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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 % 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}. 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} 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 \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 We then construct the Tanner graph by connecting each \ac{cn} to
the \acp{vn} that make up the corresponding parity check the \acp{vn} that make up the corresponding parity check
\cite[Sec.~5.1.2]{ryan_channel_2009}. \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. construction for the [7,4,3]-Hamming code.
% %
\begin{figure}[t] \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 We typically evaluate the performance of LDPC codes using the
\ac{ber} or the \ac{fer} (a \textit{frame} referes to one whole \ac{ber} or the \ac{fer} (a \textit{frame} referes to one whole
transmitted block in this context). 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 qualitative performance characteristic of an \ac{ldpc} code
\cite[Fig.~1]{costello_spatially_2014}. We talk of the \cite[Fig.~1]{costello_spatially_2014}. We talk of the
\textit{waterfall} and the \textit{error floor} regions. \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 where $K \in \mathbb{N}$ is the \textit{coupling width} and $L \in
\mathbb{N}$ is the number of spatial positions. \mathbb{N}$ is the number of spatial positions.
This construction results in a Tanner graph as depicted in This construction results in a Tanner graph as depicted in
\autoref{fig:sc-ldpc-tanner}. \Cref{fig:sc-ldpc-tanner}.
\begin{figure}[t] \begin{figure}[t]
\centering \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 Let us now examine how the observable operator $\hat{Q}$ relates to
the determinate states of the observable quantity. 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} \cite[Eq.~3.114]{griffiths_introduction_1995}
\begin{align} \begin{align}
\label{eq:gen_expr_Q_exp_lin} \label{eq:gen_expr_Q_exp_lin}
\braket{Q} = \braket{\psi \vert \hat{Q}\psi} \braket{Q} = \braket{\psi \vert \hat{Q}\psi}
.% .%
\end{align} \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. relationship.
The determinate states are inherently deterministic. The determinate states are inherently deterministic.
To relate the two, we note that since determinate states should 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 Because of the normalization of the wave function such that
$\int_{-\infty}^{\infty} \lvert \psi(x,t) \rvert^2 dx = 1$, we have $\int_{-\infty}^{\infty} \lvert \psi(x,t) \rvert^2 dx = 1$, we have
$\sum_{n=1}^{\infty} \lvert c_n \rvert ^2 = 1$. $\sum_{n=1}^{\infty} \lvert c_n \rvert ^2 = 1$.
Inserting \autoref{eq:determinate_basis} into Inserting \Cref{eq:determinate_basis} into
\autoref{eq:gen_expr_Q_exp_lin} we obtain \Cref{eq:gen_expr_Q_exp_lin} we obtain
% tex-fmt: off % tex-fmt: off
\cite[Prob.~3.35c)]{griffiths_introduction_1995} \cite[Prob.~3.35c)]{griffiths_introduction_1995}
% tex-fmt: on % tex-fmt: on
@@ -795,7 +795,7 @@ referring to the operator $\hat{Q}$.
% Projective measurements % Projective measurements
The measurements we considered in the previous section, for which 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}. \emph{projective measurements}.
For these, certain restrictions such as repeatability apply: the act For these, certain restrictions such as repeatability apply: the act
of measuring a quantum state should \emph{collapse} it onto one of 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 We can model the collapse of the original state onto one of the
superimposed basis states as a \emph{projection}. superimposed basis states as a \emph{projection}.
To see this, we use Equations \ref{eq:determinate_basis} and To see this, we use
\ref{eq:observable_eigenrelation} to compute \Cref{eq:determinate_basis,eq:observable_eigenrelation} to compute
\begin{align*} \begin{align*}
\hat{Q}\ket{\psi} = \sum_{n=1}^{\infty} c_n \hat{Q} \ket{e_n} \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} = \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*} \end{align*}
A qubit is defined to be a system with quantum state A qubit is defined to be a system with quantum state
\begin{align*} \begin{align}
\label{eq:gen_qubit_state}
\ket{\psi} = \ket{\psi} =
\begin{pmatrix} \begin{pmatrix}
\alpha \\ \alpha \\
@@ -889,7 +890,7 @@ A qubit is defined to be a system with quantum state
\end{pmatrix} \end{pmatrix}
= \alpha \ket{0} + \beta \ket{1} = \alpha \ket{0} + \beta \ket{1}
.% .%
\end{align*} \end{align}
The overall state of a composite quantum system is described using The overall state of a composite quantum system is described using
the \emph{tensor product}, denoted as $\otimes$ the \emph{tensor product}, denoted as $\otimes$
\cite[Sec.~2.2.8]{nielsen_quantum_2010}. \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 % 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 computational basis states needed to express the full composite state
is $2^n$. is $2^n$.
This is in contrast to classical systems, where the dimensionality of 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. We model state changes as operators.
Unlike classical systems, where there are only two possible states and Unlike classical systems, where there are only two possible states and
thus the only possible state change is a bit-flip, a general qubit 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. We thus technically also have an infinite number of possible state changes.
Fortunately, 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} \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 A controlled gate applies the respective operation only if the
control qubit is in state $\ket{1}$. control qubit is in state $\ket{1}$.
An example of this is the CNOT gate introduced in An example of this is the CNOT gate introduced in
\autoref{subsec:Qubits and Multi-Qubit States}, which is depicted in \Cref{subsec:Qubits and Multi-Qubit States}, which is depicted in
\autoref{fig:cnot_circuit}. \Cref{fig:cnot_circuit}.
\begin{figure}[t] \begin{figure}[t]
\centering \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. impossible to exactly copy the state of one qubit into another.
\item Qubits 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 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 \item Directly measuring the state of a qubit collapses it onto
one of the determinate states, thereby potentially destroying one of the determinate states, thereby potentially destroying
information. information.
@@ -1198,7 +1199,7 @@ whether a state belongs
% $\mathcal{C}$ or $\mathcal{F}$ with a certain probability. % $\mathcal{C}$ or $\mathcal{F}$ with a certain probability.
% } % }
to $\mathcal{C}$ or $\mathcal{F}$. 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 can be mathematically described using operators whose eigenvalues
are the possible measurement results. are the possible measurement results.
Here, we need an operator with two eigenvalues and the corresponding 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 $\ket{\psi}_\text{L}$ in such a way that the eigenvalue is indicated
by measuring the ancilla qubit instead. by measuring the ancilla qubit instead.
More specifically, using a stabilizer measurement circuit as shown in 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 three-qubit system as
\begin{align} \begin{align}
\label{eq:error_projection} \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 This is because the act of measuring the error partly collapses the
state, eliminating the uncertainty about the type of the error state, eliminating the uncertainty about the type of the error
\cite[Sec.~10.2]{nielsen_quantum_2010}. \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 $P_\mathcal{C}$ and $P_\mathcal{F}$ constitute projection operators onto
$\mathcal{C}$ and $\mathcal{F}$. $\mathcal{C}$ and $\mathcal{F}$.
E.g., $P_\mathcal{C}$ will eliminate all components of $E 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. describe which local codes are violated.
To obtain the syndrome, we simply measure the corresponding To obtain the syndrome, we simply measure the corresponding
operators $P_i$, each using a circuit as explained in 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. Note that this is an abstract representation of the syndrome extraction.
For the actual implementation in hardware, we can transform this into For the actual implementation in hardware, we can transform this into
a circuit that requires only CNOT and H-gates 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 $X$ stabilizers into a subset being made up of only $X$
operators and the rest only of $Z$ operators. operators and the rest only of $Z$ operators.
We call such codes \ac{css} codes. 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. of the Steane code.
% Construction % Construction
@@ -1514,7 +1515,7 @@ $\bm{H}_Z$ are constructed from two matrices $\bm{A}$ and $\bm{B}$ as
.% .%
\end{align*} \end{align*}
This way, we can guarantee the satisfaction of the commutativity 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. 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 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 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 % 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 with the parity information contained in the syndrome, to avoid
disturbing the quantum states of individual qubits. disturbing the quantum states of individual qubits.
This necessitates a modification of the standard \ac{bp} algorithm 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}. \cite[Sec.~3.1]{yao_belief_2024}.
Instead of attempting to find the most likely codeword directly, the Instead of attempting to find the most likely codeword directly, the
algorithm will now try to find an error pattern $\hat{\bm{e}} \in algorithm will now try to find an error pattern $\hat{\bm{e}} \in
@@ -1571,7 +1572,7 @@ indicated by the syndrome, calculating
. .
\end{align*} \end{align*}
The resulting syndrome-based \ac{bp} algorithm is shown in The resulting syndrome-based \ac{bp} algorithm is shown in
algorithm \ref{alg:syndome_bp}. \Cref{alg:syndome_bp}.
% tex-fmt: off % tex-fmt: off
\tikzexternaldisable \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, Another problem is that due to the commutativity property of the stabilizers,
quantum codes inherently contain short cycles quantum codes inherently contain short cycles
\cite[Sec.~IV.C]{babar_fifteen_2015}. \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 the violation of the independence assumption of the messages passed
during decoding, impeding performance. 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 This constrains the solution space more and more as the decoding
progresses, encouraging the algorithm to converge to one of the progresses, encouraging the algorithm to converge to one of the
solutions \cite[Sec.~5]{yao_belief_2024}. 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 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. natural constraint on the maximum number of outer iterations of the algorithm.

44
src/thesis/chapters/3_fault_tolerant_qec.tex
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@@ -53,7 +53,7 @@ indicating which errors occurred, with
\end{cases} \end{cases}
.% .%
\end{align*} \end{align*}
\autoref{fig:fault_tolerance_overview} illustrates the flow of errors. \Cref{fig:fault_tolerance_overview} illustrates the flow of errors.
Specifically for \ac{css} codes, a \ac{qec} procedure is deemed Specifically for \ac{css} codes, a \ac{qec} procedure is deemed
fault-tolerant, if \cite[Def.~4.2]{derks_designing_2025} fault-tolerant, if \cite[Def.~4.2]{derks_designing_2025}
\begin{gather*} \begin{gather*}
@@ -170,15 +170,15 @@ This is a code with check matrix
. .
\end{gather} \end{gather}
We can see that it has stabilizers $Z_1Z_2$ and $Z_2Z_3$. We can see that it has stabilizers $Z_1Z_2$ and $Z_2Z_3$.
\autoref{fig:pure_syndrome_extraction} shows the corresponding \Cref{fig:pure_syndrome_extraction} shows the corresponding
syndrome extraction circuit. syndrome extraction circuit.
We refer to the qubits carrying the logical state We refer to the qubits carrying the logical state
$\ket{\psi}_\text{L}$ as \emph{data qubits}. $\ket{\psi}_\text{L}$ as \emph{data qubits}.
Note that this is a concrete implementation using CNOT gates, as Note that this is a concrete implementation using CNOT gates, as
opposed to the system-level view introduced in opposed to the system-level view introduced in
\autoref{subsec:Stabilizer Codes}. \Cref{subsec:Stabilizer Codes}.
We visualize the different types of noise models in We visualize the different types of noise models in
\autoref{fig:noise_model_types}. \Cref{fig:noise_model_types}.
%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%
\subsection{Bit-Flip Noise} \subsection{Bit-Flip Noise}
@@ -187,7 +187,7 @@ We visualize the different types of noise models in
The simplest type of noise model is \emph{bit-flip} noise. 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 This corresponds to the classical \ac{bsc}, i.e., only $X$ errors on the
data qubits are possible \cite[Appendix~A]{gidney_new_2023}. data qubits are possible \cite[Appendix~A]{gidney_new_2023}.
This type of noise model is shown in \autoref{subfig:bit_flip}. 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 we cannot use bit-flip noise to develop fault-tolerant
systems, as it doesnt't account for errors during the syndrome extraction. systems, as it doesnt't account for errors during the syndrome extraction.
@@ -199,7 +199,7 @@ systems, as it doesnt't account for errors during the syndrome extraction.
Extending bit-flip noise to consider $X,Z$ or $Y$ instead of just $X$ Extending bit-flip noise to consider $X,Z$ or $Y$ instead of just $X$
errors, we obtain the \emph{depolarizing channel} errors, we obtain the \emph{depolarizing channel}
\cite[Sec.~7.6]{gottesman_stabilizer_1997}, depicted in \cite[Sec.~7.6]{gottesman_stabilizer_1997}, depicted in
\autoref{subfig:depolarizing}. \Cref{subfig:depolarizing}.
It is well-suited for modeling memory experiments, where data qubits It is well-suited for modeling memory experiments, where data qubits
are stored idly for some period of time and errors accumulate due to are stored idly for some period of time and errors accumulate due to
decoherence. decoherence.
@@ -223,7 +223,7 @@ locations right before each measurement \cite[Appendix~A]{gidney_new_2023}.
Note that it is enough to only consider $X$ errors at these points, Note that it is enough to only consider $X$ errors at these points,
since that is the only type of error directly affecting the since that is the only type of error directly affecting the
measurement outcomes. measurement outcomes.
This model is depicted in \autoref{subfig:phenomenological}. This model is depicted in \Cref{subfig:phenomenological}.
While not fully capturing all possible error mechanisms, While not fully capturing all possible error mechanisms,
phenomenological noise is already a significant step beyond the code phenomenological noise is already a significant step beyond the code
@@ -244,7 +244,7 @@ Specifically, we allow arbitrary $n$-qubit Pauli errors after each
$n$-qubit gate \cite[Def.~2.5]{derks_designing_2025}. $n$-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 is simply a series of correlated Pauli
errors on each related individual qubit. errors on each related individual qubit.
This type of noise model is shown in \autoref{subfig:circuit_level}. This type of noise model is shown in \Cref{subfig:circuit_level}.
While phenomenological noise is useful for some design aspects of 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
@@ -457,7 +457,7 @@ circuit, tracking which measurements they affect
We turn to our example of the three-qubit repetition code to We turn to our example of the three-qubit repetition code to
illustrate the construction of the syndrome measurement matrix. illustrate the construction of the syndrome measurement matrix.
We begin by extending our check matrix in \autoref{eq:rep_code_H} We begin by extending our check matrix in \Cref{eq:rep_code_H}
to represent three rounds of syndrome extraction. to represent three rounds of syndrome extraction.
Each round yields an additional set of syndrome bits, Each round yields an additional set of syndrome bits,
and we combine them by stacking them in a new vector and we combine them by stacking them in a new vector
@@ -476,7 +476,7 @@ additional syndrome measurement, to obtain
\end{pmatrix} \end{pmatrix}
.% .%
\end{align*} \end{align*}
\autoref{fig:rep_code_multiple_rounds_bit_flip} \Cref{fig:rep_code_multiple_rounds_bit_flip}
depicts the corresponding circuit. depicts the corresponding circuit.
Note that we have not yet introduced error locations in the syndrome 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. extraction circuitry, so we still consider only bit flip noise at this stage.
@@ -499,7 +499,7 @@ We now wish to expand the error model to phenomenological noise, though
only considering $X$ errors in this case. only considering $X$ errors in this case.
We introduce new error locations at the appropriate positions, We introduce new error locations at the appropriate positions,
arriving at the circuit depicted in arriving at the circuit depicted in
\autoref{fig:rep_code_multiple_rounds_phenomenological}. \Cref{fig:rep_code_multiple_rounds_phenomenological}.
For each additional error location, we extend $\bm{\Omega}$ by For each additional error location, we extend $\bm{\Omega}$ by
appending the corresponding syndrome vector as a column. appending the corresponding syndrome vector as a column.
\begin{gather} \begin{gather}
@@ -823,7 +823,7 @@ For two detector matrices $\bm{D}_1$ and $\bm{D}_2$, as long as
\end{gather} \end{gather}
they describe the same set of possible measurement outcomes (under they describe the same set of possible measurement outcomes (under
the absence of noise) and thus the same circuit. the absence of noise) and thus the same circuit.
In fact, as long as \autoref{eq:kern_condition} holds, the detector In fact, as long as \Cref{eq:kern_condition} holds, the detector
error matrices we construct from them can distinguish between the error matrices we construct from them can distinguish between the
same pairs of error sets \cite[Lemma~6]{derks_designing_2025}. same pairs of error sets \cite[Lemma~6]{derks_designing_2025}.
To see this, we note that we can distinguish between two circuit To see this, we note that we can distinguish between two circuit
@@ -856,7 +856,7 @@ There is, however, one way of defining the detectors that will prove useful
at a later stage. at a later stage.
To the measurement results from each syndrome extraction round we To the measurement results from each syndrome extraction round we
can add the results from the previous round, as illustrated in can add the results from the previous round, as illustrated in
\autoref{fig:detectors_from_measurements_general}. \Cref{fig:detectors_from_measurements_general}.
We thus have $D=n-k$. We thus have $D=n-k$.
Concretely, we denote the outcome of Concretely, we denote the outcome of
measurement $\ell \in \{1,\ldots,n-k\}$ in round $r \in \{1,\ldots,R\}$ by measurement $\ell \in \{1,\ldots,n-k\}$ in round $r \in \{1,\ldots,R\}$ by
@@ -912,15 +912,15 @@ with $\bm{m}^{(0)} = \bm{0}$.
\end{figure} \end{figure}
We again turn our attention to the three-qubit repetition code. We again turn our attention to the three-qubit repetition code.
In \autoref{fig:rep_code_multiple_rounds_phenomenological} we can see 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 tripped the last four measurements.
We now take those measurements and combine them according to We now take those measurements and combine them according to
\autoref{eq:measurement_combination}. \Cref{eq:measurement_combination}.
We can see this process graphically in We can see this process graphically in
\autoref{fig:detectors_from_measurements_rep_code}. \Cref{fig:detectors_from_measurements_rep_code}.
To understand why this way of defining the detectors is useful, we To understand why this way of defining the detectors is useful, we
note that the error $E_6$ in note that the error $E_6$ in
\autoref{fig:rep_code_multiple_rounds_phenomenological} has not only \Cref{fig:rep_code_multiple_rounds_phenomenological} has not only
tripped the measurements in the syndrome extraction round immediately tripped the measurements in the syndrome extraction round immediately
afterwards, but all subsequent ones as well. afterwards, but all subsequent ones as well.
To only see errors in the rounds immediately following them, we To only see errors in the rounds immediately following them, we
@@ -929,9 +929,9 @@ that effectively compute the difference between the measurements.
Each error can only trip syndrome bits that follow it. Each error can only trip syndrome bits that follow it.
This is reflected in the triangular structure of $\bm{\Omega}$ in This is reflected in the triangular structure of $\bm{\Omega}$ in
\autoref{eq:syndrome_matrix_ex}. \Cref{eq:syndrome_matrix_ex}.
Combining the measurements into detectors according to Combining the measurements into detectors according to
\autoref{eq:measurement_combination}, we are effectively performing \Cref{eq:measurement_combination}, we are effectively performing
row additions in such a way as to clear the bottom left of the matrix. row additions in such a way as to clear the bottom left of the matrix.
The detector error matrix The detector error matrix
\begin{align*} \begin{align*}
@@ -1062,7 +1062,7 @@ The overall probability of error is then
\hspace{12mm} \hspace{12mm}
\end{align} \end{align}
We approximate $p_\text{e,total}$ using a Monte Carlo simulation and We approximate $p_\text{e,total}$ using a Monte Carlo simulation and
compute the per-round-\ac{ler} using \autoref{eq:per_round_ler}. compute the per-round-\ac{ler} using \Cref{eq:per_round_ler}.
This is a common approach taken in the literature This is a common approach taken in the literature
\cite{gong_toward_2024}\cite{wang_fully_2025}. \cite{gong_toward_2024}\cite{wang_fully_2025}.
@@ -1086,7 +1086,7 @@ As it is related to the error rate through $F = 1 - 2p$, we obtain
\end{align} \end{align}
We have chosen to use the first approach, i.e., We have chosen to use the first approach, i.e.,
\autoref{eq:per_round_ler}, as the related literature is closer in \Cref{eq:per_round_ler}, as the related literature is closer in
topic to our own work. topic to our own work.
%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%
@@ -1096,7 +1096,7 @@ topic to our own work.
It is not immediately apparent how the \ac{dem} will look from looking It is not immediately apparent how the \ac{dem} will look from looking
at a code's \ac{pcm}, because it heavily depends on the exact circuit at a code's \ac{pcm}, because it heavily depends on the exact circuit
construction and choice of noise model. construction and choice of noise model.
As we noted in \autoref{subsec:Measurement Syndrome Matrix}, we can As we noted in \Cref{subsec:Measurement Syndrome Matrix}, we can
obtain a measurement syndrome matrix by propagating Pauli frames obtain a measurement syndrome matrix by propagating Pauli frames
through the circuit. through the circuit.
The standard choice of simulation tool used for this purpose is The standard choice of simulation tool used for this purpose is

6
src/thesis/main.tex
View File

@@ -27,6 +27,7 @@
\usepackage[noEnd=false]{algpseudocodex} \usepackage[noEnd=false]{algpseudocodex}
\usepackage{nicematrix} \usepackage{nicematrix}
\usepackage{colortbl} \usepackage{colortbl}
\usepackage{cleveref}
\usetikzlibrary{calc, positioning, arrows, fit} \usetikzlibrary{calc, positioning, arrows, fit}
\usetikzlibrary{external} \usetikzlibrary{external}
@@ -38,6 +39,11 @@
\setcounter{MaxMatrixCols}{20} \setcounter{MaxMatrixCols}{20}
\Crefname{equation}{}{}
\Crefname{section}{Section}{Sections}
\Crefname{subsection}{Subsection}{Subsections}
\Crefname{figure}{Figure}{Figures}
% %
% %
% Custom commands % Custom commands