694 lines
20 KiB
TeX
694 lines
20 KiB
TeX
\documentclass[overviewatsection, showsubsectionsatfirstoverview]{CELbeamer}
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\usepackage{amssymb}
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\usepackage{acro}
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\usepackage{braket}
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\usepackage{qcircuit}
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\title{Fault Tolerant Quantum Error Correction}
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\subtitle{Master's Thesis Midterm Presentation}
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\author[Tsouchlos]{Andreas Tsouchlos}
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\date[]{February 5th, 2026}
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\newcommand{\res}{src/midterm_presentation/res}
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% Acronyms
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%
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\DeclareAcronym{qec}{
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short=QEC,
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long=quantum error correction
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}
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\DeclareAcronym{css}{
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short=CSS,
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long=Calderbank Shor Steane
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}
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%
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% Document body
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\begin{document}
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\begin{frame}[title white vertical, picture=images/IMG_7801-cut]
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\titlepage
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction to Quantum Error Correction}
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\label{sec:Introduction to Quantum Error Correction}
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%%%%%%%%%%%%%%%%
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\subsection{Motivation}
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\label{subsec:Motivation}
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\begin{frame}
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\frametitle{Quantum Computing}
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% Related interesting stuff
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% - Gidney estimates we need 1399 (?) logical qubits to factor a 2048
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% bit RSA integer
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% - He goes on to estimate that to factor such an integer in less
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% than a week would require around a million physical qubits
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% [How to factor 2048 bit RSA integers with less than a million
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% noisy qubits]
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\vspace*{-15mm}
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\begin{itemize}
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\item Simulating quantum systems on classical hardware
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is exponentially complex \\
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$\rightarrow$ Can't we use quantum hardware to simulate
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quantum systems? \citereference{feynman_simulating_1982}
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\item Some problems that are ``hard'' to solve on classical
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computers we can ``easily'' solve on quantum computers
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\citereference{preskill_quantum_2018}
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\end{itemize}
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\vspace*{-5mm}
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\begin{figure}[H]
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\centering
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\includegraphics[scale=0.43]{res/google_roadmap.png}
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\vspace*{-3mm}
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\caption{
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Google Quantum AI's quantum computing roadmap
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\citereference{google_quantum_ai_quantum_nodate}.
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}
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\end{figure}
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\vspace*{3mm}
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\addreferences
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{feynman_simulating_1982}
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{preskill_quantum_2018}
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{google_quantum_ai_quantum_nodate}
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\stopreferences
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\end{frame}
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% TODO: Where should I quote Preskill? There are multiple bullet
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% points with info taken from his work
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\begin{frame}
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\frametitle{The Need for Quantum Error Correction}
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\vspace*{-17mm}
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% Related interesting stuff
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% - Qubits differ from bits in that they can be in superpositions
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% and be entangled with one another
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% - Quantum computers derive their strenght from the exponential
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% scaling of the state-space because of the way the information is
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% encoded
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% - Note that while a physical error rate of 10^{-3} may seem ok,
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% we need a couple trillion operations (~ 10^{13}) to factor a
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% 2048 bit RSA integer
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% [How to factor 2048 bit RSA integers with less than a million
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% noisy qubits]
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% - The "physical error rate" is really the value all error rates
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% in the system are set to for circuit level noise simulations
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% [High-threshold universal quantum computation on the surface code]
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% - The backlog problem is the fact that an increasing backlog of
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% syndrome data will lead to an exponential slowdown during the
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% computation
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\begin{itemize}
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\item Quantum computers represent information through
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correlations of qubits, not their values \\
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directly \citereference{preskill_quantum_2018}
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\item We want to not disturb the quantum state but need to
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interact with the system $\rightarrow$ Protect the state
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with \ac{qec}
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\item We employ more physical qubits to introduce
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redundancy and use the resulting \emph{physical} state to
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represent the \emph{logical} state
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\citereference{roffe_quantum_2019}
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\vspace*{8mm}
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\item Typical scales
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\begin{itemize}
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\item IBM recently introduced a scheme encoding $12$ logical
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qubits in $288$ physical ones
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\citereference{bravyi_high-threshold_2024}
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\item The physical error rate is typically assumed to
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be $10^{-3}$ for
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simulations (e.g.,
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\citereference{bravyi_high-threshold_2024})
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\item Decoding has to happen with ultra-low latency to avoid
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the backlog problem (about $\SI{1}{us}$ per data
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extraction round) \citereference{caune_demonstrating_2024}
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% \citereference{terhal_quantum_2015}
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\end{itemize}
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\end{itemize}
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\vspace*{7mm}
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\addreferences
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% {terhal_quantum_2015}
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{preskill_quantum_2018}
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{roffe_quantum_2019}
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{bravyi_high-threshold_2024}
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{caune_demonstrating_2024}
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\stopreferences
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\end{frame}
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%%%%%%%%%%%%%%%%
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\subsection{Fundamentals of Quantum Error Correction}
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\label{subsec:Fundamentals of Quantum Error Correction}
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% TODO: Is this all of this really necessary?
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\begin{frame}
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\frametitle{Peculiarities of the Quantum Setting}
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\vspace*{-5mm}
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% Related interesting stuff
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% - No cloning theorem -> Not replication of state, protection
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% through further entanglement
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% - States are superpositions -> We theoretically need to be able
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% to correct infinitely many different types of errors. Luckily,
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% it turns out that in actual fact we only really need to correct
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% two [Gottesman's Thesis]
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% - Mention that kets are just vectors, used here to represent the state
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% - There are actually infinitely many different errors that can
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% happen, but we can digitize them and only need to consider X and Z
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% - Not only do we only care about the coset, we specifically
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% don't want to know more than the syndrome can tell us because
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% that would mean that "we collapse the quantum mechanical state too
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% much"
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\begin{itemize}
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\item As mentioned earlier, \ac{qec} is actually able to
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protect the quantum state with all its correlations
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\item We have to consider phase flip errors in addition to
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bit flip errors \citereference{roffe_quantum_2019}
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\vspace*{-10mm}
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\begin{figure}[H]
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\centering
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\begin{subfigure}{0.5\textwidth}
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\centering
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\begin{align*}
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\ket{0} &\rightarrow \ket{1} \\
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\ket{1} &\rightarrow \ket{0}
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\end{align*}
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\caption{Bit flip (X) error}
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\end{subfigure}%
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\begin{subfigure}{0.5\textwidth}
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\centering
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\begin{align*}
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\ket{0} &\rightarrow \phantom{-}\ket{0} \\
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\ket{1} &\rightarrow -\ket{1}
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\end{align*}
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\caption{Phase flip (Z) error}
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\end{subfigure}
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\end{figure}
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\item Measuring the qubits directly destroys superpositions
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and entanglement \\
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$\rightarrow$ We generally only work with the syndrome,
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which we can measure \citereference{nielsen_quantum_2010}
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\item We don't care about restoring the specific codeword,
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only finding the coset it's in
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\end{itemize}
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\vspace*{15mm}
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\addreferences
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{nielsen_quantum_2010}
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{roffe_quantum_2019}
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\stopreferences
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\end{frame}
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\begin{frame}
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\frametitle{Stabilizer and Calderbank Shor Steane Codes}
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\vspace*{-5mm}
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% Related interesting stuff
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% - Using stabilizers to describe quantum codes is a bit like
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% using parity check equations to describe classical codes
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% -> stabilizer codes are the quantum analog of binary linear codes
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% - For CSS codes, "the parity checks for the X errors and the
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% parity checks for the Z errors can be represented independently
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% of one another"
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\begin{itemize}
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\item Stabilizer codes \citereference{nielsen_quantum_2010}
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\begin{itemize}
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\item The code space can implicitly be defined using
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\emph{stabilizer generators}
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\item We can represent them using parity
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check matrices
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\item Quantum analog of linear codes
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\end{itemize}
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\vspace*{10mm}
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\item \Ac{css} codes \citereference{nielsen_quantum_2010}
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\begin{itemize}
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\item Subset of stabilizer codes
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\item Can correct X and Z errors independently
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\item Described using two separate parity check
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matrices $\bm{H}_\text{X}$ and $\bm{H}_\text{Z}$
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\item Can be constructed from two binary linear codes
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$\mathcal{C}_1 \left[ n, k_1 \right]$ and
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$\mathcal{C}_2 \left[ n, k_2 \right]$ with
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$\mathcal{C}_2 \subset \mathcal{C}_1$
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\end{itemize}
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\vspace*{10mm}
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\item \red{Do I need to go more in depth for either
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stabilizer codes or CSS codes?}
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\end{itemize}
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\vspace*{10mm}
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\addreferences
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{nielsen_quantum_2010}
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\stopreferences
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\end{frame}
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% TODO: Do I need to show what the syndrome extraction circuitry for
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% Z errors looks like?
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\begin{frame}
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\frametitle{Syndrome Extraction Circuits}
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\vspace*{-16mm}
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\begin{itemize}
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\item We entangle the state with \emph{ancilla qubits} to
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perform syndrome measurements \citereference{nielsen_quantum_2010}
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\item Example: The 3-qubit repetition code%
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\footnote {
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Note that, for simplicity, this chosen example is a
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code that is not only able to correct X errors (bit flips)
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} %
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\red{Do I need to show what the syndrome extraction
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circuitry for Z errors looks like?}
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\end{itemize}
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\vspace*{-10mm}
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\begin{align*}
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\bm{H} =
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\begin{pmatrix}
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1 & 1 & 0 \\
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0 & 1 & 1
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\end{pmatrix}
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\end{align*}
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\vspace*{5mm}
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\begin{figure}[H]
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\centering
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\mbox{
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% tex-fmt: off
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\Qcircuit @C=1em @R=.7em {
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& & \ctrl{3} & \qw & \qw & \qw & \qw & \qw \\
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\ket{\psi}_\text{L} & & \qw & \ctrl{2} & \ctrl{3} & \qw & \qw & \qw \\
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& & \qw & \qw & \qw & \ctrl{2} & \qw & \qw \\
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\ket{0}_{\text{A}_1} & & \targ & \targ & \qw & \qw & \meter & \\
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\ket{0}_{\text{A}_2} & & \qw & \qw & \targ & \targ & \meter &
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}
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% tex-fmt: on
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}
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\vspace*{5mm}
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\caption{Syndrome extraction circuit for the 3-qubit repetition code}
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\end{figure}
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% \vspace*{5mm}
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\vspace*{-2mm}
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\addreferences
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{nielsen_quantum_2010}
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\stopreferences
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Fault Tolerance and Detector Error Models}
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\label{sec:Fault Tolerance and Detector Error Models}
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%%%%%%%%%%%%%%%%
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\subsection{Fault Tolerance}
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\label{subsec:Fault Tolerance}
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\begin{frame}
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\frametitle{Fault Tolerance}
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\begin{itemize}
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\item Quantum gates are faulty $\rightarrow$ we need QEC \\
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But we do QEC with faulty gates $\rightarrow$ we need
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fault tolerant QEC %
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\footnote{
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Designing fault-tolerant circuits using detector
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error models - Gong et al, Section 4.1
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}
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\item We generally do multiple rounds of syndrome extraction
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\item The Threshold theorem
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\setcounter{footnote}{0}
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\item Definition of fault tolerance \footnotemark
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\item \red{Different approaches to fault tolerance?}
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\end{itemize}
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\end{frame}
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% TODO: Where to we introduces the different kinds of noise models?
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\begin{frame}
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\frametitle{Noise models}
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\begin{itemize}
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\item The depolarizing channel
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\item Phenomenological noise
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\item Circuit-level noise (we generally have all error
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probabilities equal the same value \\
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for simulations \citereference{fowler_high-threshold_2009})
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\end{itemize}
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\vspace*{15mm}
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\addreferences
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{fowler_high-threshold_2009}
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\stopreferences
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\end{frame}
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%%%%%%%%%%%%%%%%
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\subsection{Detector Error Models}
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\label{subsec:Detector Error Models}
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\begin{frame}
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\frametitle{Detector Error Models}
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\begin{itemize}
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\item Idea: Go "one layer of abstraction higher" \\
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$\rightarrow$ Redefine syndrome and create new PC matrix from that
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Example: 3-Qubit Repetition Code Detector\\ Error
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Model for Circuit Level Noise}
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\begin{itemize}
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\item New Syndrome Extraction Circuitry \red{Is a
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repetition of the old circuitry needed?}
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\item New parity check matrix
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\item Highlighting of the SC-LDPC-code-like structure
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Challenges}
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\begin{itemize}
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\item \red{Multiple different errors are summarized
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$\rightarrow$ short cycles \& degeneracy}
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\footnote{
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\texttt{
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\red{https://www.math.cit.tum.de/fileadmin/w00ccg/math/\_my\_direct\_uploads/Dan\_Browne.pdf}
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}
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}
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\\
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\red{$\rightarrow$ We generally don't use "normal BP" (BP
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+ OSD, BPGD, etc.)}
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{State of the Art and Research Gap}
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\label{sec:State of the Art and Research Gap}
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%%%%%%%%%%%%%%%%
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\subsection{State of the Art}
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\label{subsec:State of the Art}
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\begin{frame}
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\frametitle{Sliding Window Decoding}
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\begin{itemize}
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% TODO: Do I have to explain BP?
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\item \red{Do I have to explain BP}
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\item Give overview of existing research
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\item Explain exactly what they do in the main paper I am
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basing my work on
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Guided Decimation Guessing Decoding}
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\begin{itemize}
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\item Update equations
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\item Key ideas
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\item Syndrome Based BP
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Memory experiments}
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\begin{itemize}
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\item What is a memory experiment?
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\item Communications Engineering view (what are my inputs and
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outpus? What do I expect?)
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Systemic overview}
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\begin{itemize}
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\item Top level overview of entire system: X and Z syndrome
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extraction, logical operator measurement, where decoding
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takes place, etc. (fig. 3 of \citereference{derks_designing_2025})
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\end{itemize}
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\vspace*{15mm}
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\addreferences
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{derks_designing_2025}
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\stopreferences
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\end{frame}
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\begin{frame}
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\frametitle{Research Gap}
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\begin{itemize}
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\item Use soft information for sliding window decoding
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$\rightarrow$ Treat as spacially coupled LDPC code
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\item Current work considers X and Z errors separately
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(probably for latency reasons) $\rightarrow$ See how
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decoding the jointly works out
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%
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\subsection{What we simulate}
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\label{subsec:What we simulate}
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\begin{frame}
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\frametitle{The lack of a Standard Evaluation System}
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\begin{itemize}
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\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)
|
|
\item Overview of variables
|
|
\end{itemize}
|
|
\end{frame}
|
|
|
|
\begin{frame}
|
|
\frametitle{Proposed Evaluation Pipeline}
|
|
|
|
\begin{itemize}
|
|
\item To what values I will fix the parameters and why
|
|
\item What figure of merit I will use and why
|
|
\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}
|
|
|
|
\vspace*{-15mm}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
\begin{subfigure}[t]{0.5\textwidth}
|
|
\centering
|
|
|
|
\includegraphics[scale=1.1]{res/architecture}
|
|
|
|
\vspace*{5mm}
|
|
|
|
\caption{Schematic workflow of surface code quantum
|
|
computation \citereference{zhang_classical_2023}.}
|
|
\end{subfigure}%
|
|
\begin{subfigure}[t]{0.5\textwidth}
|
|
\centering
|
|
|
|
\tikzset{
|
|
block/.style={
|
|
draw, rectangle,
|
|
fill = kit-blue!25,
|
|
minimum width=75mm, minimum height=15mm,
|
|
}
|
|
}
|
|
|
|
\scalebox{0.7}{
|
|
\begin{tikzpicture}[node distance=15mm and 20mm]
|
|
\node[block] (encoding) {Encoding};
|
|
\node[block, below=of encoding] (channel) {Quantum Channel};
|
|
\node[block, below=of channel] (reverse-op)
|
|
{Reverse Operation};
|
|
|
|
\node[block, right=of channel] (syn-extr)
|
|
{Syndrome Extraction};
|
|
\node[block, below=of syn-extr] (syn-dec)
|
|
{Syndrome Decoder};
|
|
|
|
\node[above=of encoding] (input) {$\ket{\phi}$};
|
|
\node[below=of reverse-op] (output)
|
|
{$\hat{\mathcal{E}}\mathcal{E}\ket{\psi}$};
|
|
|
|
\draw [-{Latex}] (encoding) -- (channel) node[midway,
|
|
right] {$\ket{\psi}$};
|
|
\draw [-{Latex}] (channel) -- (reverse-op)
|
|
node[midway, right] {$\mathcal{E}\ket{\psi}$};
|
|
\draw [-{Latex}] (channel) -- (syn-extr)
|
|
node[midway, above] {$\mathcal{E}\ket{\psi}$};
|
|
\draw [-{Latex}] (syn-extr) -- (syn-dec)
|
|
node[midway, right] {$z$};
|
|
\draw [-{Latex}] (syn-dec) -- (reverse-op)
|
|
node[midway, above] {$\hat{\mathcal{E}}$};
|
|
|
|
\draw [-{Latex}] (input) -- (encoding);
|
|
\draw [-{Latex}] (reverse-op) -- (output);
|
|
\end{tikzpicture}
|
|
}
|
|
|
|
\vspace*{5mm}
|
|
|
|
\caption{Block diagram of QEC using stabilizer codes
|
|
\citereference{miao_quaternary_2025}.}
|
|
\end{subfigure}
|
|
\end{figure}
|
|
|
|
% \vspace*{-2mm}
|
|
|
|
\addreferences
|
|
{zhang_classical_2023}
|
|
{miao_quaternary_2025}
|
|
\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}
|
|
|