an.tsouchlos/ma-thesis
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases 1 Wiki Activity

Finish first version of intro

Browse Source
This commit is contained in:
Andreas Tsouchlos 2026-01-29 16:59:05 +01:00
parent cea96986eb
commit 95631f90cf
1 changed files with 248 additions and 56 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

304
src/midterm_presentation/main.tex
View File

@ -47,6 +47,7 @@
\usepackage{amssymb}
\usepackage{acro}
\usepackage{braket}
\usepackage{qcircuit}
\title{Fault Tolerant Quantum Error Correction}
\subtitle{Master's Thesis Midterm Presentation}
@ -67,6 +68,7 @@
%
%
\newcommand{\red}[1]{\textcolor{red}{#1}}
\newcommand{\res}{src/midterm_presentation/res}
%
@ -80,6 +82,11 @@
long=quantum error correction
}
\DeclareAcronym{css}{
short=CSS,
long=Calderbank Shor Steane
}
%
%
% Document body
@ -111,7 +118,7 @@
% [How to factor 2048 bit RSA integers with less than a million
% noisy qubits]
\vspace*{-19mm}
\vspace*{-15mm}
\begin{itemize}
\item Simulating quantum systems on classical hardware
@ -121,7 +128,6 @@
\item Some problems that are ``hard'' to solve on classical
computers we can ``easily'' solve on quantum computers
\citereference{preskill_quantum_2018}
\item We are still in the early days of quantum computing
\end{itemize}
\vspace*{-5mm}
@ -138,6 +144,8 @@
}
\end{figure}
\vspace*{3mm}
\addreferences
{feynman_simulating_1982}
{preskill_quantum_2018}
@ -150,7 +158,7 @@
\begin{frame}
\frametitle{The Need for Quantum Error Correction}
\vspace*{-10mm}
\vspace*{-17mm}
% Related interesting stuff
% - Qubits differ from bits in that they can be in superpositions
@ -163,6 +171,9 @@
% 2048 bit RSA integer
% [How to factor 2048 bit RSA integers with less than a million
% noisy qubits]
% - The "physical error rate" is really the value all error rates
% in the system are set to for circuit level noise simulations
% [High-threshold universal quantum computation on the surface code]
% - The backlog problem is the fact that an increasing backlog of
% syndrome data will lead to an exponential slowdown during the
% computation
@ -172,31 +183,37 @@
correlations of qubits, not their values \\
directly \citereference{preskill_quantum_2018}
\item We want to not disturb the quantum state but need to
interact with the system \\
$\rightarrow$ Protect the state with \ac{qec}
interact with the system $\rightarrow$ Protect the state
with \ac{qec}
\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}
\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}
\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}
\end{itemize}
\vspace*{12mm}
\vspace*{7mm}
\addreferences
% {terhal_quantum_2015}
{caune_demonstrating_2024}
{preskill_quantum_2018}
{roffe_quantum_2019}
{bravyi_high-threshold_2024}
{caune_demonstrating_2024}
\stopreferences
\end{frame}
@ -208,6 +225,8 @@
\begin{frame}
\frametitle{Peculiarities of the Quantum Setting}
\vspace*{-5mm}
% Related interesting stuff
% - No cloning theorem -> Not replication of state, protection
% through further entanglement
@ -215,53 +234,164 @@
% to correct infinitely many different types of errors. Luckily,
% it turns out that in actual fact we only really need to correct
% two [Gottesman's Thesis]
% - Mention that kets are just vectors, used here to represent the state
% - There are actually infinitely many different errors that can
% happen, but we can digitize them and only need to consider X and Z
% - Not only do we only care about the coset, we specifically
% don't want to know more than the syndrome can tell us because
% that would mean that "we collapse the quantum mechanical state too
% much"
\begin{itemize}
\item Measuring the system collapses the quantum state
$\rightarrow$ Loss of benefit of quantum system \\
$\rightarrow$ For BP, we work with the syndrome and not
the variable nodes \textcolor{red}{This can't be here,
it's before introducing how QEC works}
\item X and Z errors
\item With QEC we are able to restore the quantum state, not
"just the bits"
\item We don't care about the specific error, only the coset
its in $\rightarrow$ We only really care about the syndrome
\item As mentioned earlier, \ac{qec} is actually able to
protect the quantum state with all its correlations
\item We have to consider phase flip errors in addition to
bit flip errors \citereference{roffe_quantum_2019}
\vspace*{-10mm}
\begin{figure}[H]
\centering
\begin{subfigure}{0.5\textwidth}
\centering
\begin{align*}
\ket{0} &\rightarrow \ket{1} \\
\ket{1} &\rightarrow \ket{0}
\end{align*}
\caption{Bit flip (X) error}
\end{subfigure}%
\begin{subfigure}{0.5\textwidth}
\centering
\begin{align*}
\ket{0} &\rightarrow \phantom{-}\ket{0} \\
\ket{1} &\rightarrow -\ket{1}
\end{align*}
\caption{Phase flip (Z) error}
\end{subfigure}
\end{figure}
\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 We don't care about restoring the specific codeword,
only finding the coset it's in
\end{itemize}
\vspace*{15mm}
\addreferences
{nielsen_quantum_2010}
{roffe_quantum_2019}
\stopreferences
\end{frame}
\begin{frame}
\frametitle{Fundamentals of Quantum Error Correction}
\frametitle{Stabilizer and Calderbank Shor Steane Codes}
\vspace*{-5mm}
% Related interesting stuff
% - Using stabilizers to describe quantum codes is a bit like
% using parity check equations to describe classical codes
% -> stabilizer codes are the quantum analog of binary linear codes
% - For CSS codes, "the parity checks for the X errors and the
% parity checks for the Z errors can be represented independently
% of one another"
\begin{itemize}
\item Stabilizer codes: the quantum equivalent of binary linear codes
\item CSS codes: separate corection of X and Z errors
$\rightarrow$ simpler circuitry
\item Construction of CSS codes from binary linear codes
\textcolor{red}{Is this really necessary?}
\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
\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}
\vspace*{10mm}
\item \red{Do I need to go more in depth for either
stabilizer codes or CSS codes?}
\end{itemize}
\vspace*{10mm}
\addreferences
{nielsen_quantum_2010}
\stopreferences
\end{frame}
% TODO: Is this really necessary?
% TODO: Do I need to show what the syndrome extraction circuitry for
% Z errors looks like?
\begin{frame}
\frametitle{The Quantum Error Correction Landscape}
\frametitle{Syndrome Extraction Circuits}
\vspace*{-16mm}
\begin{itemize}
\item Give basic overview of most promising avenues of
research (as in ECCentric paper)
\item We entangle the state with \emph{ancilla qubits} to
perform syndrome measurements \citereference{nielsen_quantum_2010}
\item Example: The 3-qubit repetition code%
\footnote {
Note that, for simplicity, this chosen example is a
code that is not only able to correct X errors (bit flips)
} %
\red{Do I need to show what the syndrome extraction
circuitry for Z errors looks like?}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{An Example: The Steane Code}
\vspace*{-10mm}
\begin{align*}
\bm{H} =
\begin{pmatrix}
1 & 1 & 0 \\
0 & 1 & 1
\end{pmatrix}
\end{align*}
\vspace*{5mm}
\begin{figure}[H]
\centering
\mbox{
% tex-fmt: off
\Qcircuit @C=1em @R=.7em {
& & \ctrl{3} & \qw & \qw & \qw & \qw & \qw \\
\ket{\psi}_\text{L} & & \qw & \ctrl{2} & \ctrl{3} & \qw & \qw & \qw \\
& & \qw & \qw & \qw & \ctrl{2} & \qw & \qw \\
\ket{0}_{\text{A}_1} & & \targ & \targ & \qw & \qw & \meter & \\
\ket{0}_{\text{A}_2} & & \qw & \qw & \targ & \targ & \meter &
}
% tex-fmt: on
}
\vspace*{5mm}
\caption{Syndrome extraction circuit for the 3-qubit repetition code}
\end{figure}
% \vspace*{5mm}
\vspace*{-2mm}
\addreferences
{nielsen_quantum_2010}
\stopreferences
\begin{itemize}
\item \textcolor{red}{Give example slides grey background or something?}
\item The Steane code is the quantum equivalent of the
[7,4]-Hamming code
\item Construction from Hamming code
\item Syndrome Extraction Circuitry
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -287,10 +417,29 @@
\item The Threshold theorem
\setcounter{footnote}{0}
\item Definition of fault tolerance \footnotemark
\item \textcolor{red}{Different approaches to fault tolerance?}
\item \red{Different approaches to fault tolerance?}
\end{itemize}
\end{frame}
% TODO: Where to we introduces the different kinds of noise models?
\begin{frame}
\frametitle{Noise models}
\begin{itemize}
\item The depolarizing channel
\item Phenomenological noise
\item Circuit-level noise (we generally have all error
probabilities equal the same value \\
for simulations \citereference{fowler_high-threshold_2009})
\end{itemize}
\vspace*{15mm}
\addreferences
{fowler_high-threshold_2009}
\stopreferences
\end{frame}
%%%%%%%%%%%%%%%%
\subsection{Detector Error Models}
\label{subsec:Detector Error Models}
@ -305,10 +454,11 @@
\end{frame}
\begin{frame}
\frametitle{An Example: Steane Code Detector Error Model}
\frametitle{Example: 3-Qubit Repetition Code Detector\\ Error
Model for Circuit Level Noise}
\begin{itemize}
\item New Syndrome Extraction Circuitry \textcolor{red}{Is a
\item New Syndrome Extraction Circuitry \red{Is a
repetition of the old circuitry needed?}
\item New parity check matrix
\item Highlighting of the SC-LDPC-code-like structure
@ -319,16 +469,16 @@
\frametitle{Challenges}
\begin{itemize}
\item Multiple different errors are summarized $\rightarrow$
short cycles \& degeneracy
\item \red{Multiple different errors are summarized
$\rightarrow$ short cycles \& degeneracy}
\footnote{
\texttt{
https://www.math.cit.tum.de/fileadmin/w00ccg/math/\_my\_direct\_uploads/Dan\_Browne.pdf
\red{https://www.math.cit.tum.de/fileadmin/w00ccg/math/\_my\_direct\_uploads/Dan\_Browne.pdf}
}
}
\\
$\rightarrow$ We generally don't use "normal BP" (BP +
OSD, BPGD, etc.)
\red{$\rightarrow$ We generally don't use "normal BP" (BP
+ OSD, BPGD, etc.)}
\end{itemize}
\end{frame}
@ -344,6 +494,8 @@
\frametitle{Sliding Window Decoding}
\begin{itemize}
% TODO: Do I have to explain BP?
\item \red{Do I have to explain BP}
\item Give overview of existing research
\item Explain exactly what they do in the main paper I am
basing my work on
@ -376,8 +528,14 @@
\begin{itemize}
\item Top level overview of entire system: X and Z syndrome
extraction, logical operator measurement, where decoding
takes place, etc.
takes place, etc. (fig. 3 of \citereference{derks_designing_2025})
\end{itemize}
\vspace*{15mm}
\addreferences
{derks_designing_2025}
\stopreferences
\end{frame}
\begin{frame}
@ -400,7 +558,7 @@
\frametitle{The lack of a Standard Evaluation System}
\begin{itemize}
\item \textcolor{red}{Look into ECCentric}
\item \red{Look into ECCentric}
\item There is not even a standard figure of merit (e.g.,
FER/BER over SNR in classical case) $\rightarrow$
Multiple different kinds of plots (e.g., footprint)
@ -417,9 +575,33 @@
\end{itemize}
\end{frame}
\begin{frame}[t]
\frametitle{Questions}
\begin{minipage}[c]{0.65\textwidth}
\centering
\LARGE Thank you for your attention!\\ Any questions?
\end{minipage}%
\begin{minipage}[c]{0.35\textwidth}
\centering
\begin{figure}[H]
\centering
\begin{tikzpicture}[every node/.style={scale=10}]
\node at (0, 0)
{\textcolor{kit-blue}{{\fontfamily{phv}\selectfont ?}}};
\end{tikzpicture}
\end{figure}
\end{minipage}
\end{frame}
\appendix
\beginbackup
% TODO: Move arrow into syndrome extraction lower (branch from other
% arrow) and change caption to "modified from [MSLS25]"
\begin{frame}
\frametitle{System Level Overview}
@ -495,6 +677,16 @@
\stopreferences
\end{frame}
% TODO: Is this really necessary?
\begin{frame}
\frametitle{The Quantum Error Correction Landscape}
\begin{itemize}
\item Give basic overview of most promising avenues of
research (as in ECCentric paper)
\end{itemize}
\end{frame}
\backupend
\end{document}