From 95631f90cf66efec63439553cb58365dec4a9f35 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Thu, 29 Jan 2026 16:59:05 +0100 Subject: [PATCH] Finish first version of intro --- src/midterm_presentation/main.tex | 304 ++++++++++++++++++++++++------ 1 file changed, 248 insertions(+), 56 deletions(-) diff --git a/src/midterm_presentation/main.tex b/src/midterm_presentation/main.tex index 689e7c2..eb093b8 100644 --- a/src/midterm_presentation/main.tex +++ b/src/midterm_presentation/main.tex @@ -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}