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Incorporate Lia's corrections to fault tolerance

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Andreas Tsouchlos
2026-05-04 14:59:49 +02:00
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103
src/thesis/chapters/3_fault_tolerant_qec.tex
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@@ -25,7 +25,7 @@ introduces two new challenges \cite[Sec.~4]{gottesman_introduction_2009}:
hardware themselves.
\end{itemize}
In the literature, both of these points are viewed under the umbrella
of \emph{fault tolerance}.
of \emph{fault-tolerant} quantum computing.
We focus only on the second aspect in this work.
It was recognized early on as a challenge of \ac{qec} that the correction
@@ -188,9 +188,11 @@ We visualize the different types of noise models in
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
data qubits are possible \cite[Appendix~A]{gidney_new_2023}.
The occurrence of bit-flip errors is modeled as a Bernoulli process
$\text{Bern}(p)$.
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 bit-flip noise is not suitable for developing fault-tolerant
systems, as it does not account for errors during the syndrome extraction.
%%%%%%%%%%%%%%%%
@@ -243,12 +245,12 @@ Here we not only consider noise between syndrome extraction rounds
and at the measurements, but at each gate.
Specifically, we allow arbitrary $n$-qubit Pauli errors after each
$n$-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 can be written as a series of correlated Pauli
errors on each related individual qubit.
This type of noise model is shown in \Cref{subfig:circuit_level}.
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}.
Note that this introduces new challenges during the decoding process,
as the decoding complexity is increased considerably due to the many
@@ -286,7 +288,7 @@ fault-tolerant \ac{qec} schemes.
E.g., they can be used to easily determine whether a measurement
schedule is fault-tolerant \cite[Example~12]{derks_designing_2025}.
Other approaches of implementing fault tolerance exist, such as
Other approaches of implementing fault-tolerance circuits exist, such as
flag error correction, which uses additional ancilla qubits to detect
potentially damaging high-weight errors \cite[Sec.~1]{chamberland_flag_2018}.
However, \acp{dem} offer some unique advantages
@@ -300,8 +302,7 @@ However, \acp{dem} offer some unique advantages
treated in a unified manner. This leads to a more powerful
description of the overall circuit.
\end{itemize}
In this work, we only consider the process of decoding under the
\ac{dem} framework.
In this work, we consider the process of decoding under the \ac{dem} framework.
% Core idea
@@ -459,15 +460,22 @@ circuit, tracking which measurements they affect
We turn to our example of the three-qubit repetition code to
illustrate the construction of the syndrome measurement matrix.
We begin by extending our check matrix in \Cref{eq:rep_code_H}
to represent three rounds of syndrome extraction.
We begin by extending our check matrix $\bm{H}_Z$ in
\Cref{eq:rep_code_H} to represent three rounds of syndrome extraction.
Each round yields an additional set of syndrome bits,
and we combine them by stacking them in a new vector
$\bm{s} \in \mathbb{F}_2^{R(n-k)}$.
We thus have to replicate the rows of $\bm{\Omega}$, once for each
$\bm{s} \in \mathbb{F}_2^{R(n-k)}$, where $R \in \mathbb{N}$ is the
number of syndrome measurement rounds.
We thus have to replicate the rows of $\bm{H}_Z$, once for each
additional syndrome measurement, to obtain
\begin{align*}
\bm{\Omega} =
\bm{\Omega}_0 =
\begin{pmatrix}
\bm{H}_Z \\
\bm{H}_Z \\
\bm{H}_Z
\end{pmatrix}
=
\begin{pmatrix}
1 & 1 & 0 \\
0 & 1 & 1 \\
@@ -482,7 +490,7 @@ additional syndrome measurement, to obtain
depicts the corresponding circuit.
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.
Recall that $\bm{\Omega}$ 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
are violated.
This means that if an error exists at only a single \ac{vn}, we can
@@ -491,7 +499,7 @@ If errors occur at multiple locations, the resulting syndrome will be
the linear combination of the respective columns.
We thus have
\begin{align*}
\bm{s} \in \text{span} \{\bm{\Omega}\}
\bm{s} \in \text{span} \{\bm{\Omega}_0\}
.%
\end{align*}
@@ -502,11 +510,11 @@ only considering $X$ errors in this case.
We introduce new error locations at the appropriate positions,
arriving at the circuit depicted in
\Cref{fig:rep_code_multiple_rounds_phenomenological}.
For each additional error location, we extend $\bm{\Omega}$ by
For each additional error location, we extend $\bm{\Omega}_0$ by
appending the corresponding syndrome vector as a column.
\begin{gather}
\label{eq:syndrome_matrix_ex}
\bm{\Omega} =
\bm{\Omega}_1 =
\left(
\begin{array}{ccccccccccccccc}
1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0
@@ -523,24 +531,25 @@ appending the corresponding syndrome vector as a column.
& 0 & 1 & 1 & 0 & 1
\end{array}
\right) . \\[-6mm]
\hspace*{-58.7mm}
\hspace*{-56.7mm}
\underbrace{
\phantom{
\begin{array}{ccc}
0 & 0 & 0
\end{array}
}
}_\text{Original matrix}
}_{\bm{\Omega}_0} \nonumber
\end{gather}
Notice that the first three columns correspond to the original
measurement syndrome matrix, as these columns correspond to the error
locations on the data qubits.
measurement syndrome matrix $\bm{\Omega}_0$, as these columns
correspond to the error locations on the data qubits.
In this example, all measurements we considered were syndrome measurements.
Assuming no errors, the results of those measurements were
deterministic, irrespective of the actual logical state
$\ket{\psi}_\text{L}$, as they only depend on whether
$\ket{\psi}_\text{L} \in \mathcal{C}$, not on the concrete state.
Assuming no errors, the results of those measurements are
deterministic: They are not subject to any probabilistic behavior
despite the quantum mechanical nature of the underlying system.
They only depend on whether $\ket{\psi}_\text{L} \in \mathcal{C}$,
not on the concrete state.
It is, in general, possible to also consider non-deterministic measurements.
As an example, it is usual to consider a round of noiseless
measurements of the actual data qubit states after the last syndrome
@@ -557,7 +566,7 @@ extraction round.
\centering
\begin{tikzpicture}
\node{$%
\bm{\Omega} =
\bm{\Omega}_0 =
\begin{pmatrix}
1 & 1 & 0 \\
0 & 1 & 1 \\
@@ -667,7 +676,7 @@ extraction round.
\end{gather*}
\vspace*{-8mm}
\begin{gather*}
\bm{\Omega} =
\bm{\Omega}_1 =
\left(
\begin{array}{
cccccc%
@@ -761,10 +770,10 @@ Instead of using stabilizer measurement results directly, we
generalize the notion of what constitutes a parity check slightly.
We formally define a \emph{detector} as a deterministic parity constraint on
a set of measurement outcomes \cite[Def.~2.1]{derks_designing_2025}.
It can be seen that we will have as many linearly
independent detectors as there are separate deterministic measurements.
In the most straightforward case, we may simply use the stabilizer
measurements as detectors.
We immediately recognize that we will have as many linearly
independent detectors as there are separate deterministic measurements.
We generally aim to utilize the maximum number of linearly
independent detectors \cite[Sec.~2.2]{derks_designing_2025}.
@@ -775,8 +784,8 @@ the \emph{detector matrix} $\bm{D} \in \mathbb{F}_2^{D\times M}$
\cite[Def.~2.2]{derks_designing_2025}, with $~D\in \mathbb{N}$
denoting the number of detectors.
Similar to the way a \ac{pcm} associates bits with parity checks, the
detector matrix links measurements and detectors.
Each column corresponds to a measurement, while each rows corresponds
detector matrix links measurement outcomes and detectors.
Each column corresponds to a measurement, while each row corresponds
to a detector.
We should note at this point that the combination of measurements
into detectors has no bearing on the actual construction of the
@@ -786,12 +795,12 @@ affects the decoder.
Note that we can use the detector matrix $\bm{D}$ to describe the set
of possible measurement outcomes under the absence of noise.
The same way we use a \ac{pcm} to describe the code space as
\begin{align*}
Similar to the we use a \ac{pcm} to describe the code space as
\begin{equation*}
\mathcal{C}
= \{ \bm{x} \in \mathbb{F}_2^{n} : \bm{H}\bm{x}^\text{T} = \bm{0} \}
,%
\end{align*}
\end{equation*}
the set of possible measurement outcomes is simply $\text{kern}\{\bm{D}\}$
\cite[Sec.~2.2]{derks_designing_2025}.
@@ -915,7 +924,8 @@ with $\bm{m}^{(0)} = \bm{0}$.
We again turn our attention to the three-qubit repetition code.
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 triggered the last four
measurements.
We now take those measurements and combine them according to
\Cref{eq:measurement_combination}.
We can see this process graphically in
@@ -923,13 +933,13 @@ We can see this process graphically in
To understand why this way of defining the detectors is useful, we
note that the error $E_6$ in
\Cref{fig:rep_code_multiple_rounds_phenomenological} has not only
tripped the measurements in the syndrome extraction round immediately
triggered the measurements in the syndrome extraction round immediately
afterwards, but all subsequent ones as well.
To only see errors in the rounds immediately following them, we
consider our newly defined detectors instead of the measurements,
that effectively compute the difference between the measurements.
Each error can only trip syndrome bits that follow it.
Each error can only trigger syndrome bits that follow it.
This is reflected in the triangular structure of $\bm{\Omega}$ in
\Cref{eq:syndrome_matrix_ex}.
Combining the measurements into detectors according to
@@ -949,7 +959,7 @@ The detector error matrix
\end{array}
\right)
\end{align*}
we obtain this way has a block-diagonal structure.
obtained this way has a block-diagonal structure.
Note that we exploit the fact that each syndrome measurement round is
identical to obtain this structure.
@@ -1030,11 +1040,11 @@ measurements) models \cite[Sec.~2.1]{gidney_fault-tolerant_2021}.
These differ in the way they compute individual error probabilities
from the physical error rate.
In this work we only consider \emph{standard circuit-based depolarizing
noise}, as this is the standard approach in the literature.
We thus set the error probabilities of all error locations in the
circuit-level noise model to the same value, the physical error rate
$p_\text{phys}$.
In this work we consider the \emph{standard circuit-based depolarizing
noise} variant of circuit-level noise, as this is the standard
approach in the literature:
We set the error probabilities of all error locations to the same
value, the physical error rate $p_\text{phys}$.
%%%%%%%%%%%%%%%%
\subsection{Per-Round Logical Error Rate}
@@ -1065,13 +1075,12 @@ The overall probability of error is then
\end{align}
We approximate $p_\text{e,total}$ using a Monte Carlo simulation and
compute the per-round-\ac{ler} using \Cref{eq:per_round_ler}.
This is a common approach taken in the literature
\cite{gong_toward_2024}\cite{wang_fully_2025}.
This is the approach taken in \cite{gong_toward_2024}\cite{wang_fully_2025}.
Another common approach \cite{chen_exponential_2021}%
Another approach \cite{chen_exponential_2021}%
\cite{bausch_learning_2024}\cite{beni_tesseract_2025} is to assume an
exponential decay for the decoder's \emph{logical fidelity}
\cite[Eq.~2]{bausch_learning_2024}
\cite[Eq.~(2)]{bausch_learning_2024}
\begin{align*}
F_\text{total} = (F_\text{round})^{R}
.%
@@ -1079,7 +1088,7 @@ exponential decay for the decoder's \emph{logical fidelity}
The logical fidelity is a measure of the quality of a logical state
\cite[Appendix~E]{postler_demonstration_2024}.
As it is related to the error rate through $F = 1 - 2p$, we obtain
\cite[Eq.~4]{bausch_learning_2024}
\cite[Eq.~(4)]{bausch_learning_2024}
\begin{align}
(1 - 2p_\text{e,total}) &= (1 - 2p_\text{e,round})^{R} \nonumber\\
\implies \hspace{15mm} p_\text{e,round} &= \frac{1}{2}

4
src/thesis/main.tex
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@@ -42,7 +42,7 @@
\Crefname{equation}{}{}
\Crefname{section}{Section}{Sections}
\Crefname{subsection}{Subsection}{Subsections}
\Crefname{subsection}{Section}{Sections}
\Crefname{figure}{Figure}{Figures}
%
@@ -89,7 +89,7 @@
% \thesisHeadOfInstitute{Prof. Dr.-Ing. Peter Rost}
%\thesisHeadOfInstitute{Prof. Dr.-Ing. Peter Rost\\Prof. Dr.-Ing.
% Laurent Schmalen}
\thesisSupervisor{Jonathan Mandelbaum}
\thesisSupervisor{M.Sc. Jonathan Mandelbaum}
\thesisStartDate{01.11.2025}
\thesisEndDate{04.05.2026}
\thesisSignatureDate{04.05.2026}