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Andreas Tsouchlos
2026-02-04 13:56:33 +01:00
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src/midterm_presentation/main.tex
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@@ -1,4 +1,4 @@
\documentclass[overviewatsection, showsubsectionsatfirstoverview]{CELbeamer}
\documentclass[overviewatsection]{CELbeamer}
%
%
@@ -53,7 +53,7 @@
\title{Fault Tolerant Quantum Error Correction}
\subtitle{Master's Thesis Midterm Presentation}
\author[Tsouchlos]{Andreas Tsouchlos}
\date[]{February 5th, 2026}
\date[]{}
\DeclareFieldFormat{note}{}
\DeclareFieldFormat{issn}{}
@@ -72,6 +72,11 @@
\newcommand{\red}[1]{\textcolor{red}{#1}}
\newcommand{\res}{src/midterm_presentation/res}
\newcommand{\X}{\textcolor{kit-blue}{\bm{X}}}
\newcommand{\Z}{\textcolor{kit-orange}{\bm{Z}}}
\newcommand{\Y}{\textcolor{kit-red}{\bm{Y}}}
\newcommand{\I}{\bm{I}}
%
%
% Acronyms
@@ -85,7 +90,7 @@
\DeclareAcronym{css}{
short=CSS,
long=Calderbank Shor Steane
long=Calderbank -- Shor -- Steane
}
\DeclareAcronym{bb}{
@@ -110,12 +115,12 @@
\DeclareAcronym{qldpc}{
short=QLDPC,
long=quantum low density parity check,
long=quantum low - density parity - check,
}
\DeclareAcronym{scldpc}{
short=SC-LDPC,
long=spatially coupled low density parity check
long=spatially - coupled low - density parity - check
}
%
@@ -154,25 +159,20 @@
\begin{itemize}
\item Simulating quantum systems on classical hardware
is exponentially complex \\
$\rightarrow$ Can't we use quantum hardware to simulate
quantum systems? \citereference{feynman_simulating_1982}
\item Some problems that are ``hard'' to solve on classical
computers we can ``easily'' solve on quantum computers
$\rightarrow$ Use quantum hardware to simulate quantum
systems \citereference{feynman_simulating_1982}
\item ``Hard'' to solve problems on classical computers can
be ``easy'' on quantum computers
\citereference{preskill_quantum_2018}
\item Google Quantum AI's quantum computing roadmap
\citereference{google_quantum_ai_quantum_nodate}
\end{itemize}
\vspace*{-5mm}
\vspace*{3mm}
\begin{figure}[H]
\centering
\includegraphics[scale=0.43]{res/google_roadmap.png}
\vspace*{-3mm}
\caption{
Google Quantum AI's quantum computing roadmap
\citereference{google_quantum_ai_quantum_nodate}.
}
\end{figure}
\vspace*{3mm}
@@ -189,7 +189,7 @@
\begin{frame}
\frametitle{The Need for Quantum Error Correction}
\vspace*{-17mm}
\vspace*{-15mm}
% Related interesting stuff
% - Qubits differ from bits in that they can be in superpositions
@@ -210,38 +210,61 @@
% computation
\begin{itemize}
\item Quantum computers represent information through
correlations of qubits, not their values \\
directly \citereference{preskill_quantum_2018}
\item Errors during quantum computation are inevitable
because quantum systems are fragile
\item We want to interact with the quantum state but not disturb it
\item We employ more physical qubits to introduce
redundancy and use the resulting \emph{physical} state to
represent the \emph{logical} state
\citereference{roffe_quantum_2019}
\vspace*{8mm}
\item Typical scales
\begin{itemize}
\item IBM recently introduced a scheme encoding $12$ logical
qubits in $288$ physical ones
\citereference{bravyi_high-threshold_2024}
\item The physical error rate is typically assumed to
be $10^{-3}$ for
simulations (e.g.,
\citereference{bravyi_high-threshold_2024})
\item Decoding has to happen with ultra-low latency to avoid
the backlog problem (about $\SI{1}{us}$ per data
extraction round) \citereference{caune_demonstrating_2024}
% \citereference{terhal_quantum_2015}
\end{itemize}
% \item Quantum computers represent information through
% correlations of qubits, not their values \\
% directly \citereference{preskill_quantum_2018}
\item Quantum systems are inherently fragile
\item Interacting with the quantum state disturbs it
\item Idea: Represent \schlagwort{logical qubits} using more
\schlagwort{physical qubits} \citereference{roffe_quantum_2019}
\vspace*{2mm}
\begin{figure}[H]
\centering
\begin{tikzpicture}
\node[
rectangle,
draw, fill=kit-blue!25,
minimum height=15mm,
]
(enc) {Three-qubit encoder};
\node[left=of enc] (in)
{$\ket{\psi} = \alpha\ket{0} + \beta\ket{1}$};
\node[right=of enc,yshift=6mm] (out)
{$\alpha\overbrace{\ket{000}}^{\ket{0}_\text{L}}
+\; \beta\overbrace{\ket{111}}^{\ket{1}_\text{L}} =
\ket{\psi}_\text{L}$};
\draw[-{Latex}] (in) -- (enc);
\draw[-{Latex}] (enc) -- (enc -| out.west);
\end{tikzpicture}
\end{figure}
\vspace*{5mm}
\visible<2>{
\item Typical scales
\begin{itemize}
\item Recent scheme by IBM encodes $12$ logical
qubits in $288$ physical ones
\citereference{bravyi_high-threshold_2024}
\item Physical error rate typically set to $10^{-3}$
for simulations (e.g.,
\citereference{bravyi_high-threshold_2024})
\item Decode with ultra-low latency to avoid
\schlagwort{backlog problem} (about
$\SI{1}{\micro s}$ per data \\
extraction round)
\citereference{caune_demonstrating_2024}
\end{itemize}
}
\end{itemize}
\vspace*{7mm}
\vspace*{10mm}
\addreferences
% {terhal_quantum_2015}
{preskill_quantum_2018}
{roffe_quantum_2019}
{bravyi_high-threshold_2024}
{caune_demonstrating_2024}
@@ -256,7 +279,7 @@
\begin{frame}
\frametitle{Peculiarities of the Quantum Setting}
\vspace*{-18mm}
\vspace*{-13mm}
% Related interesting stuff
% - No cloning theorem -> Not replication of state, protection
@@ -274,9 +297,10 @@
% much"
\begin{itemize}
\item \Ac{qec} is actually able to protect the actual quantum state
\item Similar to bits and gates, quantum systems are built on
top of qubits and quantum gates
% \item \Ac{qec} is actually able to protect the actual
% quantum state
\item Classical systems built with bits and gates, quantum
systems with qubits and quantum gates
\item We have to consider phase flip errors in addition to
bit flip errors \citereference{roffe_quantum_2019}
\end{itemize}
@@ -289,47 +313,49 @@
\centering
\begin{align*}
\ket{0} &\rightarrow \ket{1} \\
\ket{1} &\rightarrow \ket{0}
\ket{0} &\mapsto \ket{1} \\
\ket{1} &\mapsto \ket{0}
\end{align*}
\caption{Bit flip (X) error}
\caption{Bit flip ($\X$) error}
\end{subfigure}%
\begin{subfigure}{0.32\textwidth}
\centering
\begin{align*}
\ket{0} &\rightarrow \phantom{-}\ket{0} \\
\ket{1} &\rightarrow -\ket{1}
\ket{0} &\mapsto \phantom{-}\ket{0} \\
\ket{1} &\mapsto -\ket{1}
\end{align*}
\caption{Phase flip (Z) error}
\caption{Phase flip ($\Z$) error}
\end{subfigure}%
\begin{subfigure}{0.32\textwidth}
\centering
\begin{align*}
\ket{0} &\rightarrow \phantom{-j}\ket{1} \\
\ket{1} &\rightarrow -j\ket{0}
\ket{0} &\mapsto \phantom{-j}\ket{1} \\
\ket{1} &\mapsto -j\ket{0}
\end{align*}
\caption{Y error: Combination of X and Z}
\caption{$\Y$ error}
\end{subfigure}
\end{figure}
\vspace*{-3mm}
\begin{itemize}
\item Measuring the qubits directly destroys superpositions
and entanglement \\
$\rightarrow$ We generally only work with the syndrome,
which we can measure \citereference{nielsen_quantum_2010}
\item Sometimes superposition permits multiple equivalent
solutions to the decoding problem (\emph{quantum
degeneracy}) \citereference{roffe_decoding_2020}
\visible<2->{
\item Measuring the qubits directly destroys superpositions
and entanglement \\
$\rightarrow$ Use syndrome for decoding
\citereference{nielsen_quantum_2010}
}
\visible<3>{
\item Superposition $\rightarrow$ multiple solutions to the
decoding problem
(\schlagwort{quantum degeneracy})
\citereference{roffe_decoding_2020}}
\end{itemize}
\vspace*{7mm}
\vspace*{12mm}
\addreferences
{nielsen_quantum_2010}
@@ -354,24 +380,25 @@
\begin{itemize}
\item Stabilizer codes \citereference{nielsen_quantum_2010}
\begin{itemize}
\item The code space can implicitly be defined using
\emph{stabilizer generators}
\item We can represent them using parity
check matrices
\item Quantum analog of linear codes
\item Implicitly defined using \schlagwort{stabilizer
generators}
\item Can be represented using parity check matrices
\item Quantum analog of linear block codes
\end{itemize}
\vspace*{10mm}
\item \Ac{css} codes \citereference{nielsen_quantum_2010}
\begin{itemize}
\item Subset of stabilizer codes
\item Can correct X and Z errors independently
\item Described using two separate parity check
matrices $\bm{H}_\text{X}$ and $\bm{H}_\text{Z}$
\item Can be constructed from two binary linear codes
$\mathcal{C}_1 \left[ n, k_1 \right]$ and
$\mathcal{C}_2 \left[ n, k_2 \right]$ with
$\mathcal{C}_2 \subset \mathcal{C}_1$
\end{itemize}
\visible<2->{
\item \Acf{css} codes \citereference{nielsen_quantum_2010}
\begin{itemize}
\item Subset of stabilizer codes
\item Able to correct $\X$ and $\Z$ errors independently
\item Described using two separate parity check
matrices $\bm{H}_X$ and $\bm{H}_Z$
\item Can be constructed from two binary linear codes
$\mathcal{C}_1 \left[ n, k_1 \right]$ and
$\mathcal{C}_2 \left[ n, k_2 \right]$ with
$\mathcal{C}_2 \subset \mathcal{C}_1$
\end{itemize}
}
\end{itemize}
\vspace*{20mm}
@@ -386,21 +413,16 @@
\begin{frame}
\frametitle{Syndrome Extraction Circuits}
\vspace*{-16mm}
\vspace*{-10mm}
\begin{itemize}
\item We entangle the state with \emph{ancilla qubits} to
perform syndrome measurements \citereference{nielsen_quantum_2010}
% \item \red{Do I need to show what the syndrome extraction
% circuitry for Z errors looks like?}
\item Example: The 3-qubit repetition code%
\footnote {
Note that, for simplicity, this chosen example is a
code that is only able to correct X errors (bit flips)
} %
\item Entangle the state $\ket{\psi}$ with
\schlagwort{ancilla qubits} to perform syndrome
measurements \citereference{nielsen_quantum_2010}
\item Example: The 3-qubit repetition code for $\X$ errors
\end{itemize}
\vspace*{-10mm}
\vspace*{-5mm}
\begin{align*}
\bm{H} =
@@ -410,8 +432,6 @@
\end{pmatrix}
\end{align*}
\vspace*{5mm}
\begin{figure}[H]
% \newcommand{\anyerrgate}{\gate[style={fill=red!20}]{\mathcal{E}_\text{XYZ}}}
\newcommand{\preperr}{\gate[style={fill=orange!20}]{\phantom{1}}}
@@ -421,18 +441,17 @@
\centering
% tex-fmt: off
\begin{quantikz}%[row sep=4mm, column sep=4mm]
& \ctrl{3} & & & & & \\
\lstick{$\ket{\psi}$} & & \ctrl{2} & \ctrl{3} & & & \\
& & & & \ctrl{2} & & \\
\lstick{$\ket{0}_{\text{A}_1}$} & \targ{} & \targ{} & & & \meter{} \\
\lstick{$\ket{0}_{\text{A}_2}$} & & & \targ{} & \targ{} & \meter{}
& \ctrl{3} & & & & & \\
\lstick{$\ket{\psi}$} & & \ctrl{2} & \ctrl{3} & & & \\
& & & & \ctrl{2} & & \\
\lstick{$\ket{0}_{\text{A}_1}$} & \targ{} & \targ{} & & & \meter{} & \setwiretype{c} \\
\lstick{$\ket{0}_{\text{A}_2}$} & & & \targ{} & \targ{} & \meter{} & \setwiretype{c}
\end{quantikz}
% tex-fmt: on
% \caption{Circuit-level noise model for the 3-qubit repetition code}
\end{figure}
% \vspace*{5mm}
\vspace*{-2mm}
\vspace*{5mm}
\addreferences
{nielsen_quantum_2010}
@@ -451,19 +470,18 @@
\begin{frame}
\frametitle{Fault Tolerance}
\vspace*{-18mm}
\vspace*{-10mm}
\begin{itemize}
\item The quantum gates we use for syndrome extraction are
\item Quantum gates used for syndrome extraction are
faulty themselves \\
$\rightarrow$ We need \emph{fault-tolerant} \ac{qec}
\item A \ac{qec} procedure is said to be fault tolerant if,
in addition to correcting \emph{input errors}, the spread
of \emph{internal errors} is sufficiently limited
$\rightarrow$ Need for \schlagwort{fault-tolerant} \acf{qec}
\item In addition to correcting \schlagwort{input errors},
limit spread of \schlagwort{internal errors}
\citereference{derks_designing_2025}
\end{itemize}
% \vspace*{3mm}
\vspace*{3mm}
\begin{figure}[H]
\centering
@@ -487,22 +505,20 @@
\node[above] at (internal.north) {\small QEC};
\node[above] at (output.north) {\small Output State};
\end{tikzpicture}
\caption{Overview of the flow of errors in a \ac{qec} system.
Adapted from \citereference{derks_designing_2025}.}
\end{figure}
% \vspace*{3mm}
\vspace*{3mm}
\begin{itemize}
\item We have to modify the syndrome extraction circuitry to
be fault tolerant (e.g., by using specially prepared
multi-qubit states for each ancilla
\citereference{shor_fault-tolerant_1997})
\item We generally perform multiple rounds of syndrome extraction
\visible<2->{
\item Modify syndrome extraction circuitry (e.g., multi-qubit
states for each ancilla
\citereference{shor_fault-tolerant_1997})
\item Multiple rounds of syndrome extraction
}
\end{itemize}
\vspace*{8mm}
\vspace*{15mm}
\addreferences
{shor_fault-tolerant_1997}
@@ -520,43 +536,108 @@
\vspace*{-18mm}
\begin{itemize}
\item Each column of the \emph{measurement syndrome matrix}
$\bm{\Omega}$ corresponds to a measurement pattern an
error produces \citereference{derks_designing_2025}
\item \schlagwort{Measurement syndrome matrix} $\bm{\Omega}$ \\
contains error patterns \citereference{derks_designing_2025}
\item Example: 3-qubit repetition code \\
(Only bit flips on data qubits)
\end{itemize}
\vspace*{-28mm}
\vspace*{-35mm}
\centering
\only<1>{
\begin{minipage}{0.4\textwidth}
\centering
\begin{align*}
\bm{\Omega} =
\left(
\begin{array}{ccc}
1 & 1 & 0 \\
0 & 1 & 1 \\
1 & 1 & 0 \\
0 & 1 & 1 \\
1 & 1 & 0 \\
0 & 1 & 1
\end{array}\right)
\end{align*}
\vspace*{40mm}
\begin{tikzpicture}
\node{$%
\bm{\Omega} =
\left(
\begin{array}{ccc}
1 & 1 & 0 \\
0 & 1 & 1 \\
1 & 1 & 0 \\
0 & 1 & 1 \\
1 & 1 & 0 \\
0 & 1 & 1
\end{array}
\right)$
};
\draw [
line width=1pt,
decorate,
decoration={brace,mirror,amplitude=3mm,raise=5mm}
]
(2.4,1.2) -- (2.5,2.85)
node[midway,right,xshift=10mm]{$\text{SE}_1$};
\draw [
line width=1pt,
decorate,
decoration={brace,mirror,amplitude=3mm,raise=5mm}
]
(2.4,-0.75) -- (2.5,0.9)
node[midway,right,xshift=10mm]{$\text{SE}_2$};
\draw [
line width=1pt,
decorate,
decoration={brace,mirror,amplitude=3mm,raise=5mm}
]
(2.4,-2.7) -- (2.5,-1.1)
node[midway,right,xshift=10mm]{$\text{SE}_3$};
\end{tikzpicture}
\vspace*{-10mm}
\begin{gather*}
\bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\}
\end{gather*}
\end{minipage}%
\begin{minipage}{0.6\textwidth}
\begin{figure}[H]
\newcommand{\preperr}[1]{
\gate[style={fill=orange!20}]{\scriptstyle ##1}
}
\newcommand{\measerr}{\gate[style={fill=blue!20}]{\phantom{1}}}
\centering
% tex-fmt: off
\begin{quantikz}[row sep=4mm, column sep=4mm, wire types={q,q,q,q,q,n,n,n,n}]
\begin{quantikz}[
row sep=4mm, column sep=4mm,
wire types={q,q,q,q,q,n,n,n,n},
execute at end picture={
\draw [
line width=1pt,
decorate,
decoration={brace,amplitude=3mm,raise=5mm}
]
(\tikzcdmatrixname-4-19.north east)
--
(\tikzcdmatrixname-5-19.south east)
node[midway,right,xshift=10mm]{$\text{SE}_1$};
\draw [
line width=1pt,
decorate,
decoration={brace,amplitude=3mm,raise=5mm}
]
(\tikzcdmatrixname-6-19.north east)
--
(\tikzcdmatrixname-7-19.south east)
node[midway,right,xshift=10mm]{$\text{SE}_2$};
\draw [
line width=1pt,
decorate,
decoration={brace,amplitude=3mm,raise=5mm}
]
(\tikzcdmatrixname-8-19.north east)
--
(\tikzcdmatrixname-9-19.south east)
node[midway,right,xshift=10mm]{$\text{SE}_3$};
}
]
% tex-fmt: off
& \preperr{E_0} & \ctrl{3} & & & & & & \ctrl{5} & & & & & & \ctrl{7} & & & & \\
\lstick{$\ket{\psi}$} & \preperr{E_1} & & \ctrl{2} & \ctrl{3} & & & & & \ctrl{4} & \ctrl{5} & & & & & \ctrl{6} & \ctrl{7} & & \\
& \preperr{E_2} & & & & \ctrl{2} & & & & & & \ctrl{4} & & & & & & \ctrl{6} & \\
@@ -566,26 +647,61 @@
& & & & & & \lstick{$\ket{0}_{\text{A}_4}$} & \setwiretype{q} & & & \targ{} & \targ{} & & & & & & & \meter{} \\
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_5}$} & \setwiretype{q} & \targ{} & \targ{} & & & \meter{} \\
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_6}$} & \setwiretype{q} & & & \targ{} & \targ{} & \meter{}
% tex-fmt: on
\end{quantikz}
% tex-fmt: on
\end{figure}
\end{minipage}
}
\only<2>{
\begin{minipage}{0.4\textwidth}
\centering
\begin{align*}
\bm{\Omega} =
\left(
\begin{array}{>{\columncolor{red!20}}ccc}
1 & 1 & 0 \\
0 & 1 & 1 \\
1 & 1 & 0 \\
0 & 1 & 1 \\
1 & 1 & 0 \\
0 & 1 & 1
\end{array}\right)
\end{align*}
\vspace*{40mm}
\begin{tikzpicture}
\node{$%
\bm{\Omega} =
\left(
\begin{array}{>{\columncolor{red!20}}ccc}
1 & 1 & 0 \\
0 & 1 & 1 \\
1 & 1 & 0 \\
0 & 1 & 1 \\
1 & 1 & 0 \\
0 & 1 & 1
\end{array}
\right)$
};
\draw [
line width=1pt,
decorate,
decoration={brace,mirror,amplitude=3mm,raise=5mm}
]
(2.4,1.2) -- (2.5,2.85)
node[midway,right,xshift=10mm]{$\text{SE}_1$};
\draw [
line width=1pt,
decorate,
decoration={brace,mirror,amplitude=3mm,raise=5mm}
]
(2.4,-0.75) -- (2.5,0.9)
node[midway,right,xshift=10mm]{$\text{SE}_2$};
\draw [
line width=1pt,
decorate,
decoration={brace,mirror,amplitude=3mm,raise=5mm}
]
(2.4,-2.7) -- (2.5,-1.1)
node[midway,right,xshift=10mm]{$\text{SE}_3$};
\end{tikzpicture}
\vspace*{-10mm}
\begin{gather*}
\bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\}
\end{gather*}
\end{minipage}%
\begin{minipage}{0.6\textwidth}
\begin{figure}[H]
@@ -615,7 +731,7 @@
\tikzset{
noisy/.style={
starburst,
starburst point height=2.5mm,
starburst point height=2mm,
fill=red!25, draw=red!85!black,
line width=2pt,
inner xsep=-2pt, inner ysep=-2pt
@@ -624,8 +740,40 @@
\centering
% tex-fmt: off
\begin{quantikz}[row sep=4mm, column sep=4mm, wire types={q,q,q,q,q,n,n,n,n}]
\begin{quantikz}[
row sep=4mm, column sep=4mm,
wire types={q,q,q,q,q,n,n,n,n},
execute at end picture={
\draw [
line width=1pt,
decorate,
decoration={brace,amplitude=3mm,raise=5mm}
]
(\tikzcdmatrixname-4-19.north east)
--
(\tikzcdmatrixname-5-19.south east)
node[midway,right,xshift=10mm]{$\text{SE}_1$};
\draw [
line width=1pt,
decorate,
decoration={brace,amplitude=3mm,raise=5mm}
]
(\tikzcdmatrixname-6-19.north east)
--
(\tikzcdmatrixname-7-19.south east)
node[midway,right,xshift=10mm]{$\text{SE}_2$};
\draw [
line width=1pt,
decorate,
decoration={brace,amplitude=3mm,raise=5mm}
]
(\tikzcdmatrixname-8-19.north east)
--
(\tikzcdmatrixname-9-19.south east)
node[midway,right,xshift=10mm]{$\text{SE}_3$};
}
]
% tex-fmt: off
& \noise\redwire{17} & \redctrl{3} & & & & & & \redctrl{5} & & & & & & \redctrl{7} & & & & \\
\lstick{$\ket{\psi}$} & \preperr{E_1} & & \ctrl{2} & \ctrl{3} & & & & & \ctrl{4} & \ctrl{5} & & & & & \ctrl{6} & \ctrl{7} & & \\
& \preperr{E_2} & & & & \ctrl{2} & & & & & & \ctrl{4} & & & & & & \ctrl{6} & \\
@@ -635,13 +783,13 @@
& & & & & & \lstick{$\ket{0}_{\text{A}_4}$} & \setwiretype{q} & & & \targ{} & \targ{} & & & & & & & \meter{} \\
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_5}$} & \setwiretype{q} & \redtarg\redwire{4} & \targ{} & & & \redmeter \\
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_6}$} & \setwiretype{q} & & & \targ{} & \targ{} & \meter{}
% tex-fmt: on
\end{quantikz}
% tex-fmt: on
\end{figure}
\end{minipage}
}
\vspace*{2mm}
\vspace*{8mm}
\addreferences
{derks_designing_2025}
@@ -649,29 +797,29 @@
\end{frame}
\begin{frame}[fragile]
\frametitle{The Measurement Syndromemani Matrix II}
\frametitle{The Measurement Syndrome Matrix II}
\vspace*{-18mm}
\begin{itemize}
\item Each column of the \emph{measurement syndrome matrix}
$\bm{\Omega}$ corresponds to a measurement pattern an
error produces \citereference{derks_designing_2025}
\item
Example: 3-qubit repetition code \\
\item \schlagwort{Measurement syndrome matrix} $\bm{\Omega}$ \\
contains error patterns \citereference{derks_designing_2025}
\item Example: 3-qubit repetition code \\
(Phenomenological noise \citereference{derks_designing_2025})
\end{itemize}
\vspace*{-28mm}
\vspace*{-40mm}
\centering
\only<1>{
\begin{minipage}{0.4\textwidth}
\centering
\vspace*{61mm}
\hspace*{-75mm}
\scalebox{0.85}{
\parbox{.5\linewidth}{%
\begin{align*}
\begin{gather*}
\bm{\Omega} =
\left(
\begin{array}{ccccccccccccccc}
@@ -687,8 +835,11 @@
& 1 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0
& 0 & 1 & 1 & 0 & 1
\end{array}\right)
\end{align*}
\end{array}
\right) \\[10mm]
\hspace*{50mm} %
\bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\}
\end{gather*}
}
}
\end{minipage}%
@@ -749,11 +900,11 @@
\begin{minipage}{0.4\textwidth}
\centering
\newcommand{\pz}{\phantom{0}}
\vspace*{61mm}
\hspace*{-75mm}
\scalebox{0.85}{
\parbox{.5\linewidth}{%
\begin{align*}
\begin{gather*}
\bm{\Omega} =
\left(
\begin{array}{
@@ -773,8 +924,11 @@
& 1 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0
& 0 & 1 & 1 & 0 & 1
\end{array}\right)
\end{align*}
\end{array}
\right) \\[10mm]
\hspace*{50mm} %
\bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\}
\end{gather*}
}
}
\end{minipage}%
@@ -835,11 +989,11 @@
\begin{minipage}{0.4\textwidth}
\centering
\newcommand{\pz}{\phantom{0}}
\vspace*{61mm}
\hspace*{-75mm}
\scalebox{0.85}{
\parbox{.5\linewidth}{%
\begin{align*}
\begin{gather*}
\bm{\Omega} =
\left(
\begin{array}{
@@ -859,8 +1013,11 @@
& 1 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0
& 0 & 1 & 1 & 0 & 1
\end{array}\right)
\end{align*}
\end{array}
\right) \\[10mm]
\hspace*{50mm} %
\bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\}
\end{gather*}
}
}
\end{minipage}%
@@ -918,7 +1075,7 @@
\end{minipage}
}
\vspace*{2mm}
\vspace*{3mm}
\addreferences
{derks_designing_2025}
@@ -934,7 +1091,7 @@
\begin{itemize}
\item A detector is a parity constraint on a set of
measurement outcomes \citereference{derks_designing_2025}
\item Each column of the \emph{detector error matrix} $\bm{H}$
\item Each column of the \schlagwort{detector error matrix} $\bm{H}$
corresponds to a detector pattern an error produces
\item We can mitigate the propagation of errors into
subsequent rounds by XORing the measurements, i.e.,
@@ -1056,6 +1213,9 @@
% - The difference between an n-qubit error and multiple
% simultaneous single-qubit errors is that in the n-qubit case,
% the errors can be correlated (e.g., XX more probable than XI)
% - There is also work on using soft information at the
% measurement outputs (may translate to not-just-X-errors at the
% measurements)
\vspace*{-15mm}
@@ -1068,19 +1228,19 @@
\begin{minipage}{0.60\textwidth}
\begin{itemize}
\item The \emph{depolarizing channel} considers
\item The \schlagwort{depolarizing channel} considers
\citereference{nielsen_quantum_2010}
\begin{itemize}
\item X, Y or Z errors on the data qubits
\end{itemize}
\item \emph{Phenomenological noise} considers
\item \schlagwort{Phenomenological noise} considers
\citereference{derks_designing_2025}
\begin{itemize}
\item X errors on data qubits before each \\
measurement round
\item X errors on measurement outcomes
\end{itemize}
\item \emph{Circuit-level noise} considers
\item \schlagwort{Circuit-level noise} considers
\citereference{derks_designing_2025}
\begin{itemize}
\item \colorbox{orange!20}{X, Y or Z errors after
@@ -1523,6 +1683,20 @@
\stopreferences
\end{frame}
\begin{frame}
\frametitle{Guided Decimation Guessing Decoding}
\begin{itemize}
\item \red{Explain paper}
\end{itemize}
\vspace*{25mm}
\addreferences
{gong_toward_2024}
\stopreferences
\end{frame}
% TODO: Is this really necessary?
% \begin{frame}
% \frametitle{The Quantum Error Correction Landscape}