measure the eigenvalue of
+        i.e., $[P_i, P_j] = 0$.
-        a stabilizer without disturbing the eigenvectors of the
+        This property is important to be able to measure the
-        others \cite[Sec.~1.2]{gottesman_stabilizer_1997}.
+        eigenvalue of a stabilizer without disturbing the
+        eigenvectors of the others \cite[Sec.~1.2]{gottesman_stabilizer_1997}.
 \end{itemize}
 Formally, we define the \emph{stabilizer group} $\mathcal{S}$ as
 \cite[Sec.~4.1]{roffe_quantum_2019}
@@ -1272,10 +1303,8 @@ Formally, we define the \emph{stabilizer group} $\mathcal{S}$ as
     \mathcal{S} = \left\{P_i \in \mathcal{G}_n ~:~ P_i \ket{\psi}_\text{L} =
         (+1)\ket{\psi}_\text{L} \forall \ket{\psi}_\text{L} ~\cap~
     [P_i,P_j] = 0 \forall i,j\right\}
-    ,%
+    .%
 \end{align*}
-where $[P_i,P_j] := P_iP_j - P_j P_i$ is called the \emph{commutator}
-of $P_i$ and $P_j$.
 We care in particular about the commuting properties of stabilizers
 with respect to possible errors.
 The measurement circuit for an arbitrary stabilizer $P_i$ modifies
@@ -1344,9 +1373,17 @@ physical state corresponds to \cite[Sec.~2.6]{derks_designing_2025}.
 
 % TODO: Do I have to introduce before that stabilizers only need X
 % and Z operators?
-Stabilizer codes are the quantum analog of classical linear block codes.
+We can represent stabilizer codes using a \emph{check matrix}
-As such, we can represent them using a \emph{check matrix}
+\cite[Sec.~10.5.1]{nielsen_quantum_2010}
-\cite[Sec.~10.5.1]{nielsen_quantum_2010}.
+\begin{align*}
+    \bm{H} = \left[
+        \begin{array}{c|c}
+            \bm{H}_X & \bm{H}_Z
+        \end{array}
+    \right]
+    ,%
+\end{align*}
+with $\bm{H} \in \mathbb{F}_2^{(n-k)\times(2n)}$.
 This is similar to a classical \ac{pcm} in that it contains $n-k$
 rows, each describing one constraint. Each constraint restricts an additional
 degree of freedom of the higher-dimensional space we use to introduce
@@ -1358,7 +1395,7 @@ Take for example the Steane code \cite[Eq.~10.83]{nielsen_quantum_2010}.
 We can describe it using the check matrix
 \begin{align}
     \label{eq:steane}
-    \left[
+    \bm{H}_\text{Steane} = \left[
         \begin{array}{ccccccc|ccccccc}
             0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
             0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
@@ -1498,7 +1535,7 @@ We can then construct $\bm{A}$ and $\bm{B}$ as bivariate polynomials
 \end{align*}
 where $\bm{A}_i$ and $\bm{B}_i$ are powers of $\bm{x}$ or $\bm{y}$.
 \ac{bb} codes have large minimum distance $d_\text{min}$ and high rate,
-offering a more than 10-time reduction of encoding overhead over the
+offering a more than 10-fold reduction of encoding overhead over the
 surface code.
 Additionally, they posess short-depth syndrome measurement circuits,
 leading to lower time requirements for the syndrome extraction
@@ -1508,7 +1545,7 @@ and thus lower error rates \cite[Sec.~1]{bravyi_high-threshold_2024}.
 
 As we saw in \autoref{subsec:Stabilizer Measurements}, we work only
 with the parity information contained in the syndrome, to avoid
-disturbing the quantum states of indivudual qubits.
+disturbing the quantum states of individual qubits.
 This necessitates a modification of the standard \ac{bp} algorithm
 introduced in \autoref{subsec:Iterative Decoding}
 \cite[Sec.~3.1]{yao_belief_2024}.
@@ -1521,7 +1558,8 @@ algorithm will now try to find an error pattern $\hat{\bm{e}} \in
 \end{align*}
 To this end, we initialize the channel \acp{llr} as
 \begin{align*}
-    \tilde{L}_i = \log{\frac{P(X_i = 0)}{P(X_i = 1)}} = \frac{1 - p_i}{p_i}
+    \tilde{L}_i = \log{\frac{P(X_i = 0)}{P(X_i = 1)}} = \log{\frac{1
+    - p_i}{p_i}}
     ,%
 \end{align*}
 where $p_i$ is the prior probability of error of \ac{vn} $i$.
@@ -1590,7 +1628,7 @@ algorithm \ref{alg:syndome_bp}.
 
 % Degeneracy and short cycles
 
-Decoding \ac{qldpc} codes brings with it some unique challenges.
+Decoding \ac{qldpc} codes poses some unique challenges.
 One issue is that of \emph{quantum degeneracy}.
 Because errors that differ by a stabilizer have the same impact on
 all codewords, there can be multiple minimum-weight solutions to the
@@ -1609,7 +1647,7 @@ during decoding, impeding performance.
 
 The aforementioned issues both manifest themselves as convergence problems
 of the \ac{bp} algorithm, and different ways of modifying the algorithm
-to aide with convergence exist.
+to aid with convergence exist.
 One approach is to use \ac{bp} with guided decimation (\acs{bpgd})
 \cite[Alg.~1]{yao_belief_2024}.
 Here, a number $T\in \mathbb{N}$ of \ac{bp} iterations are performed,
@@ -1620,7 +1658,7 @@ progresses, encouraging the algorithm to converge to one of the
 solutions \cite[Sec.~5]{yao_belief_2024}.
 Algorithm \ref{alg:bpgd} shows this process.
 Note that as the Tanner graph only has $n$ \acp{vn}, this is a
-natural constraint on the maximum number of iterations.
+natural constraint on the maximum number of outer iterations of the algorithm.
 
 % TODO: Explain that setting the channel LLR to infinity is the same
 % as a hard decision and ignoring the VN in the further decoding

src/thesis/clean_bibliography.sh

@@ -0,0 +1,2 @@
+sed -i "s/Świerkowska/{\\\\'S}wierkowska/" bibliography.bib
+sed -Ezi "s/\s(abstract|note|urldate|url|keywords|file) = \{[^}]*(\{[^}]*\}[^}]*)*\},?\n//g" bibliography.bib