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Incorporate Jonathan's corrections to Fault Tolerance Chapter

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Andreas Tsouchlos
2026-05-04 19:45:15 +02:00
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src/thesis/chapters/3_fault_tolerant_qec.tex
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@@ -16,17 +16,19 @@ using qubits.
While the use of error correcting codes may facilitate this, it also While the use of error correcting codes may facilitate this, it also
introduces two new challenges \cite[Sec.~4]{gottesman_introduction_2009}: introduces two new challenges \cite[Sec.~4]{gottesman_introduction_2009}:
\begin{itemize} \begin{itemize}
\item We must be able to perform operations on the encoded state \item For realizing a quantum algorithm, we must be able to
in such a way that we do not lose the protection against errors. perform operations on the encoded state in such a way that we
\item \ac{qec} systems are themselves partially implemented in do not lose the protection against errors.
quantum hardware. In addition to the errors we have \item \ac{qec} systems, in particular the syndrome extraction
originally introduced them for, these systems must circuit, are themselves partially implemented in
be able to account for the fact they are implemented on noisy quantum hardware.
hardware themselves. In addition to the errors we have originally introduced them
for, these systems must be able to account for the fact they
are implemented on noisy hardware themselves.
\end{itemize} \end{itemize}
In the literature, both of these points are viewed under the umbrella In the literature, both of these points are viewed under the umbrella
of \emph{fault-tolerant} quantum computing. of \emph{fault-tolerant} quantum computing.
We focus only on the second aspect in this work. In this thesis, we focus only on the second aspect.
It was recognized early on as a challenge of \ac{qec} that the correction It was recognized early on as a challenge of \ac{qec} that the correction
machinery itself may introduce new faults \cite[Sec.~III]{shor_scheme_1995}. machinery itself may introduce new faults \cite[Sec.~III]{shor_scheme_1995}.
@@ -43,16 +45,16 @@ address both.
We model the possible occurrence of errors during any processing We model the possible occurrence of errors during any processing
stage as different \emph{error locations} $E_i,~i\in [1:N]$ stage as different \emph{error locations} $E_i,~i\in [1:N]$
in the circuit. in the circuit.
$N \in \mathbb{N}$ is the total number of considered error locations. The parameter $N \in \mathbb{N}$ is the total number of considered
error locations.
The \emph{circuit error vector} $\bm{e} \in \{0,1\}^N$ is a vector The \emph{circuit error vector} $\bm{e} \in \{0,1\}^N$ is a vector
indicating which errors occurred, with indicating which errors occurred, with
\begin{align*} \begin{align*}
e_i := e_i :=
\begin{cases} \begin{cases}
1, & \text{Error $E_i$ occurred} \\ 1, & \text{error $E_i$ occurred}, \\
0, & \text{otherwise} 0, & \text{otherwise}.
\end{cases} \end{cases}
.%
\end{align*} \end{align*}
\Cref{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
@@ -72,12 +74,14 @@ fault-tolerant, if \cite[Def.~4.2]{derks_designing_2025}
where $t = \lfloor (d_\text{min} -1)/2 \rfloor$ is the number of where $t = \lfloor (d_\text{min} -1)/2 \rfloor$ is the number of
errors the code is able to correct. errors the code is able to correct.
The vectors $\bm{e}_{\text{output},X}$ and $\bm{e}_{\text{output},Z}$ The vectors $\bm{e}_{\text{output},X}$ and $\bm{e}_{\text{output},Z}$
denote only $X$ and $Z$ errors respectively. denote only $X$ and $Z$ errors, respectively.
% TODO: Properly introduce d_min for QEC, specifically for CSS codes % TODO: Properly introduce d_min for QEC, specifically for CSS codes
In order to deal with internal errors that flip syndrome bits, In order to deal with internal errors that flip syndrome bits,
multiple rounds of syndrome measurements must be performed. multiple rounds of syndrome measurements are performed.
Typically, the number of syndrome extraction rounds is chosen as $d_\text{min}$. Typically, the number of syndrome extraction rounds is chosen as
$d_\text{min}$, e.g., \cite{gong_toward_2024}
\cite{koutsioumpas_automorphism_2025}.
% % This is the definition of a fault-tolerant QEC gadget % % This is the definition of a fault-tolerant QEC gadget
% A \ac{qec} procedure is deemed fault tolerant if % A \ac{qec} procedure is deemed fault tolerant if
@@ -150,7 +154,7 @@ Typically, the number of syndrome extraction rounds is chosen as $d_\text{min}$.
% Intro % Intro
We collect the probabilities of error at each location in the We collect the probabilities of error at each location in the
\emph{noise model}, a vector $\bm{p} \in [0,1]^N$. \emph{noise model}, represented by a vector $\bm{p} \in [0,1]^N$.
There are different types of noise models, each allowing for There are different types of noise models, each allowing for
different error locations in the circuit. different error locations in the circuit.
@@ -178,8 +182,7 @@ $\ket{\psi}_\text{L}$ 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
\Cref{subsec:Stabilizer Codes}. \Cref{subsec:Stabilizer Codes}.
We visualize the different types of noise models in \Cref{fig:noise_model_types} visualizes the different types of noise models.
\Cref{fig:noise_model_types}.
%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%
\subsection{Bit-Flip Noise} \subsection{Bit-Flip Noise}
@@ -190,7 +193,7 @@ 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}.
The occurrence of bit-flip errors is modeled as a Bernoulli process The occurrence of bit-flip errors is modeled as a Bernoulli process
$\text{Bern}(p)$. $\text{Bern}(p)$.
This type of noise model is shown in \Cref{subfig:bit_flip}. \Cref{subfig:bit_flip} shows this type of noise model.
Note that bit-flip noise is not suitable for developing fault-tolerant Note that bit-flip noise is not suitable for developing fault-tolerant
systems, as it does not account for errors during the syndrome extraction. systems, as it does not account for errors during the syndrome extraction.
@@ -223,7 +226,7 @@ Here, we consider multiple rounds of syndrome measurements with a
depolarizing channel before each round. depolarizing channel before each round.
Additionally, we allow for measurement errors by having $X$ error Additionally, we allow for measurement errors by having $X$ error
locations right before each measurement \cite[Appendix~A]{gidney_new_2023}. 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 before measuring,
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 \Cref{subfig:phenomenological}. This model is depicted in \Cref{subfig:phenomenological}.
@@ -253,7 +256,7 @@ 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
always be used \cite[Sec.~4.2]{derks_designing_2025}. always be used \cite[Sec.~4.2]{derks_designing_2025}.
Note that this introduces new challenges during the decoding process, Note that this introduces new challenges during the decoding process,
as the decoding complexity is increased considerably due to the many as the decoding complexity is considerably increased due to the many
error locations. error locations.
\begin{figure}[t] \begin{figure}[t]
@@ -284,11 +287,11 @@ error locations.
framework for framework for
passing information about a circuit used for \ac{qec} to a decoder. passing information about a circuit used for \ac{qec} to a decoder.
They are also useful as a theoretical tool to aid in the design of They are also useful as a theoretical tool to aid in the design of
fault-tolerant \ac{qec} schemes. fault-tolerant \ac{qec} schemes, e.g., they can be used to easily
E.g., they can be used to easily determine whether a measurement determine whether a measurement schedule is fault-tolerant
schedule is fault-tolerant \cite[Example~12]{derks_designing_2025}. \cite[Example~12]{derks_designing_2025}.
Other approaches of implementing fault-tolerance circuits exist, such as Other approaches of implementing fault-tolerance circuits exist, e.g.,
flag error correction, which uses additional ancilla qubits to detect flag error correction, which uses additional ancilla qubits to detect
potentially damaging high-weight errors \cite[Sec.~1]{chamberland_flag_2018}. potentially damaging high-weight errors \cite[Sec.~1]{chamberland_flag_2018}.
However, \acp{dem} offer some unique advantages However, \acp{dem} offer some unique advantages
@@ -310,7 +313,7 @@ To achieve fault tolerance, the goal we strive towards is to
consider the internal errors in addition to the input errors during consider the internal errors in addition to the input errors during
the decoding process. the decoding process.
The core idea behind detector error models is to do this by defining The core idea behind detector error models is to do this by defining
a new \emph{circuit code} that describes the circuit. a new \emph{circuit code} describing the whole circuit.
Each \ac{vn} of this new code corresponds to an error location in the Each \ac{vn} of this new code corresponds to an error location in the
circuit and each \ac{cn} corresponds to a syndrome measurement. circuit and each \ac{cn} corresponds to a syndrome measurement.
% This circuit code, combined with the prior probabilities of error % This circuit code, combined with the prior probabilities of error
@@ -446,12 +449,11 @@ matrix} $\bm{\Omega} \in \mathbb{F}_2^{M\times N}$, with
\begin{align*} \begin{align*}
\Omega_{\ell,i} = \Omega_{\ell,i} =
\begin{cases} \begin{cases}
1, & \text{Error $i$ flips measurement $\ell$}\\ 1, & \text{error $i$ flips measurement $\ell$},\\
0, & \text{otherwise} 0, & \text{otherwise},
\end{cases} \end{cases}
,%
\end{align*} \end{align*}
where $M \in \mathbb{N}$ is the number of measurements. where $M \in \mathbb{N}$ is the number of performed syndrome measurements.
To obtain $\bm{\Omega}$, we must propagate Pauli errors through the To obtain $\bm{\Omega}$, we must propagate Pauli errors through the
circuit, tracking which measurements they affect circuit, tracking which measurements they affect
\cite[Sec.~2.4]{derks_designing_2025}. \cite[Sec.~2.4]{derks_designing_2025}.
@@ -466,8 +468,8 @@ 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
$\bm{s} \in \mathbb{F}_2^{R(n-k)}$, where $R \in \mathbb{N}$ is the $\bm{s} \in \mathbb{F}_2^{R(n-k)}$, where $R \in \mathbb{N}$ is the
number of syndrome measurement rounds. number of syndrome measurement rounds.
We thus have to replicate the rows of $\bm{H}_Z$, once for each Thus, we have to replicate the rows of $\bm{H}_Z$, once for each
additional syndrome measurement, to obtain additional syndrome measurement, and obtain
\begin{align*} \begin{align*}
\bm{\Omega}_0 = \bm{\Omega}_0 =
\begin{pmatrix} \begin{pmatrix}
@@ -493,11 +495,11 @@ extraction circuitry, so we still consider only bit flip noise at this stage.
Recall that $\bm{\Omega}_0$ describes which \ac{vn} is connected to Recall that $\bm{\Omega}_0$ describes which \ac{vn} is connected to
which parity check and the syndrome indicates which parity checks which parity check and the syndrome indicates which parity checks
are violated. are violated.
This means that if an error exists at only a single \ac{vn}, we can Therefore, if an error occurs that corresponds to a single \ac{vn},
read off the syndrome in the corresponding column. the measured syndrome is the corresponding column.
If errors occur at multiple locations, the resulting syndrome will be If errors occur at multiple locations, the resulting syndrome will be
the linear combination of the respective columns. the linear combination of the respective columns.
We thus have Thus, we have
\begin{align*} \begin{align*}
\bm{s} \in \text{span} \{\bm{\Omega}_0\} \bm{s} \in \text{span} \{\bm{\Omega}_0\}
.% .%
@@ -505,13 +507,13 @@ We thus have
% Expand to phenomenological % Expand to phenomenological
We now wish to expand the error model to phenomenological noise, though Next, we 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 resulting in the circuit depicted in
\Cref{fig:rep_code_multiple_rounds_phenomenological}. \Cref{fig:rep_code_multiple_rounds_phenomenological}.
For each additional error location, we extend $\bm{\Omega}_0$ by For each additional error location, we extend $\bm{\Omega}_0$ by
appending the corresponding syndrome vector as a column. appending the corresponding syndrome vector as a column, yielding
\begin{gather} \begin{gather}
\label{eq:syndrome_matrix_ex} \label{eq:syndrome_matrix_ex}
\bm{\Omega}_1 = \bm{\Omega}_1 =
@@ -790,15 +792,14 @@ to a detector.
We should note at this point that the combination of measurements We should note at this point that the combination of measurements
into detectors has no bearing on the actual construction of the into detectors has no bearing on the actual construction of the
syndrome extraction circuitry. syndrome extraction circuitry.
It is something that happens ``virtually'' after the fact and only It is something that happens ``virtually'' and only affects the decoder.
affects the decoder.
Note that we can use the detector matrix $\bm{D}$ to describe the set Note that we can use the detector matrix $\bm{D}$ to describe the set
of possible measurement outcomes under the absence of noise. of possible measurement outcomes under the absence of noise.
Similar to the we use a \ac{pcm} to describe the code space as Similar to the we use a \ac{pcm} to describe the code space as
\begin{equation*} \begin{equation*}
\mathcal{C} \mathcal{C}
= \{ \bm{x} \in \mathbb{F}_2^{n} : \bm{H}\bm{x}^\text{T} = \bm{0} \} = \{ \bm{x} \in \mathbb{F}_2^{n} : \bm{H}\bm{x}^\mathsf{T} = \bm{0} \}
,% ,%
\end{equation*} \end{equation*}
the set of possible measurement outcomes is simply $\text{kern}\{\bm{D}\}$ the set of possible measurement outcomes is simply $\text{kern}\{\bm{D}\}$
@@ -815,7 +816,7 @@ affect the measurements (through $\bm{\Omega}$), and we know how the
measurements relate to the detectors (through $\bm{D}$). measurements relate to the detectors (through $\bm{D}$).
For decoding, we are interested in the effect of the errors on the For decoding, we are interested in the effect of the errors on the
detectors directly. detectors directly.
We thus construct the \emph{detector error matrix} $\bm{H} \in Thus, we construct the \emph{detector error matrix} $\bm{H} \in
\mathbb{F}_2^{D\times N}$ \cite[Def.~2.9]{derks_designing_2025} as \mathbb{F}_2^{D\times N}$ \cite[Def.~2.9]{derks_designing_2025} as
\begin{align*} \begin{align*}
\bm{H} := \bm{D}\bm{\Omega} \bm{H} := \bm{D}\bm{\Omega}
@@ -859,7 +860,7 @@ It may, however, change the decoding performance when using a practical decoder.
What constitutes a good set of detectors is difficult to assess What constitutes a good set of detectors is difficult to assess
without performing explicit decoding simulations, since it ultimately without performing explicit decoding simulations, since it ultimately
depends on the decoder employed. depends on the employed decoder.
For iterative decoders, high sparsity is generally beneficial, but For iterative decoders, high sparsity is generally beneficial, but
finding detectors that maximize sparsity is an NP-complete problem finding detectors that maximize sparsity is an NP-complete problem
\cite[Sec.~2.6]{derks_designing_2025}. \cite[Sec.~2.6]{derks_designing_2025}.
@@ -868,7 +869,7 @@ 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
\Cref{fig:detectors_from_measurements_general}. \Cref{fig:detectors_from_measurements_general}.
We thus have $D=n-k$. Thus, we have $D=n-k$.
Concretely, we denote the outcome of Concretely, we denote the outcome of
measurement $\ell \in [1:n-k]$ in round $r \in [1:R]$ by measurement $\ell \in [1:n-k]$ in round $r \in [1:R]$ by
$m_\ell^{(r)} \in \mathbb{F}_2$ $m_\ell^{(r)} \in \mathbb{F}_2$
@@ -935,17 +936,18 @@ note that the error $E_6$ in
\Cref{fig:rep_code_multiple_rounds_phenomenological} has not only \Cref{fig:rep_code_multiple_rounds_phenomenological} has not only
triggered the measurements in the syndrome extraction round immediately triggered 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 the effect of errors in the syndrome measurement round
consider our newly defined detectors instead of the measurements, immediately following them, we consider our newly defined detectors
that effectively compute the difference between the measurements. instead of the measurements, that effectively compute the difference
between the measurements.
Each error can only trigger syndrome bits that follow it. Hereby, each error can only trigger 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
\Cref{eq:syndrome_matrix_ex}. \Cref{eq:syndrome_matrix_ex}.
Combining the measurements into detectors according to Combining the measurements into detectors according to
\Cref{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 resulting detector error matrix
\begin{align*} \begin{align*}
\bm{H} = \bm{H} =
\left( \left(
@@ -959,7 +961,7 @@ The detector error matrix
\end{array} \end{array}
\right) \right)
\end{align*} \end{align*}
obtained this way has a block-diagonal structure. has a block-diagonal structure.
Note that we exploit the fact that each syndrome measurement round is Note that we exploit the fact that each syndrome measurement round is
identical to obtain this structure. identical to obtain this structure.
@@ -1008,9 +1010,8 @@ error matrix $\bm{H}$ and the noise model $\bm{p}$.
\cite[Sec.~6]{derks_designing_2025}. \cite[Sec.~6]{derks_designing_2025}.
It serves as an abstract representation of a circuit and can be used It serves as an abstract representation of a circuit and can be used
both to transfer information to a decoder but also to aid in the both to transfer information to a decoder but also to aid in the
design of fault-tolerant systems. design of fault-tolerant systems, e.g., it can be used to investigate
E.g., it can be used to investigate the properties of a circuit with the properties of a circuit with respect to fault tolerance.
respect to fault tolerance.
It contains all information necessary for the decoding process. It contains all information necessary for the decoding process.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -1052,7 +1053,7 @@ value, the physical error rate $p_\text{phys}$.
% Per-round LER % Per-round LER
Another aspect that is important to consider is the meaning of the Another important aspect to consider is the meaning of the
\ac{ler} in the context of a \ac{qec} system with multiple \ac{ler} in the context of a \ac{qec} system with multiple
rounds of syndrome measurements. rounds of syndrome measurements.
In order to facilitate the comparability of results obtained from In order to facilitate the comparability of results obtained from
@@ -1063,7 +1064,7 @@ The simplest way of calculating the per-round \ac{ler} is by modeling
each round as an independent experiment. each round as an independent experiment.
For each experiment, an error might occur with a certain probability For each experiment, an error might occur with a certain probability
$p_\text{e,round}$. $p_\text{e,round}$.
The overall probability of error is then Then the overall probability of error is
\begin{align} \begin{align}
\hspace{-12mm} \hspace{-12mm}
p_\text{e,total} &= 1 - (1 - p_\text{e,round})^{R} \nonumber\\ p_\text{e,total} &= 1 - (1 - p_\text{e,round})^{R} \nonumber\\
@@ -1073,13 +1074,14 @@ 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 To this end, we approximate $p_\text{e,total}$ using a Monte Carlo
compute the per-round-\ac{ler} using \Cref{eq:per_round_ler}. simulation and
compute the per-round-\ac{ler} according to \Cref{eq:per_round_ler}.
This is the approach taken in \cite{gong_toward_2024}\cite{wang_fully_2025}. This is the approach taken in \cite{gong_toward_2024}\cite{wang_fully_2025}.
Another approach \cite{chen_exponential_2021}% Another approach \cite{chen_exponential_2021}%
\cite{bausch_learning_2024}\cite{beni_tesseract_2025} is to assume an \cite{bausch_learning_2024}\cite{beni_tesseract_2025} is to assume an
exponential decay for the decoder's \emph{logical fidelity} exponential decay for the \emph{logical fidelity} of the decoder
\cite[Eq.~(2)]{bausch_learning_2024} \cite[Eq.~(2)]{bausch_learning_2024}
\begin{align*} \begin{align*}
F_\text{total} = (F_\text{round})^{R} F_\text{total} = (F_\text{round})^{R}
@@ -1104,10 +1106,10 @@ topic to our own work.
\subsection{Stim} \subsection{Stim}
\label{subsec:Stim} \label{subsec:Stim}
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
at a code's \ac{pcm}, because it heavily depends on the exact circuit considering the \ac{pcm} of a code, because it heavily depends on the
construction and choice of noise model. exact circuit construction and choice of noise model.
As we noted in \Cref{subsec:Measurement Syndrome Matrix}, we can As we noted in \Cref{subsec:Measurement Syndrome Matrix}, we
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
@@ -1118,16 +1120,16 @@ pypi package.
In fact, it was in this tool that the concept of the \ac{dem} was In fact, it was in this tool that the concept of the \ac{dem} was
first introduced. first introduced.
One capability of stim, and \acp{dem} in general, that we didn't go One capability of stim, and \acp{dem} in general, that we did not
into detail about in this chapter is the merging of error mechanisms. explain in detail about in this chapter, is the merging of error mechanisms.
Since \acp{dem} differentiate errors based on their effect on the Since \acp{dem} differentiate errors based on their effect on the
measurements and not on their Pauli type and location measurements and not on their Pauli type and location
\cite[Sec.~1.4.3]{higgott_practical_2024}, it is natural to group \cite[Sec.~1.4.3]{higgott_practical_2024}, it is natural to group
errors that have the same effect. errors that have the same effect, i.e., syndrome.
This slightly lowers the computational complexity of decoding, as the This slightly lowers the computational complexity of decoding, as the
number of resulting \acp{vn} is reduced. number of resulting \acp{vn} is reduced.
While stim is a useful tool for circuit simulation, it doesn't While stim is a useful tool for circuit simulation, it does not
include many utilities for building syndrome extraction circuitry automatically. include many utilities for building syndrome extraction circuitry automatically.
The user has to define most, if not all, of the circuit manually, The user has to define most, if not all, of the circuit manually,
depending on the code in question. depending on the code in question.