Finish QEC intro, most of stabilizer measurements, add outline for rest

src/thesis/chapters/2_fundamentals.tex

@@ -815,23 +815,23 @@ using \emph{projection operators} \cite[Eq.~3.160]{griffiths_introduction_1995}
 \end{align*}%
 These project a vector onto the subspace spanned by $\ket{e_n}$.
 
-% Using projection operators to measure if a state has a component
+% % Using projection operators to measure if a state has a component
-% along a basis vector
+% % along a basis vector
-
+%
-A particularly interesting property of projection operators is that
+% A particularly interesting property of projection operators is that
-\begin{align*}
+% \begin{align*}
-    \hat{P}_n (\hat{P}_n \ket{\psi}) = \hat{P}_n^2 \ket{\psi}
+%     \hat{P}_n (\hat{P}_n \ket{\psi}) = \hat{P}_n^2 \ket{\psi}
-    = \hat{P}_n \ket{\psi},
+%     = \hat{P}_n \ket{\psi},
-\end{align*}%
+% \end{align*}%
-and the only way this can hold for any $\ket{\psi}$ is if $\hat{P}_n$
+% and the only way this can hold for any $\ket{\psi}$ is if $\hat{P}_n$
-only has the eigenvalues $0$ or $1$
+% only has the eigenvalues $0$ or $1$
-% tex-fmt: off
+% % tex-fmt: off
-\cite[Prob.~3.57a)]{griffiths_introduction_1995}.
+% \cite[Prob.~3.57a)]{griffiths_introduction_1995}.
-% tex-fmt: on
+% % tex-fmt: on
-The eigenvalues can again be interpreted as possible measurement results.
+% The eigenvalues can again be interpreted as possible measurement results.
-We can thus use $\hat{P}$ as an observable and treat
+% We can thus use $\hat{P}$ as an observable and treat
-the eigenvalue as an indicator of the state having a component along
+% the eigenvalue as an indicator of the state having a component along
-the related basis vector.
+% the related basis vector.
 
 %%%%%%%%%%%%%%%%
 \subsection{Qubits and Multi-Qubit States}
@@ -892,7 +892,7 @@ We have
     \end{split}
 \end{align}
 We call $\ket{x_0, \ldots, x_n}~, x_i \in \{0,1\}$ the
-\emph{computational basis states} \cite[Sec.~4.6]{wohlin_guidelines_2014}.
+\emph{computational basis states} \cite[Sec.~4.6]{nielsen_quantum_2010}.
 
 % Entanglement
 
@@ -937,7 +937,7 @@ thus the only possible state change is a bit-flip, a gerenal 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
-\emph{Pauli operators}
+\emph{Pauli operators} \cite[Sec.~2.2]{gottesman_stabilizer_1997}
 \cite[Sec.~2.2]{roffe_quantum_2019}
 \begin{align*}
     \begin{array}{c}
@@ -1010,21 +1010,342 @@ Other important operators include the \emph{Hadamard} and
 \noindent Many more operators relevant to quantum computing exist, but they are
 not covered here as they are not central to this work.
 
+\indent\red{[We only need to consider X and Z errors]
+\cite[Equation~8]{roffe_quantum_2019}} \\
+\indent\red{[Explain commuting/anticommuting property of operators]
+[Journal~p.~46]}
+
 %%%%%%%%%%%%%%%%
 \subsection{Quantum Circuits}
 \label{Quantum Circuits}
 
-\red{[TODO] \cite[Sec.~1.3.4]{nielsen_quantum_2010}}
+\noindent\indent\red{[Controlled operations]
+\cite[Sec.~4.3]{nielsen_quantum_2010}} \\
+\indent\red{[In case this reference is needed: Measurements
+\cite[Sec.~4.4]{nielsen_quantum_2010}]} \\
+\indent\red{[General circuit stuff] \cite[Sec.~1.3.4]{nielsen_quantum_2010}}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Quantum Error Correction}
 \label{sec:Quantum Error Correction}
 
-% Why we need quantum error correction
+% TODO: Use this for the introduction as well
 
-% The unique challenges of QEC compared to classical FEC
+% General motivation behind QEC
 
-% The unique challenges of QEC compared to classical FEC
+One of the major barriers on the road to building a functioning
+quantum computer is the inevitability of errors during quantum
+computation 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
+eventual \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
+correction.
+
+% The unique challenges of QEC
+
+The problem setting of \ac{qec} differs slightly from the classical case, as
+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
+        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
+        one of the determinate states, thereby potentially destroying
+        information.
+\end{itemize}
+
+% General idea (logical vs. physical gates) + notation
+
+Much like in classical error correction, in \ac{qec} information
+is protected by mapping it onto codewords in an expanded space,
+thereby introducing redundancy.
+To this end, $k \in \mathbb{N}$ \emph{logical qubits} are mapped onto
+$n \in \mathbb{N},~n>k$ \emph{physical qubits}.
+We circumvent the no-cloning restriction by not copying the state of
+the $k$ logical qubits, but rather spreading it out over all $n$
+physical ones \cite[Intro.]{calderbank_good_1996}
+To differentiate a quantum codes from classical ones, we denote a
+code with parameters $k,n$ and minimum distance $d_\text{min}$ using
+double brackets, as $[[ n,k,d_\text{min} ]]$ \cite[Sec.~4]{roffe_quantum_2019}.
+
+%%%%%%%%%%%%%%%%
+\subsection{Stabilizer Measurements}
+\label{subsec:Stabilizer Measurements}
+
+% Setting the stage
+
+Before we move on to the description of entire codes, we introduce
+the notion of the \emph{stabilizer measurement}.
+Consider the two-qubit repetition code
+\cite[Sec.~2.4]{roffe_quantum_2019}, where we map
+\begin{align*}
+    \ket{\psi} = \alpha \ket{0} + \beta \ket{1}
+    \hspace*{3mm} \mapsto \hspace*{3mm}
+    \ket{\psi}_\text{L}
+    = \alpha \underbrace{\ket{00}}_{=:\ket{0}_\text{L}} + \beta
+    \underbrace{\ket{11}}_{=:\ket{1}_\text{L}}
+    .%
+\end{align*}
+We call $\ket{\psi}_L$ the logical state.
+We define the \emph{codespace} as $\mathcal{C} := \text{span}\mleft\{
+\ket{00}, \ket{11} \mright\}$ and the \emph{error subspace} as
+$\mathcal{F} := \text{span} \mleft\{\ket{01}, \ket{10} \mright\}$.
+Note that this code is only able to detect single $X$-type errors.
+
+% Measuring stabilizers
+
+To determine if an error occurred, we want to measure
+whether a state belongs
+% TODO: Remove footnote?
+% \footnote{
+%     It is possible for a state to not completely lie in either subspace.
+%     In this case, we can interpret it as being in
+%     $\mathcal{C}$ or $\mathcal{F}$ with a certain probability.
+% }
+to $\mathcal{C}$ or $\mathcal{F}$.
+As explained in \autoref{subsec:Observables}, physical measurements
+can be mathematically described using operators whose eigenvalues
+are the possible measurement results.
+Here, we need an operator with two eigenvalues and the corresponding
+eigenspaces should be $\mathcal{C}$ and $\mathcal{F}$ respectively.
+For the two-qubit code, $Z_1Z_2$ is such an operator:
+\begin{align}
+    Z_1Z_2 E \ket{\psi}_\text{L} &= (+1) E \ket{\psi}_\text{L}
+    \hspace*{3mm} \forall
+    E \ket{\psi}_\text{L} \in \mathcal{C} \\
+    Z_1Z_2 E \ket{\psi}_\text{L} &= (-1) E \ket{\psi}_\text{L}
+    \hspace*{3mm} \forall
+    E \ket{\psi}_\text{L} \in \mathcal{F}
+    .%
+\end{align}
+$E \in \left\{ X,I \right\}$ is an operator describing a possible
+error and $E \ket{\psi}_\text{L}$ is the resulting state after that error.
+By measuring the corresponding eigenvalue, we can determine if
+$E\ket{\psi}_\text{L}$ lies in $\mathcal{C}$ or $\mathcal{F}$.
+% TODO: If necessary, cite \cite[Sec.~3]{roffe_quantum_2019} for the
+% non-compromising meausrement of the information
+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 we entangle it with
+$\ket{\psi}_\text{L}$ in such a way that the eigenvalue is indicated
+by measuring that 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
+\begin{align}
+    \label{eq:error_projection}
+    E\ket{\psi}_\text{L} \ket{0}_\text{A} \hspace*{3mm} \rightarrow
+    \hspace*{3mm}
+    \underbrace{\frac{1}{2} \mleft( I_1I_2 + Z_1Z_2 \mright)}_{=:
+    P_\mathcal{C}} E\ket{\psi}_\text{L}
+    \ket{0}_\text{A}
+    + \underbrace{\frac{1}{2} \mleft( I_1I_2 - Z_1Z_2 \mright)}_{=:
+    P_\mathcal{F}}
+    E\ket{\psi}_\text{L} \ket{1}_\text{A}
+    .%
+\end{align}
+If $E \ket{\psi}_\text{L} \in \mathcal{C}$, the second term will
+cancel and we will deterministically measure $\ket{0}_\text{A}$ for
+the ancilla qubit. Similarly, if $E \ket{\psi}_\text{L} \in
+\mathcal{F}$, we will deterministically measure $\ket{1}_\text{A}$.
+
+\begin{figure}[t]
+    \centering
+
+    \tikzset{
+        meter/.append style={
+            draw, rectangle,
+            font=\vphantom{A}, minimum width=8mm, minimum height=8mm,
+            path picture={
+                \draw[black]
+                ([shift={(.1,.3)}]path picture bounding box.south west)
+                to[bend left=50]
+                ([shift={(-.1,.3)}]path picture bounding box.south east);
+                \draw[black,-latex]
+                ([shift={(0,.1)}]path picture bounding box.south)
+                -- ([shift={(.3,-.1)}]path picture bounding box.north);
+            }
+        }
+    }
+
+    \begin{tikzpicture}
+        \node[rectangle, minimum width=2cm, minimum height=3cm, draw]
+        (ZZ) {$Z_1Z_2$};
+
+        \coordinate (qi1) at (-3, 1);
+        \coordinate (qi2) at (-3, -1);
+        \coordinate (qi3) at (-3, -3);
+
+        \coordinate (qo1) at (4, 1);
+        \coordinate (qo2) at (4, -1);
+        \coordinate (qo3) at (4, -3);
+
+        \node[rectangle, minimum width=8mm, minimum height=8mm, draw]
+        (H1) at ($(qo3 -| ZZ) + (-2, 0)$) {H};
+        \node[rectangle, minimum width=8mm, minimum height=8mm, draw]
+        (H2) at ($(qo3 -| ZZ) + (2, 0)$) {H};
+        \node[circle, fill] (not) at (H1 -| ZZ) {};
+        \node[meter, right=5mm of H2] (mes) {};
+
+        \draw (qi1) -- (ZZ.west |- qi1);
+        \draw (qi2) -- (ZZ.west |- qi2);
+
+        \draw (qo1) -- (ZZ.east |- qo1);
+        \draw (qo2) -- (ZZ.east |- qo2);
+
+        \draw (qi3) -- (H1) -- (not) -- (H2) -- (mes);
+
+        \draw (not) -- (ZZ);
+
+        \coordinate (qo3u) at ($(qo3) + (0, .5mm)$);
+        \coordinate (qo3d) at ($(qo3) + (0, -.5mm)$);
+
+        \draw (mes.east |- qo3u) -- (qo3u);
+        \draw (mes.east |- qo3d) -- (qo3d);
+
+        \node[left] at (qi3) {$\ket{0}_\text{A}$};
+        \node[left] at ($(qi1)!.5!(qi2)$) {$E\ket{\psi}_\text{L}$};
+        \node[right] at ($(qo1)!.5!(qo2)$) {$E\ket{\psi}_\text{L}$};
+
+    \end{tikzpicture}
+
+    \caption{Stabilizer measurement circuit for the two-qubit repetition code.}
+    \label{fig:stabilizer_measurement}
+\end{figure}
+
+% Digitization of errors
+
+Note that it is possible for a vector $E\ket{\psi}$ to not completely
+lie in either subspace.
+In this case, we can interpret it as being in $\mathcal{C}$ or
+$\mathcal{F}$ with a certain probability.
+However, when we measure the stabilizer, we will find that the vector
+lies either in one or the other.
+This is because the act of measuring the error partly collapses the
+state, eliminating the uncertainty about the type of the error
+\cite[Sec.~10.2]{nielsen_quantum_2010}.
+This can be seen in \autoref{eq:error_projection}, as the expressions
+$P_\mathcal{C}$ and $P_\mathcal{F}$ constitute projection operators onto
+$\mathcal{C}$ and $\mathcal{F}$.
+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} scheme only needs to be able to
+correct $X$ and $Z$ errors.
+This effect is referred to as error \emph{digitization}
+\cite[Sec.~2.2]{roffe_quantum_2019}.
+
+% The stabilizer group
+
+\indent\red{[Conditions for the stabilizer group
+\cite[Sec.~4.1]{roffe_quantum_2019}]} \\
+\indent\red{[Why we care about (anti-)commutativity of the
+        stabilizers with errors + Z-type operators for X type errors and vice
+versa ]} \\
+\indent\red{[(?) Stabilizer generators]}
+
+% A general stabilizer measurement circuit
+
+\indent\red{[General stabilizer measurement circuit
+\cite[Figure~4]{roffe_quantum_2019}]}
+
+%%%%%%%%%%%%%%%%
+\subsection{Stabilizer Codes}
+\label{subsec:Stabilizer Codes}
+
+% Structure of a stabilizer code
+
+For classical binary linear block codes, we use $n-k$ parity-checks
+to reduce the degrees of freedom introduced by the encoding operation.
+Effectively, each parity-check defines a local code, splitting the
+vector space in half, with only one half containing valid codewords.
+The global code is the intersection of all local codes.
+We can do the same in the quantum case.
+Each split is represented using stabilizer, whose eigenvalues signify
+whether a candidate vector lies in the local codespace or local error subspace.
+It is only a valid codeword if it lies in the codespace of all local codes.
+We call codes constructed this way \emph{stabilizer codes}.
+
+% Syndrome extraction circuitry
+
+Similar to the classical case, we can use a syndrome vector to
+describe which local codes are violated.
+To obtain the syndrome, we simply measure the corresponding
+operators, each using a circuit as explained in
+\autoref{subsec:Stabilizer Measurements}.
+A full \emph{syndrome extraction circuit} is depicted in \autoref{fig:sec}.
+
+% TODO: Move this further up to the commutativity of operators?
+\indent\red{[Fixing the error after finding it
+\cite[Sec.~10.5.5]{nielsen_quantum_2010}]} \\
+\indent\red{[Logical operators \cite[Sec.~4.2]{roffe_quantum_2019}]} \\
+\indent\red{[Measuring logical operators gives yields the outcomes of
+the encoded computations \cite[Sec.~2.6]{derks_designing_2025}]} \\
+\indent\red{[X and Z measurements can be performed with only CNOT and
+Hadamard gates \cite[Sec.~10.5.8]{nielsen_quantum_2010}]} \\
+\indent\red{[Parity-check matrix \cite[Sec.~10.5.1]{nielsen_quantum_2010}]}
+
+\begin{figure}[t]
+    \centering
+
+    \tikzset{
+        meter/.append style={
+            draw, rectangle,
+            font=\vphantom{A}, minimum width=8mm, minimum height=8mm,
+            path picture={
+                \draw[black]
+                ([shift={(.1,.3)}]path picture bounding box.south west)
+                to[bend left=50]
+                ([shift={(-.1,.3)}]path picture bounding box.south east);
+                \draw[black,-latex]
+                ([shift={(0,.1)}]path picture bounding box.south)
+                -- ([shift={(.3,-.1)}]path picture bounding box.north);
+            }
+        }
+    }
+
+    \red{Hier könnte Ihre Werbung stehen.}
+
+    \caption{
+        \red{Illustration of a general syndrome extraction circuit.
+        Adapted from \cite[Figure~4]{roffe_quantum_2019}.}
+    }
+    \label{fig:sec}
+\end{figure}
+
+%%%%%%%%%%%%%%%%
+\subsection{Calderbank-Shor-Steane Codes}
+\label{subsec:Calderbank-Shor-Steane Codes}
+
+Stabilizer codes are especially practical to work with when they can
+handle $X$- and $Z$-type errors independently.
+We can then separate the stabilizer generators into some with only
+$Z$ operators and some with only $X$ operators.
+
+\indent\red{[Construction from two binary linear codes
+\cite[p.~452,469]{nielsen_quantum_2010}]}
+
+\subsection{Quantum Low-Density Parity-Check Codes}
+
+\noindent\red{[Constant overhead scaling]} \\
+\noindent\red{[Scaling of minimum distance with code length]} \\
+\noindent\red{[Bivariate Bicycle codes]} \\
+\noindent\red{[Decoding QLDPC codes (syndrome-based BP)]} \\
+\noindent\red{[Degeneracy -> BP+OSD, BPGD]} \\
+\noindent\red{[``The task of decoding is therefore to infer, from a
+        measured syndrome, the most likely error coset rather than the exact
+        physical error.''
+% tex-fmt: off
+\cite[Sec.~II~B)]{koutsioumpas_colour_2025}]}%
+% tex-fmt: on
+        \\
 
         \red{
             \textbf{General Notes:}
@@ -1035,27 +1356,32 @@ not covered here as they are not central to this work.
                 \item Degeneracy
                 \item The QEC decoding problem (considering degeneracy)
                     \cite[Sec.~2.3]{yao_belief_2024}
-        \item Why we need commutativity of the stabilizers [Journal,
-            p.~51], [Got97, p.~6]
             \end{itemize}
             \textbf{Content:}
             \begin{itemize}
                 \item  General context
                     \begin{itemize}
-                \item Why we need QEC (correcting errors due to noisy gates)
+                        \item Why we need QEC (correcting errors due
+                            to noisy gates)
                         \item Main challenges of QEC compared to classical
                             error correction
-                \item Logical vs physical states, logical vs physical operators
+                        \item Logical vs physical states, logical vs
+                            physical operators
                     \end{itemize}
                 \item Stabilizer codes
                     \begin{itemize}
                         \item Definition of a stabilizer code
                         \item The stabilizer its generators (note somewhere
-                        that the generators have to commute to be able to
+                                that the generators have to commute
+                                to be able to
                             be measured without disturbing each other)
+                            (Why we need commutativity of the
+                                stabilizers [Journal,
+                            p.~51], [Got97, p.~6])
                         \item syndrome extraction circuit
                         \item Stabilizer codes are effectively the QM
-                    % TODO: Actually binary linear codes or just linear codes?
+                            % TODO: Actually binary linear codes or
+                            % just linear codes?
                             equivalent of binary linear codes (e.g.,
                             expressible via check matrix)
                         \item Similar to parity checks, quantum states can be
@@ -1070,7 +1396,3 @@ not covered here as they are not central to this work.
             \end{itemize}
         }
 
-\subsection{Stabilizer Codes}
-\subsection{CSS Codes}
-\subsection{Quantum Low-Density Parity-Check Codes}
-