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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} \title{Fault Tolerant Quantum Error Correction}
\subtitle{Master's Thesis Midterm Presentation} \subtitle{Master's Thesis Midterm Presentation}
\author[Tsouchlos]{Andreas Tsouchlos} \author[Tsouchlos]{Andreas Tsouchlos}
\date[]{February 5th, 2026} \date[]{}
\DeclareFieldFormat{note}{} \DeclareFieldFormat{note}{}
\DeclareFieldFormat{issn}{} \DeclareFieldFormat{issn}{}
@@ -72,6 +72,11 @@
\newcommand{\red}[1]{\textcolor{red}{#1}} \newcommand{\red}[1]{\textcolor{red}{#1}}
\newcommand{\res}{src/midterm_presentation/res} \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 % Acronyms
@@ -85,7 +90,7 @@
\DeclareAcronym{css}{ \DeclareAcronym{css}{
short=CSS, short=CSS,
long=Calderbank Shor Steane long=Calderbank -- Shor -- Steane
} }
\DeclareAcronym{bb}{ \DeclareAcronym{bb}{
@@ -110,12 +115,12 @@
\DeclareAcronym{qldpc}{ \DeclareAcronym{qldpc}{
short=QLDPC, short=QLDPC,
long=quantum low density parity check, long=quantum low - density parity - check,
} }
\DeclareAcronym{scldpc}{ \DeclareAcronym{scldpc}{
short=SC-LDPC, short=SC-LDPC,
long=spatially coupled low density parity check long=spatially - coupled low - density parity - check
} }
% %
@@ -154,25 +159,20 @@
\begin{itemize} \begin{itemize}
\item Simulating quantum systems on classical hardware \item Simulating quantum systems on classical hardware
is exponentially complex \\ is exponentially complex \\
$\rightarrow$ Can't we use quantum hardware to simulate $\rightarrow$ Use quantum hardware to simulate quantum
quantum systems? \citereference{feynman_simulating_1982} systems \citereference{feynman_simulating_1982}
\item Some problems that are ``hard'' to solve on classical \item ``Hard'' to solve problems on classical computers can
computers we can ``easily'' solve on quantum computers be ``easy'' on quantum computers
\citereference{preskill_quantum_2018} \citereference{preskill_quantum_2018}
\item Google Quantum AI's quantum computing roadmap
\citereference{google_quantum_ai_quantum_nodate}
\end{itemize} \end{itemize}
\vspace*{-5mm} \vspace*{3mm}
\begin{figure}[H] \begin{figure}[H]
\centering \centering
\includegraphics[scale=0.43]{res/google_roadmap.png} \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} \end{figure}
\vspace*{3mm} \vspace*{3mm}
@@ -189,7 +189,7 @@
\begin{frame} \begin{frame}
\frametitle{The Need for Quantum Error Correction} \frametitle{The Need for Quantum Error Correction}
\vspace*{-17mm} \vspace*{-15mm}
% Related interesting stuff % Related interesting stuff
% - Qubits differ from bits in that they can be in superpositions % - Qubits differ from bits in that they can be in superpositions
@@ -210,38 +210,61 @@
% computation % computation
\begin{itemize} \begin{itemize}
\item Quantum computers represent information through % \item Quantum computers represent information through
correlations of qubits, not their values \\ % correlations of qubits, not their values \\
directly \citereference{preskill_quantum_2018} % directly \citereference{preskill_quantum_2018}
\item Errors during quantum computation are inevitable \item Quantum systems are inherently fragile
because quantum systems are fragile \item Interacting with the quantum state disturbs it
\item We want to interact with the quantum state but not disturb it \item Idea: Represent \schlagwort{logical qubits} using more
\item We employ more physical qubits to introduce \schlagwort{physical qubits} \citereference{roffe_quantum_2019}
redundancy and use the resulting \emph{physical} state to
represent the \emph{logical} state \vspace*{2mm}
\citereference{roffe_quantum_2019}
\vspace*{8mm} \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 \item Typical scales
\begin{itemize} \begin{itemize}
\item IBM recently introduced a scheme encoding $12$ logical \item Recent scheme by IBM encodes $12$ logical
qubits in $288$ physical ones qubits in $288$ physical ones
\citereference{bravyi_high-threshold_2024} \citereference{bravyi_high-threshold_2024}
\item The physical error rate is typically assumed to \item Physical error rate typically set to $10^{-3}$
be $10^{-3}$ for for simulations (e.g.,
simulations (e.g.,
\citereference{bravyi_high-threshold_2024}) \citereference{bravyi_high-threshold_2024})
\item Decoding has to happen with ultra-low latency to avoid \item Decode with ultra-low latency to avoid
the backlog problem (about $\SI{1}{us}$ per data \schlagwort{backlog problem} (about
extraction round) \citereference{caune_demonstrating_2024} $\SI{1}{\micro s}$ per data \\
% \citereference{terhal_quantum_2015} extraction round)
\citereference{caune_demonstrating_2024}
\end{itemize} \end{itemize}
}
\end{itemize} \end{itemize}
\vspace*{7mm} \vspace*{10mm}
\addreferences \addreferences
% {terhal_quantum_2015}
{preskill_quantum_2018}
{roffe_quantum_2019} {roffe_quantum_2019}
{bravyi_high-threshold_2024} {bravyi_high-threshold_2024}
{caune_demonstrating_2024} {caune_demonstrating_2024}
@@ -256,7 +279,7 @@
\begin{frame} \begin{frame}
\frametitle{Peculiarities of the Quantum Setting} \frametitle{Peculiarities of the Quantum Setting}
\vspace*{-18mm} \vspace*{-13mm}
% Related interesting stuff % Related interesting stuff
% - No cloning theorem -> Not replication of state, protection % - No cloning theorem -> Not replication of state, protection
@@ -274,9 +297,10 @@
% much" % much"
\begin{itemize} \begin{itemize}
\item \Ac{qec} is actually able to protect the actual quantum state % \item \Ac{qec} is actually able to protect the actual
\item Similar to bits and gates, quantum systems are built on % quantum state
top of qubits and quantum gates \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 \item We have to consider phase flip errors in addition to
bit flip errors \citereference{roffe_quantum_2019} bit flip errors \citereference{roffe_quantum_2019}
\end{itemize} \end{itemize}
@@ -289,47 +313,49 @@
\centering \centering
\begin{align*} \begin{align*}
\ket{0} &\rightarrow \ket{1} \\ \ket{0} &\mapsto \ket{1} \\
\ket{1} &\rightarrow \ket{0} \ket{1} &\mapsto \ket{0}
\end{align*} \end{align*}
\caption{Bit flip (X) error} \caption{Bit flip ($\X$) error}
\end{subfigure}% \end{subfigure}%
\begin{subfigure}{0.32\textwidth} \begin{subfigure}{0.32\textwidth}
\centering \centering
\begin{align*} \begin{align*}
\ket{0} &\rightarrow \phantom{-}\ket{0} \\ \ket{0} &\mapsto \phantom{-}\ket{0} \\
\ket{1} &\rightarrow -\ket{1} \ket{1} &\mapsto -\ket{1}
\end{align*} \end{align*}
\caption{Phase flip (Z) error} \caption{Phase flip ($\Z$) error}
\end{subfigure}% \end{subfigure}%
\begin{subfigure}{0.32\textwidth} \begin{subfigure}{0.32\textwidth}
\centering \centering
\begin{align*} \begin{align*}
\ket{0} &\rightarrow \phantom{-j}\ket{1} \\ \ket{0} &\mapsto \phantom{-j}\ket{1} \\
\ket{1} &\rightarrow -j\ket{0} \ket{1} &\mapsto -j\ket{0}
\end{align*} \end{align*}
\caption{Y error: Combination of X and Z} \caption{$\Y$ error}
\end{subfigure} \end{subfigure}
\end{figure} \end{figure}
\vspace*{-3mm}
\begin{itemize} \begin{itemize}
\visible<2->{
\item Measuring the qubits directly destroys superpositions \item Measuring the qubits directly destroys superpositions
and entanglement \\ and entanglement \\
$\rightarrow$ We generally only work with the syndrome, $\rightarrow$ Use syndrome for decoding
which we can measure \citereference{nielsen_quantum_2010} \citereference{nielsen_quantum_2010}
\item Sometimes superposition permits multiple equivalent }
solutions to the decoding problem (\emph{quantum \visible<3>{
degeneracy}) \citereference{roffe_decoding_2020} \item Superposition $\rightarrow$ multiple solutions to the
decoding problem
(\schlagwort{quantum degeneracy})
\citereference{roffe_decoding_2020}}
\end{itemize} \end{itemize}
\vspace*{7mm} \vspace*{12mm}
\addreferences \addreferences
{nielsen_quantum_2010} {nielsen_quantum_2010}
@@ -354,24 +380,25 @@
\begin{itemize} \begin{itemize}
\item Stabilizer codes \citereference{nielsen_quantum_2010} \item Stabilizer codes \citereference{nielsen_quantum_2010}
\begin{itemize} \begin{itemize}
\item The code space can implicitly be defined using \item Implicitly defined using \schlagwort{stabilizer
\emph{stabilizer generators} generators}
\item We can represent them using parity \item Can be represented using parity check matrices
check matrices \item Quantum analog of linear block codes
\item Quantum analog of linear codes
\end{itemize} \end{itemize}
\vspace*{10mm} \vspace*{10mm}
\item \Ac{css} codes \citereference{nielsen_quantum_2010} \visible<2->{
\item \Acf{css} codes \citereference{nielsen_quantum_2010}
\begin{itemize} \begin{itemize}
\item Subset of stabilizer codes \item Subset of stabilizer codes
\item Can correct X and Z errors independently \item Able to correct $\X$ and $\Z$ errors independently
\item Described using two separate parity check \item Described using two separate parity check
matrices $\bm{H}_\text{X}$ and $\bm{H}_\text{Z}$ matrices $\bm{H}_X$ and $\bm{H}_Z$
\item Can be constructed from two binary linear codes \item Can be constructed from two binary linear codes
$\mathcal{C}_1 \left[ n, k_1 \right]$ and $\mathcal{C}_1 \left[ n, k_1 \right]$ and
$\mathcal{C}_2 \left[ n, k_2 \right]$ with $\mathcal{C}_2 \left[ n, k_2 \right]$ with
$\mathcal{C}_2 \subset \mathcal{C}_1$ $\mathcal{C}_2 \subset \mathcal{C}_1$
\end{itemize} \end{itemize}
}
\end{itemize} \end{itemize}
\vspace*{20mm} \vspace*{20mm}
@@ -386,21 +413,16 @@
\begin{frame} \begin{frame}
\frametitle{Syndrome Extraction Circuits} \frametitle{Syndrome Extraction Circuits}
\vspace*{-16mm} \vspace*{-10mm}
\begin{itemize} \begin{itemize}
\item We entangle the state with \emph{ancilla qubits} to \item Entangle the state $\ket{\psi}$ with
perform syndrome measurements \citereference{nielsen_quantum_2010} \schlagwort{ancilla qubits} to perform syndrome
% \item \red{Do I need to show what the syndrome extraction measurements \citereference{nielsen_quantum_2010}
% circuitry for Z errors looks like?} \item Example: The 3-qubit repetition code for $\X$ errors
\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)
} %
\end{itemize} \end{itemize}
\vspace*{-10mm} \vspace*{-5mm}
\begin{align*} \begin{align*}
\bm{H} = \bm{H} =
@@ -410,8 +432,6 @@
\end{pmatrix} \end{pmatrix}
\end{align*} \end{align*}
\vspace*{5mm}
\begin{figure}[H] \begin{figure}[H]
% \newcommand{\anyerrgate}{\gate[style={fill=red!20}]{\mathcal{E}_\text{XYZ}}} % \newcommand{\anyerrgate}{\gate[style={fill=red!20}]{\mathcal{E}_\text{XYZ}}}
\newcommand{\preperr}{\gate[style={fill=orange!20}]{\phantom{1}}} \newcommand{\preperr}{\gate[style={fill=orange!20}]{\phantom{1}}}
@@ -424,15 +444,14 @@
& \ctrl{3} & & & & & \\ & \ctrl{3} & & & & & \\
\lstick{$\ket{\psi}$} & & \ctrl{2} & \ctrl{3} & & & \\ \lstick{$\ket{\psi}$} & & \ctrl{2} & \ctrl{3} & & & \\
& & & & \ctrl{2} & & \\ & & & & \ctrl{2} & & \\
\lstick{$\ket{0}_{\text{A}_1}$} & \targ{} & \targ{} & & & \meter{} \\ \lstick{$\ket{0}_{\text{A}_1}$} & \targ{} & \targ{} & & & \meter{} & \setwiretype{c} \\
\lstick{$\ket{0}_{\text{A}_2}$} & & & \targ{} & \targ{} & \meter{} \lstick{$\ket{0}_{\text{A}_2}$} & & & \targ{} & \targ{} & \meter{} & \setwiretype{c}
\end{quantikz} \end{quantikz}
% tex-fmt: on % tex-fmt: on
% \caption{Circuit-level noise model for the 3-qubit repetition code} % \caption{Circuit-level noise model for the 3-qubit repetition code}
\end{figure} \end{figure}
% \vspace*{5mm} \vspace*{5mm}
\vspace*{-2mm}
\addreferences \addreferences
{nielsen_quantum_2010} {nielsen_quantum_2010}
@@ -451,19 +470,18 @@
\begin{frame} \begin{frame}
\frametitle{Fault Tolerance} \frametitle{Fault Tolerance}
\vspace*{-18mm} \vspace*{-10mm}
\begin{itemize} \begin{itemize}
\item The quantum gates we use for syndrome extraction are \item Quantum gates used for syndrome extraction are
faulty themselves \\ faulty themselves \\
$\rightarrow$ We need \emph{fault-tolerant} \ac{qec} $\rightarrow$ Need for \schlagwort{fault-tolerant} \acf{qec}
\item A \ac{qec} procedure is said to be fault tolerant if, \item In addition to correcting \schlagwort{input errors},
in addition to correcting \emph{input errors}, the spread limit spread of \schlagwort{internal errors}
of \emph{internal errors} is sufficiently limited
\citereference{derks_designing_2025} \citereference{derks_designing_2025}
\end{itemize} \end{itemize}
% \vspace*{3mm} \vspace*{3mm}
\begin{figure}[H] \begin{figure}[H]
\centering \centering
@@ -487,22 +505,20 @@
\node[above] at (internal.north) {\small QEC}; \node[above] at (internal.north) {\small QEC};
\node[above] at (output.north) {\small Output State}; \node[above] at (output.north) {\small Output State};
\end{tikzpicture} \end{tikzpicture}
\caption{Overview of the flow of errors in a \ac{qec} system.
Adapted from \citereference{derks_designing_2025}.}
\end{figure} \end{figure}
% \vspace*{3mm} \vspace*{3mm}
\begin{itemize} \begin{itemize}
\item We have to modify the syndrome extraction circuitry to \visible<2->{
be fault tolerant (e.g., by using specially prepared \item Modify syndrome extraction circuitry (e.g., multi-qubit
multi-qubit states for each ancilla states for each ancilla
\citereference{shor_fault-tolerant_1997}) \citereference{shor_fault-tolerant_1997})
\item We generally perform multiple rounds of syndrome extraction \item Multiple rounds of syndrome extraction
}
\end{itemize} \end{itemize}
\vspace*{8mm} \vspace*{15mm}
\addreferences \addreferences
{shor_fault-tolerant_1997} {shor_fault-tolerant_1997}
@@ -520,20 +536,22 @@
\vspace*{-18mm} \vspace*{-18mm}
\begin{itemize} \begin{itemize}
\item Each column of the \emph{measurement syndrome matrix} \item \schlagwort{Measurement syndrome matrix} $\bm{\Omega}$ \\
$\bm{\Omega}$ corresponds to a measurement pattern an contains error patterns \citereference{derks_designing_2025}
error produces \citereference{derks_designing_2025}
\item Example: 3-qubit repetition code \\ \item Example: 3-qubit repetition code \\
(Only bit flips on data qubits) (Only bit flips on data qubits)
\end{itemize} \end{itemize}
\vspace*{-28mm} \vspace*{-35mm}
\centering \centering
\only<1>{ \only<1>{
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
\centering \centering
\begin{align*}
\vspace*{40mm}
\begin{tikzpicture}
\node{$%
\bm{\Omega} = \bm{\Omega} =
\left( \left(
\begin{array}{ccc} \begin{array}{ccc}
@@ -543,20 +561,83 @@
0 & 1 & 1 \\ 0 & 1 & 1 \\
1 & 1 & 0 \\ 1 & 1 & 0 \\
0 & 1 & 1 0 & 1 & 1
\end{array}\right) \end{array}
\end{align*} \right)$
};
\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}% \end{minipage}%
\begin{minipage}{0.6\textwidth} \begin{minipage}{0.6\textwidth}
\begin{figure}[H] \begin{figure}[H]
\newcommand{\preperr}[1]{ \newcommand{\preperr}[1]{
\gate[style={fill=orange!20}]{\scriptstyle ##1} \gate[style={fill=orange!20}]{\scriptstyle ##1}
} }
\newcommand{\measerr}{\gate[style={fill=blue!20}]{\phantom{1}}}
\centering \centering
\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 % tex-fmt: off
\begin{quantikz}[row sep=4mm, column sep=4mm, wire types={q,q,q,q,q,n,n,n,n}]
& \preperr{E_0} & \ctrl{3} & & & & & & \ctrl{5} & & & & & & \ctrl{7} & & & & \\ & \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} & & \\ \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} & \\ & \preperr{E_2} & & & & \ctrl{2} & & & & & & \ctrl{4} & & & & & & \ctrl{6} & \\
@@ -566,15 +647,18 @@
& & & & & & \lstick{$\ket{0}_{\text{A}_4}$} & \setwiretype{q} & & & \targ{} & \targ{} & & & & & & & \meter{} \\ & & & & & & \lstick{$\ket{0}_{\text{A}_4}$} & \setwiretype{q} & & & \targ{} & \targ{} & & & & & & & \meter{} \\
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_5}$} & \setwiretype{q} & \targ{} & \targ{} & & & \meter{} \\ & & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_5}$} & \setwiretype{q} & \targ{} & \targ{} & & & \meter{} \\
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_6}$} & \setwiretype{q} & & & \targ{} & \targ{} & \meter{} & & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_6}$} & \setwiretype{q} & & & \targ{} & \targ{} & \meter{}
\end{quantikz}
% tex-fmt: on % tex-fmt: on
\end{quantikz}
\end{figure} \end{figure}
\end{minipage} \end{minipage}
} }
\only<2>{ \only<2>{
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
\centering \centering
\begin{align*}
\vspace*{40mm}
\begin{tikzpicture}
\node{$%
\bm{\Omega} = \bm{\Omega} =
\left( \left(
\begin{array}{>{\columncolor{red!20}}ccc} \begin{array}{>{\columncolor{red!20}}ccc}
@@ -584,8 +668,40 @@
0 & 1 & 1 \\ 0 & 1 & 1 \\
1 & 1 & 0 \\ 1 & 1 & 0 \\
0 & 1 & 1 0 & 1 & 1
\end{array}\right) \end{array}
\end{align*} \right)$
};
\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}% \end{minipage}%
\begin{minipage}{0.6\textwidth} \begin{minipage}{0.6\textwidth}
\begin{figure}[H] \begin{figure}[H]
@@ -615,7 +731,7 @@
\tikzset{ \tikzset{
noisy/.style={ noisy/.style={
starburst, starburst,
starburst point height=2.5mm, starburst point height=2mm,
fill=red!25, draw=red!85!black, fill=red!25, draw=red!85!black,
line width=2pt, line width=2pt,
inner xsep=-2pt, inner ysep=-2pt inner xsep=-2pt, inner ysep=-2pt
@@ -624,8 +740,40 @@
\centering \centering
\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 % tex-fmt: off
\begin{quantikz}[row sep=4mm, column sep=4mm, wire types={q,q,q,q,q,n,n,n,n}]
& \noise\redwire{17} & \redctrl{3} & & & & & & \redctrl{5} & & & & & & \redctrl{7} & & & & \\ & \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} & & \\ \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} & \\ & \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{$\ket{0}_{\text{A}_4}$} & \setwiretype{q} & & & \targ{} & \targ{} & & & & & & & \meter{} \\
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_5}$} & \setwiretype{q} & \redtarg\redwire{4} & \targ{} & & & \redmeter \\ & & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_5}$} & \setwiretype{q} & \redtarg\redwire{4} & \targ{} & & & \redmeter \\
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_6}$} & \setwiretype{q} & & & \targ{} & \targ{} & \meter{} & & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_6}$} & \setwiretype{q} & & & \targ{} & \targ{} & \meter{}
\end{quantikz}
% tex-fmt: on % tex-fmt: on
\end{quantikz}
\end{figure} \end{figure}
\end{minipage} \end{minipage}
} }
\vspace*{2mm} \vspace*{8mm}
\addreferences \addreferences
{derks_designing_2025} {derks_designing_2025}
@@ -649,29 +797,29 @@
\end{frame} \end{frame}
\begin{frame}[fragile] \begin{frame}[fragile]
\frametitle{The Measurement Syndromemani Matrix II} \frametitle{The Measurement Syndrome Matrix II}
\vspace*{-18mm} \vspace*{-18mm}
\begin{itemize} \begin{itemize}
\item Each column of the \emph{measurement syndrome matrix} \item \schlagwort{Measurement syndrome matrix} $\bm{\Omega}$ \\
$\bm{\Omega}$ corresponds to a measurement pattern an contains error patterns \citereference{derks_designing_2025}
error produces \citereference{derks_designing_2025} \item Example: 3-qubit repetition code \\
\item
Example: 3-qubit repetition code \\
(Phenomenological noise \citereference{derks_designing_2025}) (Phenomenological noise \citereference{derks_designing_2025})
\end{itemize} \end{itemize}
\vspace*{-28mm} \vspace*{-40mm}
\centering \centering
\only<1>{ \only<1>{
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
\centering \centering
\vspace*{61mm}
\hspace*{-75mm}
\scalebox{0.85}{ \scalebox{0.85}{
\parbox{.5\linewidth}{% \parbox{.5\linewidth}{%
\begin{align*} \begin{gather*}
\bm{\Omega} = \bm{\Omega} =
\left( \left(
\begin{array}{ccccccccccccccc} \begin{array}{ccccccccccccccc}
@@ -687,8 +835,11 @@
& 1 & 1 & 0 & 1 & 0 \\ & 1 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0
& 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 1
\end{array}\right) \end{array}
\end{align*} \right) \\[10mm]
\hspace*{50mm} %
\bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\}
\end{gather*}
} }
} }
\end{minipage}% \end{minipage}%
@@ -749,11 +900,11 @@
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
\centering \centering
\newcommand{\pz}{\phantom{0}} \vspace*{61mm}
\hspace*{-75mm}
\scalebox{0.85}{ \scalebox{0.85}{
\parbox{.5\linewidth}{% \parbox{.5\linewidth}{%
\begin{align*} \begin{gather*}
\bm{\Omega} = \bm{\Omega} =
\left( \left(
\begin{array}{ \begin{array}{
@@ -773,8 +924,11 @@
& 1 & 1 & 0 & 1 & 0 \\ & 1 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0
& 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 1
\end{array}\right) \end{array}
\end{align*} \right) \\[10mm]
\hspace*{50mm} %
\bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\}
\end{gather*}
} }
} }
\end{minipage}% \end{minipage}%
@@ -835,11 +989,11 @@
\begin{minipage}{0.4\textwidth} \begin{minipage}{0.4\textwidth}
\centering \centering
\newcommand{\pz}{\phantom{0}} \vspace*{61mm}
\hspace*{-75mm}
\scalebox{0.85}{ \scalebox{0.85}{
\parbox{.5\linewidth}{% \parbox{.5\linewidth}{%
\begin{align*} \begin{gather*}
\bm{\Omega} = \bm{\Omega} =
\left( \left(
\begin{array}{ \begin{array}{
@@ -859,8 +1013,11 @@
& 1 & 1 & 0 & 1 & 0 \\ & 1 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0
& 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 1
\end{array}\right) \end{array}
\end{align*} \right) \\[10mm]
\hspace*{50mm} %
\bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\}
\end{gather*}
} }
} }
\end{minipage}% \end{minipage}%
@@ -918,7 +1075,7 @@
\end{minipage} \end{minipage}
} }
\vspace*{2mm} \vspace*{3mm}
\addreferences \addreferences
{derks_designing_2025} {derks_designing_2025}
@@ -934,7 +1091,7 @@
\begin{itemize} \begin{itemize}
\item A detector is a parity constraint on a set of \item A detector is a parity constraint on a set of
measurement outcomes \citereference{derks_designing_2025} 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 corresponds to a detector pattern an error produces
\item We can mitigate the propagation of errors into \item We can mitigate the propagation of errors into
subsequent rounds by XORing the measurements, i.e., subsequent rounds by XORing the measurements, i.e.,
@@ -1056,6 +1213,9 @@
% - The difference between an n-qubit error and multiple % - The difference between an n-qubit error and multiple
% simultaneous single-qubit errors is that in the n-qubit case, % simultaneous single-qubit errors is that in the n-qubit case,
% the errors can be correlated (e.g., XX more probable than XI) % 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} \vspace*{-15mm}
@@ -1068,19 +1228,19 @@
\begin{minipage}{0.60\textwidth} \begin{minipage}{0.60\textwidth}
\begin{itemize} \begin{itemize}
\item The \emph{depolarizing channel} considers \item The \schlagwort{depolarizing channel} considers
\citereference{nielsen_quantum_2010} \citereference{nielsen_quantum_2010}
\begin{itemize} \begin{itemize}
\item X, Y or Z errors on the data qubits \item X, Y or Z errors on the data qubits
\end{itemize} \end{itemize}
\item \emph{Phenomenological noise} considers \item \schlagwort{Phenomenological noise} considers
\citereference{derks_designing_2025} \citereference{derks_designing_2025}
\begin{itemize} \begin{itemize}
\item X errors on data qubits before each \\ \item X errors on data qubits before each \\
measurement round measurement round
\item X errors on measurement outcomes \item X errors on measurement outcomes
\end{itemize} \end{itemize}
\item \emph{Circuit-level noise} considers \item \schlagwort{Circuit-level noise} considers
\citereference{derks_designing_2025} \citereference{derks_designing_2025}
\begin{itemize} \begin{itemize}
\item \colorbox{orange!20}{X, Y or Z errors after \item \colorbox{orange!20}{X, Y or Z errors after
@@ -1523,6 +1683,20 @@
\stopreferences \stopreferences
\end{frame} \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? % TODO: Is this really necessary?
% \begin{frame} % \begin{frame}
% \frametitle{The Quantum Error Correction Landscape} % \frametitle{The Quantum Error Correction Landscape}