Finish writing stabilizer codes

src/thesis/acronyms.tex

@@ -18,6 +18,11 @@
     long=sum-product algorithm
 }
 
+\DeclareAcronym{css}{
+    short=CSS,
+    long=Calderbank-Shor-Steane
+}
+
 \DeclareAcronym{llr}{
     short=LLR,
     long=log-likelihood ratio

src/thesis/chapters/2_fundamentals.tex

@@ -1155,8 +1155,8 @@ Consider the two-qubit repetition code
     \underbrace{\ket{11}}_{=:\ket{1}_\text{L}}
     .%
 \end{align*}
-We call $\ket{\psi}_L$ the logical state.
+We call $\ket{\psi}_L$ the logical state, and
-We define the \emph{codespace} as $\mathcal{C} := \text{span}\mleft\{
+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.
@@ -1195,7 +1195,7 @@ $E\ket{\psi}_\text{L}$ lies in $\mathcal{C}$ or $\mathcal{F}$.
 % 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
+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.
 More specifically, using a stabilizer measurement circuit as shown in
@@ -1258,7 +1258,7 @@ This effect is referred to as error \emph{digitization}
 % The stabilizer group
 
 Operators such as $Z_1Z_2$ above are called \emph{stabilizers}.
-An operator $P_i \in \mathcal{G}_n$ is called a stabilizer of an
+More generally, an operator $P_i \in \mathcal{G}_n$ is called a stabilizer of an
 $[[n, k, d_\text{min}]]$ code $\mathcal{C}$, if
 \begin{itemize}
     \item It stabilizes all logical states, i.e.,
@@ -1307,11 +1307,11 @@ and the stabilizer measurement returns 1.
 
 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
+Effectively, each parity-check defines a local code splitting the
-vector space in half, with only one half containing valid codewords.
+vector space in half, with only one part 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
+Each split is represented using a 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}.
@@ -1323,19 +1323,60 @@ describe which local codes are violated.
 To obtain the syndrome, we simply measure the corresponding
 operators $P_i$, each using a circuit as explained in
 \autoref{subsec:Stabilizer Measurements}.
-A full \emph{syndrome extraction circuit} is depicted in \autoref{fig:sec}.
+Note that this is an abstract representation of the syndrome extraction.
+For the actual implementation in hardware, we can transform this into
+a circuit that requires only CNOT and H-gates
+\cite[Sec.~10.5.8]{nielsen_quantum_2010}.
 
-% TODO: Move this further up to the commutativity of operators?
+% Logical operators
-\indent\red{[Fixing the error after finding it
+
-        \cite[Sec.~10.5.5]{nielsen_quantum_2010} -> This may require
+In order to modify the logical state encoded using the physical
-introducing the gates as unitary]} \\
+qubits, we can use \emph{logical operators} \cite[Sec.~4.2]{roffe_quantum_2019}.
-\indent\red{[Logical operators \cite[Sec.~4.2]{roffe_quantum_2019}]} \\
+For each qubit, there are two logical operators, $X_i$ and $Z_j$.
-\indent\red{[Measuring logical operators yields the outcomes of
+These are operators that
-the encoded computations \cite[Sec.~2.6]{derks_designing_2025}]} \\
+\begin{itemize}
-\indent\red{[X and Z measurements can be performed with only CNOT and
+    \item Commute with all the stabilizers in $\mathcal{S}$.
-Hadamard gates \cite[Sec.~10.5.8]{nielsen_quantum_2010}]} \\
+    \item Anti-commute with one another, i.e., $[ \overline{X}_i,
-\indent\red{[(?) Stabilizer generators]} \\
+        \overline{Z}_i ]_{+} = \overline{X}_i \overline{Z}_i +
-\indent\red{[Parity-check matrix \cite[Sec.~10.5.1]{nielsen_quantum_2010}]}
+        \overline{Z}_i \overline{X}_i = 0$.
+\end{itemize}
+We can also measure these operators to find out the logical state a
+physical state corresponds to \cite[Sec.~2.6]{derks_designing_2025}.
+
+% Parity-check matrix
+
+% 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.
+As such, we can represent them using a \emph{check matrix}
+\cite[Sec.~10.5.1]{nielsen_quantum_2010}.
+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
+redundancy.
+In contrast to the classical case, this matrix now has $2n$ columns,
+as we have to consider both the $X$ and $Z$ type operators that make up
+the stabilizers.
+Take for example the Steane code \cite[Eq.~10.83]{nielsen_quantum_2010}.
+We can describe it using the check matrix
+\begin{align*}
+    \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 \\
+            1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
+            0 & 0 & 0 & 0 & 0 & 0 & 0 & 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 & 1 & 0 & 1 & 0 & 1 & 0 & 1
+        \end{array}
+    \right]
+    .%
+\end{align*}
+We can understand each row as defining a stabilizer.
+The first $n$ columns correspond to $X$ operators acting on the
+corresponding physical qubit, the rest to the $Z$ operators.
+
+% TODO: Write
 
 \begin{figure}[t]
     \centering
@@ -1352,7 +1393,7 @@ Hadamard gates \cite[Sec.~10.5.8]{nielsen_quantum_2010}]} \\
     % tex-fmt: on
 
     \caption{
-        Illustration of a general syndrome extraction circuit.
+        Illustration of a full syndrome extraction circuit.
         Adapted from \cite[Figure~4]{roffe_quantum_2019}.
     }
     \label{fig:sec}
@@ -1366,13 +1407,17 @@ 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.
+We call such codes \ac{css} codes.
 
 \indent\red{[Z-type operators for X type errors and vice versa ]} \\
 \indent\red{[Construction from two binary linear codes
 \cite[p.~452,469]{nielsen_quantum_2010}]} \\
 
+%%%%%%%%%%%%%%%%
 \subsection{Quantum Low-Density Parity-Check Codes}
+\label{subsec:Quantum Low-Density Parity-Check Codes}
 
+\noindent\red{[(?) Mention color and surface codes]} \\
 \noindent\red{[Constant overhead scaling]} \\
 \noindent\red{[Scaling of minimum distance with code length]} \\
 \noindent\red{[Bivariate Bicycle codes]} \\
@@ -1382,37 +1427,19 @@ $Z$ operators and some with only $X$ operators.
         measured syndrome, the most likely error coset rather than the exact
         physical error.''
 % tex-fmt: off
-\cite[Sec.~II~B)]{koutsioumpas_colour_2025}]}%
+\cite[Sec.~II~B)]{koutsioumpas_colour_2025}%
 % tex-fmt: on
-        \\
+]} \\
 
-        \red{
+%%%%%%%%%%%%%%%%
-            \textbf{General Notes:}
+\subsection{Decoding Quantum Codes}
-            \begin{itemize}
+\label{subsec:Decoding Quantum Codes}
-                \item Note that there are other codes than stabilizer codes
+
-                    (and research and give some examples), but only
+\noindent\indent\red{[The QEC decoding problem
-                    stabilizer codes are considered in this work
+\cite[Sec.~2.3]{yao_belief_2024}]} \\
-                \item Degeneracy
+\indent\red{[+ Degeneracy]} \\
-                \item The QEC decoding problem (considering degeneracy)
+\indent\red{[+ Short cycles]} \\
-                    \cite[Sec.~2.3]{yao_belief_2024}
+\indent\red{[Fixing the error after finding it
-            \end{itemize}
+        \cite[Sec.~10.5.5]{nielsen_quantum_2010} -> This
-            \textbf{Content:}
+may require introducing the gates as unitary]}
-            \begin{itemize}
-                \item Stabilizer codes
-                    \begin{itemize}
-                        \item syndrome extraction circuit
-                        \item Stabilizer codes are effectively the QM
-                            % 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
-                            more conveniently described using stabilizers
-                            rather than working with the states directly
-                            \cite[Sec.~10.5.1]{nielsen_quantum_2010}
-                    \end{itemize}
-                \item Color codes?
-                \item Surface codes?
-            \end{itemize}
-        }
 

src/thesis/main.tex

@@ -30,6 +30,8 @@
 \renewcommand{\todo}[2][]{\tikzexternaldisable\@todo[#1]{#2}\tikzexternalenable}
 \makeatother
 
+\setcounter{MaxMatrixCols}{20}
+
 %
 %
 % Custom commands