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Replace autoref by cref

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Andreas Tsouchlos
2026-04-29 20:56:41 +02:00
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src/thesis/chapters/2_fundamentals.tex
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@@ -106,7 +106,7 @@ exponentially with $n$, in contrast to keeping track of all codewords directly.
% The decoding problem
%
Figure \ref{fig:Diagram of a transmission system} visualizes the
\Cref{fig:Diagram of a transmission system} visualizes the
communication process \cite[Sec.~1.1]{ryan_channel_2009}.
An input message $\bm{u}\in \mathbb{F}_2^k$ is mapped onto a codeword $\bm{x}
\in \mathbb{F}_2^n$. This is passed on to a modulator, which
@@ -197,7 +197,7 @@ bits, and \acp{cn}, corresponding to individual parity checks.
We then construct the Tanner graph by connecting each \ac{cn} to
the \acp{vn} that make up the corresponding parity check
\cite[Sec.~5.1.2]{ryan_channel_2009}.
Figure \ref{PCM and Tanner graph of the Hamming code} shows this
\Cref{PCM and Tanner graph of the Hamming code} shows this
construction for the [7,4,3]-Hamming code.
%
\begin{figure}[t]
@@ -286,7 +286,7 @@ $\mathcal{N}_\text{C} (j) = \left\{ i \in \mathcal{I} : \bm{H}_{j,i}
We typically evaluate the performance of LDPC codes using the
\ac{ber} or the \ac{fer} (a \textit{frame} referes to one whole
transmitted block in this context).
Considering an \ac{awgn} channel, \autoref{fig:ldpc-perf} shows a
Considering an \ac{awgn} channel, \Cref{fig:ldpc-perf} shows a
qualitative performance characteristic of an \ac{ldpc} code
\cite[Fig.~1]{costello_spatially_2014}. We talk of the
\textit{waterfall} and the \textit{error floor} regions.
@@ -415,7 +415,7 @@ This is achieved by connecting some \acp{vn} of one spatial position to
where $K \in \mathbb{N}$ is the \textit{coupling width} and $L \in
\mathbb{N}$ is the number of spatial positions.
This construction results in a Tanner graph as depicted in
\autoref{fig:sc-ldpc-tanner}.
\Cref{fig:sc-ldpc-tanner}.
\begin{figure}[t]
\centering
@@ -701,14 +701,14 @@ 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 algebra as
We begin by translating \Cref{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}
\braket{Q} = \braket{\psi \vert \hat{Q}\psi}
.%
\end{align}
\autoref{eq:gen_expr_Q_exp_lin} expresses an inherently probabilistic
\Cref{eq:gen_expr_Q_exp_lin} expresses an inherently probabilistic
relationship.
The determinate states are inherently deterministic.
To relate the two, we note that since determinate states should
@@ -757,8 +757,8 @@ We can use the determinate states for this purpose, expressing the state as%
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
Inserting \Cref{eq:determinate_basis} into
\Cref{eq:gen_expr_Q_exp_lin} we obtain
% tex-fmt: off
\cite[Prob.~3.35c)]{griffiths_introduction_1995}
% tex-fmt: on
@@ -795,7 +795,7 @@ referring to the operator $\hat{Q}$.
% Projective measurements
The measurements we considered in the previous section, for which
\autoref{eq:gen_expr_Q_exp_lin} holds, belong to the category of
\Cref{eq:gen_expr_Q_exp_lin} holds, belong to the category of
\emph{projective measurements}.
For these, certain restrictions such as repeatability apply: the act
of measuring a quantum state should \emph{collapse} it onto one of
@@ -809,8 +809,8 @@ they are not relevant to this work.
We can model the collapse of the original state onto one of the
superimposed basis states as a \emph{projection}.
To see this, we use Equations \ref{eq:determinate_basis} and
\ref{eq:observable_eigenrelation} to compute
To see this, we use
\Cref{eq:determinate_basis,eq:observable_eigenrelation} to compute
\begin{align*}
\hat{Q}\ket{\psi} = \sum_{n=1}^{\infty} c_n \hat{Q} \ket{e_n}
= \sum_{n=1}^{\infty} \lambda_n c_n \ket{e_n}
@@ -881,7 +881,8 @@ We fix an orthonormal basis of $\mathbb{C}^2$ to be
.%
\end{align*}
A qubit is defined to be a system with quantum state
\begin{align*}
\begin{align}
\label{eq:gen_qubit_state}
\ket{\psi} =
\begin{pmatrix}
\alpha \\
@@ -889,7 +890,7 @@ A qubit is defined to be a system with quantum state
\end{pmatrix}
= \alpha \ket{0} + \beta \ket{1}
.%
\end{align*}
\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}.
@@ -950,7 +951,7 @@ information is stored in the correlations between the qubits
% The size of the vector space
As we can see in \autoref{eq:product_state}, the number of
As we can see in \Cref{eq:product_state}, the number of
computational basis states needed to express the full composite state
is $2^n$.
This is in contrast to classical systems, where the dimensionality of
@@ -968,7 +969,7 @@ we now shift our focus to describing the evolution of their states.
We model state changes as operators.
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.
state as shown in \Cref{eq:gen_qubit_state} lives on a continuum of values.
We thus technically also have an infinite number of possible state changes.
Fortunately, we can express any operator as a linear combination of the
\emph{Pauli operators} \cite[Sec.~2.2]{gottesman_stabilizer_1997}
@@ -1083,8 +1084,8 @@ the gate to the corresponding qubit, where a filled dot is placed.
A controlled gate applies the respective operation only if the
control qubit is in state $\ket{1}$.
An example of this is the CNOT gate introduced in
\autoref{subsec:Qubits and Multi-Qubit States}, which is depicted in
\autoref{fig:cnot_circuit}.
\Cref{subsec:Qubits and Multi-Qubit States}, which is depicted in
\Cref{fig:cnot_circuit}.
\begin{figure}[t]
\centering
@@ -1127,7 +1128,7 @@ Three main restrictions apply \cite[Sec.~2.4]{roffe_quantum_2019}:
impossible to exactly copy the state of one qubit into another.
\item Qubits are susceptible to more types of errors than
just bit-flips, as we saw in
\autoref{subsec:Qubits and Multi-Qubit States}.
\Cref{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.
@@ -1198,7 +1199,7 @@ whether a state belongs
% $\mathcal{C}$ or $\mathcal{F}$ with a certain probability.
% }
to $\mathcal{C}$ or $\mathcal{F}$.
As explained in \autoref{subsec:Observables}, physical measurements
As explained in \Cref{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
@@ -1225,7 +1226,7 @@ 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 the ancilla qubit instead.
More specifically, using a stabilizer measurement circuit as shown in
\autoref{fig:stabilizer_measurement}, we transform the state of the
\Cref{fig:stabilizer_measurement}, we transform the state of the
three-qubit system as
\begin{align}
\label{eq:error_projection}
@@ -1270,7 +1271,7 @@ 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
This can be seen in \Cref{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
@@ -1348,7 +1349,7 @@ 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 $P_i$, each using a circuit as explained in
\autoref{subsec:Stabilizer Measurements}.
\Cref{subsec:Stabilizer Measurements}.
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
@@ -1444,7 +1445,7 @@ vice versa, this property translates into being able to split the
stabilizers into a subset being made up of only $X$
operators and the rest only of $Z$ operators.
We call such codes \ac{css} codes.
We can see this property in \autoref{eq:steane} in the check matrix
We can see this property in \Cref{eq:steane} in the check matrix
of the Steane code.
% Construction
@@ -1514,7 +1515,7 @@ $\bm{H}_Z$ are constructed from two matrices $\bm{A}$ and $\bm{B}$ as
.%
\end{align*}
This way, we can guarantee the satisfaction of the commutativity
condition (\autoref{eq:css_condition}).
condition (\Cref{eq:css_condition}).
To define $\bm{A}$ and $\bm{B}$ we first introduce some additional notation.
We denote the identity matrix as $\bm{I_l} \in \mathbb{F}^{l\times l}$ and
the \emph{cyclic shift matrix} as $\bm{S_l} \in \mathbb{F}^{l\times
@@ -1543,11 +1544,11 @@ and thus lower error rates \cite[Sec.~1]{bravyi_high-threshold_2024}.
% Syndrome-based BP
As we saw in \autoref{subsec:Stabilizer Measurements}, we work only
As we saw in \Cref{subsec:Stabilizer Measurements}, we work only
with the parity information contained in the syndrome, to avoid
disturbing the quantum states of individual qubits.
This necessitates a modification of the standard \ac{bp} algorithm
introduced in \autoref{subsec:Iterative Decoding}
introduced in \Cref{subsec:Iterative Decoding}
\cite[Sec.~3.1]{yao_belief_2024}.
Instead of attempting to find the most likely codeword directly, the
algorithm will now try to find an error pattern $\hat{\bm{e}} \in
@@ -1571,7 +1572,7 @@ indicated by the syndrome, calculating
.
\end{align*}
The resulting syndrome-based \ac{bp} algorithm is shown in
algorithm \ref{alg:syndome_bp}.
\Cref{alg:syndome_bp}.
% tex-fmt: off
\tikzexternaldisable
@@ -1639,7 +1640,7 @@ direction to proceed in \cite[Sec.~5]{yao_belief_2024}.
Another problem is that due to the commutativity property of the stabilizers,
quantum codes inherently contain short cycles
\cite[Sec.~IV.C]{babar_fifteen_2015}.
As discussed in \autoref{subsec:Iterative Decoding}, these lead to
As discussed in \Cref{subsec:Iterative Decoding}, these lead to
the violation of the independence assumption of the messages passed
during decoding, impeding performance.
@@ -1656,7 +1657,7 @@ a hard decision and excluding it from further decoding.
This constrains the solution space more and more as the decoding
progresses, encouraging the algorithm to converge to one of the
solutions \cite[Sec.~5]{yao_belief_2024}.
Algorithm \ref{alg:bpgd} shows this process.
\Cref{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 outer iterations of the algorithm.