2215 lines
95 KiB
TeX
2215 lines
95 KiB
TeX
\documentclass[overviewatsection]{CELbeamer}
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\groupname{Communications Engineering Lab (CEL)}
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% Document setup
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\usetikzlibrary{spy, external, intersections, positioning, tikzmark}
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\usepackage{amsmath}
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\usepackage{graphicx}
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\usepackage{calc}
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\usepackage{amssymb}
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\usepackage{acro}
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\usepackage{braket}
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\usepackage{quantikz}
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\usepackage{nicematrix}
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\usepackage{xpatch}
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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[ Andreas]{Andreas Tsouchlos}
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\date[]{}
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\DeclareFieldFormat[article,book,inproceedings]{urldate}{}
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\addbibresource{MA.bib}
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% Custom commands
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\newcommand{\red}[1]{\textcolor{red}{#1}}
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\newcommand{\res}{src/midterm_presentation/res}
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\newcommand{\X}{\textcolor{kit-blue}{\bm{X}}}
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\newcommand{\Z}{\textcolor{kit-orange}{\bm{Z}}}
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\newcommand{\Y}{\textcolor{kit-red}{\bm{Y}}}
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\newcommand{\I}{\bm{I}}
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\xpatchcmd{\fullwire}
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{\arrow[arrows,start anchor=#2startone,end anchor=#2endone]
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{#1}\arrow[arrows,start anchor=#2starttwo,end anchor=#2endtwo,#4] {#1}}
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{\arrow[arrows,start anchor=#2startone,end anchor=#2endone,#4]
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{#1}\arrow[arrows,start anchor=#2starttwo,end anchor=#2endtwo,#4] {#1}}
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{\typeout{Successfully patched fullwire for classical wires}}
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{\typeout{Failed to patch fullwire for classical wires}}
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\makeatletter
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\newcommand{\addreferencesmanual}{%
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\begin{scriptsize}
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\begin{tabular}{lp{0.88\textwidth}}
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\@addreferencesimanual
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}
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\newcommand\@addreferencesimanual{\@ifnextchar\stopreferencesmanual{\@addreferencesendmanual}{\@addreferencesiimanual}}
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\newcommand\@addreferencesiimanual[2]{%
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\textcolor{kit-green100}{[#1]} & \textcolor{kit-green100}{#2} \\
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\@addreferencesimanual % restart the recursion
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}
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\newcommand\@addreferencesendmanual[1]{% The argument is \stopimages
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\end{tabular}
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\end{scriptsize}
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}
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\makeatother
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\newcommand{\citereferencemanual}[1]{\textcolor{kit-green100}{\textbf{\scriptsize{[#1]}}}}
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%
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%
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% Acronyms
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%
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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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\DeclareAcronym{bb}{
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short=BB,
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long=bivariate bicycle
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}
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\DeclareAcronym{dem}{
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short=DEM,
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long=detector error model
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}
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\DeclareAcronym{bp}{
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short=BP,
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long=belief propagation
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}
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\DeclareAcronym{osd}{
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short=OSD,
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long=ordered statistics decoding,
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}
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\DeclareAcronym{qldpc}{
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short=QLDPC,
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long=quantum low-density parity-check,
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}
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\DeclareAcronym{scldpc}{
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short=SC-LDPC,
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long=spatially-coupled low-density parity-check
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}
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\DeclareAcronym{ler}{
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short=LER,
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long=logical error rate,
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}
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%
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%
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% Document body
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%
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%
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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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$\rightarrow$ Use quantum hardware to simulate quantum
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systems \citereferencemanual{Fey82}
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\item ``Hard'' to solve problems on classical computers can
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be ``easy'' on quantum computers
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\citereferencemanual{Pre18}
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\item Google Quantum AI's quantum computing roadmap
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\citereferencemanual{Goo23}
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\end{itemize}
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\vspace*{3mm}
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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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\end{figure}
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\vspace*{3mm}
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\addreferencesmanual
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{Fey82}{
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R. P. Feynman, ``Simulating physics with computers,'',
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\emph{International Journal of Theoretical Physics}, 1982.
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}
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{Pre18}{
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J. Preskill, ``Quantum Computing in the NISQ era and
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beyond,'' \emph{Quantum}, 2018.
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}
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{Goo23}{
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Google Quantum AI, \emph{Quantum Computing Roadmap}, URL:
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\url{https://quantumai.google/qecmilestone2023}, 2023.
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}
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\stopreferencesmanual
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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*{-12mm}
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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 Quantum systems are inherently fragile
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\item Interacting with the quantum state disturbs it
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\item Idea: Represent \schlagwort{logical qubits} using more
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\schlagwort{physical qubits} \citereferencemanual{Rof19}
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\vspace*{2mm}
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\node[
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rectangle,
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draw, fill=kit-blue!25,
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minimum height=15mm,
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]
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(enc) {Three-qubit encoder};
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\node[left=of enc] (in)
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{$\ket{\psi} = \alpha\ket{0} + \beta\ket{1}$};
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\node[right=of enc,yshift=6mm] (out)
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{$\alpha\overbrace{\ket{000}}^{\ket{0}_\text{L}}
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+\; \beta\overbrace{\ket{111}}^{\ket{1}_\text{L}} =
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\ket{\psi}_\text{L}$};
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\draw[-{Latex}] (in) -- (enc);
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\draw[-{Latex}] (enc) -- (enc -| out.west);
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\end{tikzpicture}
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\end{figure}
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\vspace*{5mm}
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\visible<2>{
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\item Typical scales
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\begin{itemize}
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\item Recent scheme by IBM encodes $12$ logical
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qubits in $288$ physical ones
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\citereferencemanual{BCG$^+$24}
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\item Physical error rate typically set to $10^{-3}$
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for simulations (e.g.,
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\citereferencemanual{BCG$^+$24})
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\item Decode with ultra-low latency to avoid
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\schlagwort{backlog problem} (about
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$\SI{1}{\micro s}$ per data \\
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extraction round)
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\citereferencemanual{CSB$^+$24}
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\end{itemize}
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}
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\end{itemize}
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\vspace*{10mm}
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\addreferencesmanual
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{Rof19}{
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J. Roffe, ``Quantum error correction: An introductory
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guide,'' \emph{Contemporary Physics}, 2019.
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}
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{BCG$^+$24}{
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S. Bravyi et al., ``High-threshold and low-overhead
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fault-tolerant quantum memory,'' \emph{Nature}, 2024.
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}
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{CSB$^+$24}{
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L. Caune et al., ``Demonstrating real-time and low-latency
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quantum error correction with superconducting qubits'',
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\emph{arXiv:2410.05202}, 2024.
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}
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\stopreferencesmanual
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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*{-10mm}
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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 \Ac{qec} is actually able to protect the actual
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% quantum state
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\item Classical systems built with bits and gates, quantum
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systems with qubits and quantum gates
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\item We have to consider phase flip errors in addition to
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bit flip errors \citereferencemanual{Rof19}
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\end{itemize}
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\vspace*{-3mm}
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\begin{figure}[H]
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\centering
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\begin{subfigure}{0.32\textwidth}
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\centering
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\begin{align*}
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\ket{0} &\mapsto \ket{1} \\
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\ket{1} &\mapsto \ket{0}
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\end{align*}
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\caption{Bit flip ($\X$) error}
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\end{subfigure}%
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\begin{subfigure}{0.32\textwidth}
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\centering
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\begin{align*}
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\ket{0} &\mapsto \phantom{-}\ket{0} \\
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\ket{1} &\mapsto -\ket{1}
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\end{align*}
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\caption{Phase flip ($\Z$) error}
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\end{subfigure}%
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\begin{subfigure}{0.32\textwidth}
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\centering
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\begin{align*}
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\ket{0} &\mapsto \phantom{-j}\ket{1} \\
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\ket{1} &\mapsto -j\ket{0}
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\end{align*}
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\caption{$\Y$ error}
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\end{subfigure}
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\end{figure}
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\begin{itemize}
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\visible<2->{
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\item Measuring the qubits directly destroys superpositions
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and entanglement \\
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$\rightarrow$ Use syndrome for decoding
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\citereferencemanual{NC10}
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}
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\visible<3>{
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\item Superposition $\rightarrow$ Multiple solutions to the
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decoding problem
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(\schlagwort{quantum degeneracy})
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\citereferencemanual{RWB$^+$20}}
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\end{itemize}
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\vspace*{12mm}
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\addreferencesmanual
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{Rof19}{
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J. Roffe, ``Quantum error correction: An introductory
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guide,'' \emph{Contemporary Physics}, 2019.
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}
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{NC10}{
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M. A. Nielsen and I. L. Chuang, ``Quantum Computation and
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Quantum Information'', \emph{Cambridge University Press}, 2010.
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}
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{RWB$^+$20}{
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J. Roffe et al., ``Decoding across the quantum low-density
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parity-check code landscape,'' \emph{Physical Review Research}, 2020.
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}
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\stopreferencesmanual
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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 \citereferencemanual{NC10}
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\begin{itemize}
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\item Implicitly defined using \schlagwort{stabilizer
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generators}
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\item Can be represented using parity check matrices
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\item Quantum analog of linear block codes
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\end{itemize}
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\vspace*{10mm}
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\visible<2->{
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\item \Acf{css} codes \citereferencemanual{NC10}
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\begin{itemize}
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\item Subset of stabilizer codes
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\item Able to correct $\X$ and $\Z$ errors independently
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\item Described using two separate parity check
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matrices $\bm{H}_X$ and $\bm{H}_Z$
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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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}
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\end{itemize}
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\vspace*{20mm}
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\addreferencesmanual
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{NC10}{
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M. A. Nielsen and I. L. Chuang, ``Quantum Computation and
|
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Quantum Information'', \emph{Cambridge University Press}, 2010.
|
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}
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\stopreferencesmanual
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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*{-10mm}
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\begin{itemize}
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\item Entangle the state $\ket{\psi}$ with
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\schlagwort{ancilla qubits} to perform syndrome
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measurements \citereferencemanual{NC10}
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\item Example: The 3-qubit repetition code for $\X$ errors
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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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\hspace*{-25mm}
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\begin{subfigure}{0.4\textwidth}
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\centering
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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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% \newcommand{\anyerrgate}{\gate[style={fill=red!20}]{\mathcal{E}_\text{XYZ}}}
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\newcommand{\preperr}{\gate[style={fill=orange!20}]{\phantom{1}}}
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\newcommand{\gateerr}{\gate[style={fill=red!20}]{\phantom{1}}}
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\newcommand{\measerr}{\gate[style={fill=blue!20}]{\phantom{1}}}
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\centering
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% tex-fmt: off
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\begin{quantikz}%[row sep=4mm, column sep=4mm]
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& \ctrl{3} & & & & & \\
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\lstick{$\ket{\psi}$} & & \ctrl{2} & \ctrl{3} & & & \\
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& & & & \ctrl{2} & & \\
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\lstick{$\ket{0}_{\text{A}_1}$} & \targ{} & \targ{} & & & \meter{} & \setwiretype{c} \\
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\lstick{$\ket{0}_{\text{A}_2}$} & & & \targ{} & \targ{} & \meter{} & \setwiretype{c}
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\end{quantikz}
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% tex-fmt: on
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\end{subfigure}%
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\begin{subfigure}{0.15\textwidth}
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\centering
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\begin{align*}
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\bm{H} \bm{y}^\text{T} = \bm{H} \bm{e}^\text{T} = \bm{s}
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\end{align*}
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\vspace*{-5mm}
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\end{subfigure}
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\end{figure}
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\vspace*{10mm}
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\addreferencesmanual
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{NC10}{
|
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M. A. Nielsen and I. L. Chuang, ``Quantum Computation and
|
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Quantum Information'', \emph{Cambridge University Press}, 2010.
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}
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\stopreferencesmanual
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\end{frame}
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|
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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}
|
|
|
|
%%%%%%%%%%%%%%%%
|
|
\subsection{Fault Tolerance}
|
|
\label{subsec:Fault Tolerance}
|
|
|
|
% TODO: Fix journal for {shor_fault-tolerant_1997} not showing
|
|
\begin{frame}
|
|
\frametitle{Fault Tolerance}
|
|
|
|
\vspace*{-10mm}
|
|
|
|
\begin{itemize}
|
|
\item Quantum gates used for syndrome extraction are
|
|
faulty themselves \\
|
|
$\rightarrow$ Need for \schlagwort{fault-tolerant} \acf{qec}
|
|
\item In addition to correcting \schlagwort{input errors},
|
|
limit spread of \schlagwort{internal errors}
|
|
\citereferencemanual{DTB$^+$25}
|
|
\end{itemize}
|
|
|
|
\vspace*{3mm}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\node[rectangle, draw, fill=orange!20, minimum
|
|
height=3cm, minimum width=3.5cm, align=left] at (0,0)
|
|
(internal) {Internal\\ Errors};
|
|
|
|
\node[signal, draw, fill=Plum!20, minimum height=3cm,
|
|
minimum width=4cm, align=left, signal pointer angle=140]
|
|
at (-3.7, 0) (input) {Input\\ Errors};
|
|
|
|
\node at (2.5,0) {\huge =};
|
|
|
|
\node[rectangle, draw, fill=MidnightBlue!20, minimum height=3cm,
|
|
minimum width=3.5cm, align=left] at (5,0) (output)
|
|
{Output\\ Errors};
|
|
|
|
\node[above] at (input.north) {\small Input State};
|
|
\node[above] at (internal.north) {\small QEC};
|
|
\node[above] at (output.north) {\small Output State};
|
|
\end{tikzpicture}
|
|
\end{figure}
|
|
|
|
\vspace*{3mm}
|
|
|
|
\begin{itemize}
|
|
\visible<2->{
|
|
\item Modify syndrome extraction circuitry (e.g., multi-qubit
|
|
states for each ancilla
|
|
\citereferencemanual{Sho97})
|
|
\item Multiple rounds of syndrome extraction
|
|
}
|
|
\end{itemize}
|
|
|
|
\vspace*{15mm}
|
|
|
|
\addreferencesmanual
|
|
{DTB$^+$25}{
|
|
P.- J. H. S. Derks et al., ``Designing fault-tolerant
|
|
circuits using detector error models,'' \emph{Quantum}, 2025.
|
|
}
|
|
{Sho97}{
|
|
P. W. Shor, ``Fault-tolerant quantum computation,''
|
|
\emph{arXiv:quant-ph/9605011}, 1997.
|
|
}
|
|
\stopreferencesmanual
|
|
\end{frame}
|
|
|
|
%%%%%%%%%%%%%%%%
|
|
\subsection{Detector Error Models}
|
|
\label{subsec:Detector Error Models}
|
|
|
|
\begin{frame}[fragile]
|
|
\frametitle{The Measurement Syndrome Matrix I}
|
|
|
|
\vspace*{-18mm}
|
|
|
|
\begin{itemize}
|
|
\item \schlagwort{Measurement syndrome matrix} $\bm{\Omega}$ \\
|
|
contains error patterns \citereferencemanual{DTB$^+$25}
|
|
\item Example: 3-qubit repetition code
|
|
\end{itemize}
|
|
|
|
\vspace*{-25mm}
|
|
|
|
\centering
|
|
\only<1>{
|
|
\begin{minipage}{0.4\textwidth}
|
|
\centering
|
|
|
|
\vspace*{40mm}
|
|
\begin{tikzpicture}
|
|
\node{$%
|
|
\bm{\Omega} =
|
|
\left(
|
|
\begin{array}{ccc}
|
|
1 & 1 & 0 \\
|
|
0 & 1 & 1 \\
|
|
1 & 1 & 0 \\
|
|
0 & 1 & 1 \\
|
|
1 & 1 & 0 \\
|
|
0 & 1 & 1
|
|
\end{array}
|
|
\right)$
|
|
};
|
|
|
|
\draw [
|
|
line width=1pt,
|
|
decorate,
|
|
decoration={brace,mirror,amplitude=3mm,raise=5mm}
|
|
]
|
|
(2.5,1.2) -- (2.5,2.85)
|
|
node[midway,right,xshift=10mm]{$\text{SE}_1$};
|
|
|
|
\draw [
|
|
line width=1pt,
|
|
decorate,
|
|
decoration={brace,mirror,amplitude=3mm,raise=5mm}
|
|
]
|
|
(2.5,-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}
|
|
]
|
|
(2.5,-2.7) -- (2.5,-1.1)
|
|
node[midway,right,xshift=10mm]{$\text{SE}_3$};
|
|
\end{tikzpicture}
|
|
|
|
\vspace*{-10mm}
|
|
|
|
\begin{gather*}
|
|
\bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\}
|
|
\end{gather*}
|
|
\end{minipage}%
|
|
\begin{minipage}{0.6\textwidth}
|
|
\begin{figure}[H]
|
|
\newcommand{\preperr}[1]{
|
|
\gate[style={fill=orange!20}]{\scriptstyle ##1}
|
|
}
|
|
|
|
\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=9mm}
|
|
]
|
|
(\tikzcdmatrixname-4-19.north east)
|
|
--
|
|
(\tikzcdmatrixname-5-19.south east)
|
|
node[midway,right,xshift=14mm]{$\text{SE}_1$};
|
|
\draw [
|
|
line width=1pt,
|
|
decorate,
|
|
decoration={brace,amplitude=3mm,raise=9mm}
|
|
]
|
|
(\tikzcdmatrixname-6-19.north east)
|
|
--
|
|
(\tikzcdmatrixname-7-19.south east)
|
|
node[midway,right,xshift=14mm]{$\text{SE}_2$};
|
|
\draw [
|
|
line width=1pt,
|
|
decorate,
|
|
decoration={brace,amplitude=3mm,raise=9mm}
|
|
]
|
|
(\tikzcdmatrixname-8-19.north east)
|
|
--
|
|
(\tikzcdmatrixname-9-19.south east)
|
|
node[midway,right,xshift=14mm]{$\text{SE}_3$};
|
|
}
|
|
]
|
|
% tex-fmt: off
|
|
& \preperr{E_0} & \ctrl{3} & & & & & & \ctrl{5} & & & & & & \ctrl{7} & & & & & \\
|
|
\lstick{$\ket{\psi}$} & \preperr{E_1} & & \ctrl{2} & \ctrl{3} & & & & & \ctrl{4} & \ctrl{5} & & & & & \ctrl{6} & \ctrl{7} & & & \\
|
|
& \preperr{E_2} & & & & \ctrl{2} & & & & & & \ctrl{4} & & & & & & \ctrl{6} & & \\
|
|
\lstick{$\ket{0}_{\text{A}_1}$} & & \targ{} & \targ{} & & & & & & & & & & & & & & & \meter{} & \setwiretype{c} \\
|
|
\lstick{$\ket{0}_{\text{A}_2}$} & & & & \targ{} & \targ{} & & & & & & & & & & & & & \meter{} & \setwiretype{c} \\
|
|
& & & & & & \lstick{$\ket{0}_{\text{A}_3}$} & \setwiretype{q} & \targ{} & \targ{} & & & & & & & & & \meter{} & \setwiretype{c} \\
|
|
& & & & & & \lstick{$\ket{0}_{\text{A}_4}$} & \setwiretype{q} & & & \targ{} & \targ{} & & & & & & & \meter{} & \setwiretype{c} \\
|
|
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_5}$} & \setwiretype{q} & \targ{} & \targ{} & & & \meter{} & \setwiretype{c} \\
|
|
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_6}$} & \setwiretype{q} & & & \targ{} & \targ{} & \meter{} & \setwiretype{c}
|
|
% tex-fmt: on
|
|
\end{quantikz}
|
|
\end{figure}
|
|
\end{minipage}
|
|
}
|
|
\only<2>{
|
|
\begin{minipage}{0.4\textwidth}
|
|
\centering
|
|
|
|
\vspace*{40mm}
|
|
\begin{tikzpicture}
|
|
\node{$%
|
|
\bm{\Omega} =
|
|
\left(
|
|
\begin{array}{>{\columncolor{red!20}}ccc}
|
|
1 & 1 & 0 \\
|
|
0 & 1 & 1 \\
|
|
1 & 1 & 0 \\
|
|
0 & 1 & 1 \\
|
|
1 & 1 & 0 \\
|
|
0 & 1 & 1
|
|
\end{array}
|
|
\right)$
|
|
};
|
|
|
|
\draw [
|
|
line width=1pt,
|
|
decorate,
|
|
decoration={brace,mirror,amplitude=3mm,raise=5mm}
|
|
]
|
|
(2.5,1.2) -- (2.5,2.85)
|
|
node[midway,right,xshift=10mm]{$\text{SE}_1$};
|
|
|
|
\draw [
|
|
line width=1pt,
|
|
decorate,
|
|
decoration={brace,mirror,amplitude=3mm,raise=5mm}
|
|
]
|
|
(2.5,-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}
|
|
]
|
|
(2.5,-2.7) -- (2.5,-1.1)
|
|
node[midway,right,xshift=10mm]{$\text{SE}_3$};
|
|
\end{tikzpicture}
|
|
|
|
\vspace*{-10mm}
|
|
|
|
\begin{gather*}
|
|
\bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\}
|
|
\end{gather*}
|
|
\end{minipage}%
|
|
\begin{minipage}{0.6\textwidth}
|
|
\begin{figure}[H]
|
|
\newcommand{\preperr}[1]{
|
|
\gate[style={fill=orange!20}]{\scriptstyle ##1}
|
|
}
|
|
\newcommand{\measerr}{\gate[style={fill=blue!20}]{\phantom{1}}}
|
|
|
|
\newcommand{\noise}{
|
|
\gate[style={noisy}]{\text{\small X}}%
|
|
\setwiretype{n}%
|
|
\wire[l][1]{q}
|
|
}
|
|
\newcommand{\redwire}[1]{
|
|
\wire[r][##1][style={draw=red, line width=2pt}]{q}
|
|
}
|
|
\newcommand{\redtarg}{
|
|
\targ[style={draw=red}]{}%
|
|
\setwiretype{n}%
|
|
\wire[l][1]{q}
|
|
}
|
|
\newcommand{\redctrl}[1]{
|
|
\ctrl[style={draw=red,fill=red,line width=2pt}]{##1}
|
|
}
|
|
\newcommand{\redmeter}{\meter[style={draw=red,fill=red!20}]{}}
|
|
|
|
\tikzset{
|
|
noisy/.style={
|
|
starburst,
|
|
starburst point height=2mm,
|
|
fill=red!25, draw=red!85!black,
|
|
line width=2pt,
|
|
inner xsep=-2pt, inner ysep=-2pt
|
|
},
|
|
}
|
|
|
|
\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=9mm}
|
|
]
|
|
(\tikzcdmatrixname-4-19.north east)
|
|
--
|
|
(\tikzcdmatrixname-5-19.south east)
|
|
node[midway,right,xshift=14mm]{$\text{SE}_1$};
|
|
\draw [
|
|
line width=1pt,
|
|
decorate,
|
|
decoration={brace,amplitude=3mm,raise=9mm}
|
|
]
|
|
(\tikzcdmatrixname-6-19.north east)
|
|
--
|
|
(\tikzcdmatrixname-7-19.south east)
|
|
node[midway,right,xshift=14mm]{$\text{SE}_2$};
|
|
\draw [
|
|
line width=1pt,
|
|
decorate,
|
|
decoration={brace,amplitude=3mm,raise=9mm}
|
|
]
|
|
(\tikzcdmatrixname-8-19.north east)
|
|
--
|
|
(\tikzcdmatrixname-9-19.south east)
|
|
node[midway,right,xshift=14mm]{$\text{SE}_3$};
|
|
}
|
|
]
|
|
% tex-fmt: off
|
|
& \noise\redwire{18} & \redctrl{3} & & & & & & \redctrl{5} & & & & & & \redctrl{7} & & & & & \\
|
|
\lstick{$\ket{\psi}$} & \preperr{E_1} & & \ctrl{2} & \ctrl{3} & & & & & \ctrl{4} & \ctrl{5} & & & & & \ctrl{6} & \ctrl{7} & & & \\
|
|
& \preperr{E_2} & & & & \ctrl{2} & & & & & & \ctrl{4} & & & & & & \ctrl{6} & & \\
|
|
\lstick{$\ket{0}_{\text{A}_1}$} & & \redtarg{}\redwire{16} & \targ{} & & & & & & & & & & & & & & & \redmeter\wire[r][1][style={draw=red,double, line width=2pt}]{q} & \setwiretype{n} \\
|
|
\lstick{$\ket{0}_{\text{A}_2}$} & & & & \targ{} & \targ{} & & & & & & & & & & & & & \meter{} & \setwiretype{c} \\
|
|
& & & & & & \lstick{$\ket{0}_{\text{A}_3}$} & \setwiretype{q} & \redtarg\redwire{10} & \targ{} & & & & & & & & & \redmeter\wire[r][1][style={draw=red,double, line width=2pt}]{q} & \setwiretype{n} \\
|
|
& & & & & & \lstick{$\ket{0}_{\text{A}_4}$} & \setwiretype{q} & & & \targ{} & \targ{} & & & & & & & \meter{} & \setwiretype{c} \\
|
|
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_5}$} & \setwiretype{q} & \redtarg\redwire{4} & \targ{} & & & \redmeter\wire[r][1][style={draw=red,double, line width=2pt}]{q} & \setwiretype{n} \\
|
|
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_6}$} & \setwiretype{q} & & & \targ{} & \targ{} & \meter{} & \setwiretype{c}
|
|
% tex-fmt: on
|
|
\end{quantikz}
|
|
\end{figure}
|
|
\end{minipage}
|
|
}
|
|
|
|
\vspace*{8mm}
|
|
|
|
\addreferencesmanual
|
|
{DTB$^+$25}{
|
|
P.- J. H. S. Derks et al., ``Designing fault-tolerant
|
|
circuits using detector error models,'' \emph{Quantum}, 2025.
|
|
}
|
|
\stopreferencesmanual
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]
|
|
\frametitle{The Measurement Syndrome Matrix II}
|
|
|
|
\vspace*{-18mm}
|
|
|
|
\begin{itemize}
|
|
\item \schlagwort{Measurement syndrome matrix} $\bm{\Omega}$ \\
|
|
contains error patterns \citereferencemanual{DTB$^+$25}\
|
|
\item Example: 3-qubit repetition code
|
|
\end{itemize}
|
|
|
|
\vspace*{-29mm}
|
|
|
|
\centering
|
|
\only<1>{
|
|
\begin{minipage}{0.4\textwidth}
|
|
\centering
|
|
|
|
\vspace*{40mm}
|
|
\hspace*{-75mm}
|
|
\scalebox{0.85}{
|
|
\parbox{.5\linewidth}{%
|
|
\vspace*{22.6mm}
|
|
\begin{gather*}
|
|
\bm{\Omega} =
|
|
\left(
|
|
\begin{array}{ccccccccccccccc}
|
|
1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0
|
|
& 0 & 0 & 0 & 0 & 0 \\
|
|
0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0
|
|
& 0 & 0 & 0 & 0 & 0 \\
|
|
1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0
|
|
& 0 & 0 & 0 & 0 & 0 \\
|
|
0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1
|
|
& 0 & 0 & 0 & 0 & 0 \\
|
|
1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0
|
|
& 1 & 1 & 0 & 1 & 0 \\
|
|
0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0
|
|
& 0 & 1 & 1 & 0 & 1
|
|
\end{array}
|
|
\right) \\[10mm]
|
|
\hspace*{50mm} %
|
|
\bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\}
|
|
\end{gather*}
|
|
}
|
|
}
|
|
\end{minipage}%
|
|
\begin{minipage}{0.6\textwidth}
|
|
\begin{figure}[H]
|
|
\newcommand{\preperr}[1]{
|
|
\gate[style={fill=orange!20}]{\scriptstyle ##1}
|
|
}
|
|
\newcommand{\measerr}{\gate[style={fill=blue!20}]{\phantom{1}}}
|
|
|
|
\newcommand{\noise}{
|
|
\gate[style={noisy}]{\text{\small X}}%
|
|
\setwiretype{n}%
|
|
\wire[l][1]{q}
|
|
}
|
|
\newcommand{\redwire}[1]{
|
|
\wire[r][##1][style={draw=red, line width=2pt}]{q}
|
|
}
|
|
\newcommand{\redtarg}{
|
|
\targ[style={draw=red}]{}%
|
|
\setwiretype{n}%
|
|
\wire[l][1]{q}
|
|
}
|
|
\newcommand{\redctrl}[1]{
|
|
\ctrl[style={draw=red,fill=red,line width=2pt}]{##1}
|
|
}
|
|
\newcommand{\redmeter}{\meter[style={draw=red,fill=red!20}]{}}
|
|
|
|
\tikzset{
|
|
noisy/.style={
|
|
starburst,
|
|
starburst point height=2.5mm,
|
|
fill=red!25, draw=red!85!black,
|
|
line width=2pt,
|
|
inner xsep=-2pt, inner ysep=-2pt
|
|
},
|
|
}
|
|
|
|
\centering
|
|
|
|
% 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} & & & & \preperr{E_5} & & \ctrl{5} & & & & \preperr{E_{10}} & & \ctrl{7} & & & & & & \\
|
|
\lstick{$\ket{\psi}$} & \preperr{E_1} & & \ctrl{2} & \ctrl{3} & & \preperr{E_6} & & & \ctrl{4} & \ctrl{5} & & \preperr{E_{11}} & & & \ctrl{6} & \ctrl{7} & & & & \\
|
|
& \preperr{E_2} & & & & \ctrl{2} & \preperr{E_7} & & & & & \ctrl{4} & \preperr{E_{12}} & & & & & \ctrl{6} & & & \\
|
|
\lstick{$\ket{0}_{\text{A}_1}$} & & \targ{} & \targ{} & & & & & & & & & & & & & & & \preperr{E_3} & \meter{} & \setwiretype{c} \\
|
|
\lstick{$\ket{0}_{\text{A}_2}$} & & & & \targ{} & \targ{} & & & & & & & & & & & & & \preperr{E_4} & \meter{} & \setwiretype{c} \\
|
|
& & & & & & \lstick{$\ket{0}_{\text{A}_3}$} & \setwiretype{q} & \targ{} & \targ{} & & & & & & & & & \preperr{E_8} & \meter{} & \setwiretype{c} \\
|
|
& & & & & & \lstick{$\ket{0}_{\text{A}_4}$} & \setwiretype{q} & & & \targ{} & \targ{} & & & & & & & \preperr{E_9} & \meter{} & \setwiretype{c} \\
|
|
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_5}$} & \setwiretype{q} & \targ{} & \targ{} & & & \preperr{E_{13}} & \meter{} & \setwiretype{c} \\
|
|
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_6}$} & \setwiretype{q} & & & \targ{} & \targ{} & \preperr{E_{14}} & \meter{} & \setwiretype{c}
|
|
\end{quantikz}
|
|
% tex-fmt: on
|
|
\end{figure}
|
|
\end{minipage}
|
|
}
|
|
\only<2>{
|
|
\begin{minipage}{0.4\textwidth}
|
|
\centering
|
|
|
|
\vspace*{40mm}
|
|
\hspace*{-75mm}
|
|
\scalebox{0.85}{
|
|
\parbox{.5\linewidth}{%
|
|
\begin{gather*}
|
|
\hspace*{58.25mm}%
|
|
\begin{array}{c}
|
|
E_5 \\
|
|
\downarrow
|
|
\end{array}
|
|
\end{gather*}
|
|
\vspace*{-10mm}
|
|
\begin{gather*}
|
|
\bm{\Omega} =
|
|
\left(
|
|
\begin{array}{
|
|
ccccc%
|
|
>{\columncolor{red!20}}c%
|
|
ccccccccc
|
|
}
|
|
1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0
|
|
& 0 & 0 & 0 & 0 & 0 \\
|
|
0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0
|
|
& 0 & 0 & 0 & 0 & 0 \\
|
|
1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0
|
|
& 0 & 0 & 0 & 0 & 0 \\
|
|
0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1
|
|
& 0 & 0 & 0 & 0 & 0 \\
|
|
1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0
|
|
& 1 & 1 & 0 & 1 & 0 \\
|
|
0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0
|
|
& 0 & 1 & 1 & 0 & 1
|
|
\end{array}
|
|
\right) \\[10mm]
|
|
\hspace*{50mm} %
|
|
\bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\}
|
|
\end{gather*}
|
|
}
|
|
}
|
|
\end{minipage}%
|
|
\begin{minipage}{0.6\textwidth}
|
|
\begin{figure}[H]
|
|
\newcommand{\preperr}[1]{
|
|
\gate[style={fill=orange!20}]{\scriptstyle ##1}
|
|
}
|
|
\newcommand{\measerr}{\gate[style={fill=blue!20}]{\phantom{1}}}
|
|
|
|
\newcommand{\noise}{
|
|
\gate[style={noisy}]{\text{\small X}}%
|
|
\setwiretype{n}%
|
|
\wire[l][1]{q}
|
|
}
|
|
\newcommand{\redwire}[1]{
|
|
\wire[r][##1][style={draw=red, line width=2pt}]{q}
|
|
}
|
|
\newcommand{\redtarg}{
|
|
\targ[style={draw=red}]{}%
|
|
\setwiretype{n}%
|
|
\wire[l][1]{q}
|
|
}
|
|
\newcommand{\redctrl}[1]{
|
|
\ctrl[style={draw=red,fill=red,line width=2pt}]{##1}
|
|
}
|
|
\newcommand{\redmeter}{\meter[style={draw=red,fill=red!20}]{}}
|
|
|
|
\tikzset{
|
|
noisy/.style={
|
|
starburst,
|
|
starburst point height=2.5mm,
|
|
fill=red!25, draw=red!85!black,
|
|
line width=2pt,
|
|
inner xsep=-2pt, inner ysep=-2pt
|
|
},
|
|
}
|
|
|
|
\centering
|
|
|
|
% 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} & & & & \noise\redwire{14} & & \redctrl{5} & & & & \preperr{E_{10}} & & \redctrl{7} & & & & & & \\
|
|
\lstick{$\ket{\psi}$} & \preperr{E_1} & & \ctrl{2} & \ctrl{3} & & \preperr{E_6} & & & \ctrl{4} & \ctrl{5} & & \preperr{E_{11}} & & & \ctrl{6} & \ctrl{7} & & & & \\
|
|
& \preperr{E_2} & & & & \ctrl{2} & \preperr{E_7} & & & & & \ctrl{4} & \preperr{E_{12}} & & & & & \ctrl{6} & & & \\
|
|
\lstick{$\ket{0}_{\text{A}_1}$} & & \targ{} & \targ{} & & & & & & & & & & & & & & & \preperr{E_3} & \meter{} & \setwiretype{c} \\
|
|
\lstick{$\ket{0}_{\text{A}_2}$} & & & & \targ{} & \targ{} & & & & & & & & & & & & & \preperr{E_4} & \meter{} & \setwiretype{c} \\
|
|
& & & & & & \lstick{$\ket{0}_{\text{A}_3}$} & \setwiretype{q} & \redtarg\redwire{11} & \targ{} & & & & & & & & & \preperr{E_8} & \redmeter\wire[r][1][style={draw=red,double, line width=2pt}]{q} & \setwiretype{n} \\
|
|
& & & & & & \lstick{$\ket{0}_{\text{A}_4}$} & \setwiretype{q} & & & \targ{} & \targ{} & & & & & & & \preperr{E_9} & \meter{} & \setwiretype{c} \\
|
|
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_5}$} & \setwiretype{q} & \redtarg\redwire{5} & \targ{} & & & \preperr{E_{13}} & \redmeter\wire[r][1][style={draw=red,double, line width=2pt}]{q} & \setwiretype{n} \\
|
|
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_6}$} & \setwiretype{q} & & & \targ{} & \targ{} & \preperr{E_{14}} & \meter{} & \setwiretype{c}
|
|
\end{quantikz}
|
|
% tex-fmt: on
|
|
\end{figure}
|
|
\end{minipage}
|
|
}
|
|
\only<3>{
|
|
\begin{minipage}{0.4\textwidth}
|
|
\centering
|
|
|
|
\vspace*{40mm}
|
|
\hspace*{-75mm}
|
|
\scalebox{0.85}{
|
|
\parbox{.5\linewidth}{%
|
|
\begin{gather*}
|
|
\hspace*{65.5mm}%
|
|
\begin{array}{c}
|
|
E_6 \\
|
|
\downarrow
|
|
\end{array}
|
|
\end{gather*}
|
|
\vspace*{-10mm}
|
|
\begin{gather*}
|
|
\bm{\Omega} =
|
|
\left(
|
|
\begin{array}{
|
|
cccccc%
|
|
>{\columncolor{red!20}}c%
|
|
cccccccc
|
|
}
|
|
1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0
|
|
& 0 & 0 & 0 & 0 & 0 \\
|
|
0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0
|
|
& 0 & 0 & 0 & 0 & 0 \\
|
|
1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0
|
|
& 0 & 0 & 0 & 0 & 0 \\
|
|
0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1
|
|
& 0 & 0 & 0 & 0 & 0 \\
|
|
1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0
|
|
& 1 & 1 & 0 & 1 & 0 \\
|
|
0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0
|
|
& 0 & 1 & 1 & 0 & 1
|
|
\end{array}
|
|
\right) \\[10mm]
|
|
\hspace*{50mm} %
|
|
\bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\}
|
|
\end{gather*}
|
|
}
|
|
}
|
|
\end{minipage}%
|
|
\begin{minipage}{0.6\textwidth}
|
|
\begin{figure}[H]
|
|
\newcommand{\preperr}[1]{
|
|
\gate[style={fill=orange!20}]{\scriptstyle ##1}
|
|
}
|
|
\newcommand{\measerr}{\gate[style={fill=blue!20}]{\phantom{1}}}
|
|
|
|
\newcommand{\noise}{
|
|
\gate[style={noisy}]{\text{\small X}}%
|
|
\setwiretype{n}%
|
|
\wire[l][1]{q}
|
|
}
|
|
\newcommand{\redwire}[1]{
|
|
\wire[r][##1][style={draw=red, line width=2pt}]{q}
|
|
}
|
|
\newcommand{\redtarg}{
|
|
\targ[style={draw=red}]{}%
|
|
\setwiretype{n}%
|
|
\wire[l][1]{q}
|
|
}
|
|
\newcommand{\redctrl}[1]{
|
|
\ctrl[style={draw=red,fill=red,line width=2pt}]{##1}
|
|
}
|
|
\newcommand{\redmeter}{\meter[style={draw=red,fill=red!20}]{}}
|
|
|
|
\tikzset{
|
|
noisy/.style={
|
|
starburst,
|
|
starburst point height=2.5mm,
|
|
fill=red!25, draw=red!85!black,
|
|
line width=2pt,
|
|
inner xsep=-2pt, inner ysep=-2pt
|
|
},
|
|
}
|
|
|
|
\centering
|
|
|
|
% 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} & & & & \preperr{E_5} & & \ctrl{5} & & & & \preperr{E_{10}} & & \ctrl{7} & & & & & & \\
|
|
\lstick{$\ket{\psi}$} & \preperr{E_1} & & \ctrl{2} & \ctrl{3} & & \noise\redwire{14} & & & \redctrl{4} & \redctrl{5} & & \preperr{E_{11}} & & & \redctrl{6} & \redctrl{7} & & & & \\
|
|
& \preperr{E_2} & & & & \ctrl{2} & \preperr{E_7} & & & & & \ctrl{4} & \preperr{E_{12}} & & & & & \ctrl{6} & & & \\
|
|
\lstick{$\ket{0}_{\text{A}_1}$} & & \targ{} & \targ{} & & & & & & & & & & & & & & & \preperr{E_3} & \meter{} & \setwiretype{c} \\
|
|
\lstick{$\ket{0}_{\text{A}_2}$} & & & & \targ{} & \targ{} & & & & & & & & & & & & & \preperr{E_4} & \meter{} & \setwiretype{c} \\
|
|
& & & & & & \lstick{$\ket{0}_{\text{A}_3}$} & \setwiretype{q} & \targ{} & \redtarg\redwire{10} & & & & & & & & & \preperr{E_8} & \redmeter\wire[r][1][style={draw=red,double, line width=2pt}]{q} & \setwiretype{n} \\
|
|
& & & & & & \lstick{$\ket{0}_{\text{A}_4}$} & \setwiretype{q} & & & \redtarg\redwire{9} & \targ{} & & & & & & & \preperr{E_9} & \redmeter\wire[r][1][style={draw=red,double, line width=2pt}]{q} & \setwiretype{n} \\
|
|
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_5}$} & \setwiretype{q} & \targ{} & \redtarg\redwire{4} & & & \preperr{E_{13}} & \redmeter\wire[r][1][style={draw=red,double, line width=2pt}]{q} & \setwiretype{n} \\
|
|
& & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_6}$} & \setwiretype{q} & & & \redtarg\redwire{3} & \targ{} & \preperr{E_{14}} & \redmeter\wire[r][1][style={draw=red,double, line width=2pt}]{q} & \setwiretype{n}
|
|
\end{quantikz}
|
|
% tex-fmt: on
|
|
\end{figure}
|
|
\end{minipage}
|
|
}
|
|
|
|
\vspace*{4mm}
|
|
|
|
\addreferencesmanual
|
|
{DTB$^+$25}{
|
|
P.- J. H. S. Derks et al., ``Designing fault-tolerant
|
|
circuits using detector error models,'' \emph{Quantum}, 2025.
|
|
}
|
|
\stopreferencesmanual
|
|
\end{frame}
|
|
|
|
% TODO: Journal not showing for derks_designing_2025
|
|
\begin{frame}[fragile]
|
|
\frametitle{The Detector Error Matrix I}
|
|
|
|
\vspace*{-14mm}
|
|
|
|
\begin{itemize}
|
|
\item Repetition of errors mitigated by XORing measurements
|
|
\end{itemize}
|
|
|
|
\begin{minipage}{0.4\textwidth}
|
|
\begin{figure}[H]
|
|
\newcommand{\redwire}[1]{
|
|
\wire[r][#1][style={draw=red, line width=2pt, double}]{q}
|
|
}
|
|
\newcommand{\inwire}{
|
|
\wire[l][1][style={draw=red, line width=2pt}]{q}
|
|
}
|
|
\newcommand{\redtarg}{
|
|
\targ[style={draw=red,line width=2pt}]{}%
|
|
\setwiretype{n}%
|
|
}
|
|
\newcommand{\redctrl}[1]{
|
|
\ctrl[style={draw=red,fill=red, line width=2pt}]{0}%
|
|
\wire[d][#1][style={draw=red, line width=2pt, double}]{q}
|
|
}
|
|
\newcommand{\redmeter}{\meter[style={draw=red,fill=red!20}]{}}
|
|
\newcommand{\redgate}[1]{\gate[style={draw=red,fill=red!20}]{\textcolor{red}{#1}}}
|
|
|
|
\centering
|
|
|
|
\only<1>{
|
|
\hspace*{-46mm}
|
|
% tex-fmt: off
|
|
\begin{quantikz}[row sep=4mm, column sep=4mm, wire types={n,n,n,n,n,n}]
|
|
& \meter{}\wire[l][1]{q}\wire[r][1]{c} & \\
|
|
& \meter{}\wire[l][1]{q}\wire[r][1]{c} & \\
|
|
& \redmeter{}\inwire\redwire{1} & \\
|
|
& \redmeter{}\inwire\redwire{1} & \\
|
|
& \redmeter{}\inwire\redwire{1} & \\
|
|
& \redmeter{}\inwire\redwire{1} &
|
|
\end{quantikz}
|
|
% tex-fmt: on
|
|
}
|
|
\only<2->{
|
|
% tex-fmt: off
|
|
\begin{quantikz}[row sep=4mm, column sep=4mm, wire types={n,n,n,n,n,n}]
|
|
& \meter{}\wire[l][1]{q}\wire[r][1]{c} & \setwiretype{c} & & & \ctrl[vertical wire=c]{2} & & \gate{D_1} \\
|
|
& \meter{}\wire[l][1]{q}\wire[r][1]{c} & \setwiretype{c} & & & & \ctrl[vertical wire=c]{2} & \gate{D_2} \\
|
|
& \redmeter{}\inwire\redwire{6} & & \redctrl{2} & & \targ{} & & \redgate{D_3} \\
|
|
& \redmeter{}\inwire\redwire{6} & & & \redctrl{2} & & \targ{} & \redgate{D_4} \\
|
|
& \redmeter{}\inwire\redwire{2} & & \redtarg\wire[r][4]{c} & & & & \gate{D_5} \\
|
|
& \redmeter{}\inwire\redwire{3} & & & \redtarg\wire[r][3]{c} & & & \gate{D_6}
|
|
\end{quantikz}
|
|
% tex-fmt: on
|
|
}
|
|
\end{figure}
|
|
\end{minipage}%
|
|
\begin{minipage}{0.6\textwidth}
|
|
\newcommand\cc{\cellcolor{blue!20}}
|
|
|
|
\visible<3->{
|
|
\begin{align*}
|
|
\bm{H} =
|
|
% tex-fmt: off
|
|
\left(\begin{array}{ccccccccccccccc}
|
|
1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
|
|
0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
|
|
\cc{0} & \cc{0} & \cc{0} & \cc{1} & \cc{0} & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
|
|
\cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{1} & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
|
|
\cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{1} & \cc{0} & 1 & 1 & 0 & 1 & 0 \\
|
|
\cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{0} & \cc{1} & 0 & 1 & 1 & 0 & 1
|
|
\end{array}\right)
|
|
% tex-fmt: on
|
|
\end{align*}
|
|
}
|
|
\end{minipage}
|
|
|
|
\vspace*{5mm}
|
|
|
|
\visible<3->{
|
|
\begin{itemize}
|
|
\item A detector is a parity constraint on a set of
|
|
measurement outcomes
|
|
\item The \schlagwort{detector error matrix} $\bm{H}$ contains
|
|
modified error patterns \citereferencemanual{DTB$^+$25}\
|
|
\end{itemize}
|
|
}
|
|
|
|
\vspace*{10mm}
|
|
|
|
\addreferencesmanual
|
|
{DTB$^+$25}{
|
|
P.- J. H. S. Derks et al., ``Designing fault-tolerant
|
|
circuits using detector error models,'' \emph{Quantum}, 2025.
|
|
}
|
|
\stopreferencesmanual
|
|
\end{frame}
|
|
|
|
\begin{frame}
|
|
\frametitle{The Detector Error Matrix II}
|
|
|
|
\vspace*{-17mm}
|
|
|
|
\begin{itemize}
|
|
\item Visualization of general process
|
|
% \red{Deal with 3-qubit state (somehow represent arbitrary
|
|
% qubit state)}
|
|
\end{itemize}
|
|
|
|
\vspace*{2mm}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\tikzset{
|
|
gate/.style={
|
|
draw, %line width=1pt,
|
|
minimum height=2cm,
|
|
}
|
|
}
|
|
|
|
% tex-fmt: off
|
|
\begin{quantikz}[row sep=2mm, column sep=4mm, wire types={q,q,q,n,n,n}]
|
|
& \gate[3]{\text{SE}_1} & & \gate[3]{\text{SE}_2} & & \gate[3]{\text{SE}_3} & & \gate[3]{\text{SE}_4} & \\
|
|
\lstick{$\ket{\psi}$} & & & & & & & & & \setwiretype{n} & \ldots \\
|
|
& \wire[d][3]{c} & & \wire[d][1]{c} & & \wire[d][1]{c} & & \wire[d][1]{c} & \\
|
|
& \ctrl[wire=c]{0}\wire[r][1]{c} & \wire[d][1]{c} & \ctrl[vertical wire=c]{1}\wire[r][1]{c} & \wire[d][1]{c} & \ctrl[vertical wire=c]{1}\wire[r][1]{c} & \wire[d][1]{c} & \ctrl[vertical wire=c]{1}\wire[r][1]{c} & \\
|
|
& & \wire[r][1]{c} & \targ{}\wire[d][1]{c} & \wire[r][1]{c} & \targ{}\wire[d][1]{c} & \wire[r][1]{c} & \targ{}\wire[d][1]{c} & \\
|
|
& \gate[1]{\bm{D}_1} & & \gate[1]{\bm{D}_2} & & \gate[1]{\bm{D}_3} & & \gate[1]{\bm{D}_4} & \\
|
|
\end{quantikz}
|
|
% tex-fmt: on
|
|
\end{figure}
|
|
|
|
\begin{itemize}
|
|
\item E.g., for \ac{bb} codes under circuit-level noise
|
|
\citereferencemanual{GCR24}
|
|
\end{itemize}
|
|
|
|
\vspace*{-4mm}
|
|
|
|
\begin{align*}
|
|
\bm{H} =
|
|
\begin{pmatrix}
|
|
\bm{H}_0 & \bm{H}_1 & & & & & \\
|
|
& \bm{H}_2 & \bm{H}_0 & \bm{H}_1 & & & \\
|
|
& & & \bm{H}_2 & \bm{H}_0 & \bm{H}_1 & \\
|
|
& & & & & \bm{H}_2 & \\
|
|
& & & & & & & \ddots
|
|
\end{pmatrix}
|
|
\end{align*}
|
|
|
|
\vspace*{5mm}
|
|
|
|
\addreferencesmanual
|
|
{GCR24}{A. Gong, S. Cammerer, and J. M. Renes, ``Toward
|
|
Low-latency Iterative Decoding of QLDPC Codes Under
|
|
Circuit-Level Noise,'', 2024.
|
|
}
|
|
\stopreferencesmanual
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]
|
|
\frametitle{Noise Model}
|
|
|
|
% Related interesting stuff
|
|
% - The difference between an n-qubit error and multiple
|
|
% simultaneous single-qubit errors is that in the n-qubit case,
|
|
% the errors can be correlated (e.g., XX more probable than XI)
|
|
% - There is also work on using soft information at the
|
|
% measurement outputs (may translate to not-just-X-errors at the
|
|
% measurements)
|
|
|
|
\vspace*{-12mm}
|
|
|
|
\begin{itemize}
|
|
\item The \schlagwort{noise model} assigns probabilities to
|
|
error locations
|
|
\end{itemize}
|
|
|
|
\vspace*{1mm}
|
|
|
|
\begin{minipage}{0.60\textwidth}
|
|
\begin{itemize}
|
|
\item Noise model types
|
|
\begin{itemize}
|
|
\visible<1->{
|
|
\item The \schlagwort{depolarizing channel} considers
|
|
\citereferencemanual{NC10}
|
|
\begin{itemize}
|
|
\item $\X$, $\Y$ or $\Z$ errors on
|
|
the data qubits
|
|
\end{itemize}
|
|
}
|
|
\visible<2->{
|
|
\item \schlagwort{Phenomenological noise} considers
|
|
\citereferencemanual{DTB$^+$25}
|
|
\begin{itemize}
|
|
\item $\X$ errors on data qubits before each \\
|
|
measurement round
|
|
\item $\X$ errors on measurement outcomes
|
|
\end{itemize}
|
|
}
|
|
\visible<3->{
|
|
\item \schlagwort{Circuit-level noise} considers
|
|
\citereferencemanual{DTB$^+$25}
|
|
\begin{itemize}
|
|
\item $\X$, $\Y$ or $\Z$ errors after
|
|
state preparation
|
|
\item $n$-qubit $\X$, $\Y$ or $\Z$ errors
|
|
after any $n$-qubit gate
|
|
\item $\X$ errors on measurement outcomes
|
|
\end{itemize}
|
|
}
|
|
\end{itemize}
|
|
\end{itemize}
|
|
\end{minipage}%
|
|
\hfill%
|
|
\begin{minipage}{0.39\textwidth}
|
|
\begin{figure}[H]
|
|
\newcommand{\xerr}{\gate[style={fill=kit-blue!50}]{\phantom{1}}}
|
|
\newcommand{\xyzerr}{\gate[style={
|
|
draw=black,
|
|
fill=kit-red,
|
|
path picture={
|
|
\fill[kit-blue!60]
|
|
($(path picture bounding box.south west)+(0,0)$) --
|
|
($(path picture bounding box.north west)+(0,0)$) --
|
|
($(path picture bounding box.north
|
|
west)+(0.34,0)$) -- cycle;
|
|
\fill[kit-orange!60]
|
|
($(path picture bounding box.north east)+(0,0)$) --
|
|
($(path picture bounding box.south east)+(0,0)$) --
|
|
($(path picture bounding box.south
|
|
east)+(-0.34,0)$) -- cycle;
|
|
\fill[kit-red!60]
|
|
($(path picture bounding box.north east)+(0,0)$) --
|
|
($(path picture bounding box.south
|
|
east)+(-0.34,0)$) --
|
|
($(path picture bounding box.south west)+(0,0)$) --
|
|
($(path picture bounding box.north
|
|
west)+(0.34,0)$) -- cycle;
|
|
}
|
|
}]{\phantom{1}}}
|
|
|
|
\hspace*{-18mm}
|
|
\only<1>{
|
|
% tex-fmt: off
|
|
\begin{quantikz}[row sep=4mm, column sep=4mm]
|
|
& \xyzerr & \ctrl{3} & & & & & \\
|
|
\lstick{$\ket{\psi}$} & \xyzerr & & \ctrl{2} & \ctrl{3} & & & \\
|
|
& \xyzerr & & & & \ctrl{2} & & \\
|
|
\lstick{$\ket{0}_{\text{A}_1}$} & & \targ{} & \targ{} & & & \meter{} & \setwiretype{c} \\
|
|
\lstick{$\ket{0}_{\text{A}_2}$} & & & & \targ{} & \targ{} & \meter{} & \setwiretype{c}
|
|
\end{quantikz}
|
|
% tex-fmt: on
|
|
}
|
|
\only<2>{
|
|
% tex-fmt: off
|
|
\begin{quantikz}[row sep=4mm, column sep=4mm]
|
|
& \xerr & \ctrl{3} & & & & & & \\
|
|
\lstick{$\ket{\psi}$} & \xerr & & \ctrl{2} & \ctrl{3} & & & & \\
|
|
& \xerr & & & & \ctrl{2} & & & \\
|
|
\lstick{$\ket{0}_{\text{A}_1}$} & & \targ{} & \targ{} & & & \xerr & \meter{} & \setwiretype{c} \\
|
|
\lstick{$\ket{0}_{\text{A}_2}$} & & & & \targ{} & \targ{} & \xerr & \meter{} & \setwiretype{c}
|
|
\end{quantikz}
|
|
% tex-fmt: on
|
|
}
|
|
\only<3>{
|
|
% tex-fmt: off
|
|
\begin{quantikz}[row sep=4mm, column sep=2mm]
|
|
& \xyzerr & \ctrl{3} & \xyzerr \wire[d][3]{q} & & & & & & & & & \\
|
|
\lstick{$\ket{\psi}$} & \xyzerr & & & \ctrl{2} & \xyzerr \wire[d][2]{q} & \ctrl{3} & \xyzerr \wire[d][3]{q} & & & & & \\
|
|
& \xyzerr & & & & & & & \ctrl{2} & \xyzerr \wire[d][2]{q} & & & \\
|
|
\lstick{$\ket{0}_{\text{A}_1}$} & \xyzerr & \targ{} & \xyzerr & \targ{} & \xyzerr & & & & & \xerr & \meter{} & \setwiretype{c} \\
|
|
\lstick{$\ket{0}_{\text{A}_2}$} & \xyzerr & & & & & \targ{} & \xyzerr & \targ{} & \xyzerr & \xerr & \meter{} & \setwiretype{c}
|
|
\end{quantikz}
|
|
% tex-fmt: on
|
|
}
|
|
\end{figure}
|
|
\end{minipage}
|
|
|
|
\vspace*{12mm}
|
|
|
|
\addreferencesmanual
|
|
{NC10}{
|
|
M. A. Nielsen and I. L. Chuang, ``Quantum Computation and
|
|
Quantum Information'', \emph{Cambridge University Press}, 2010.
|
|
}
|
|
{DTB$^+$25}{
|
|
P.- J. H. S. Derks et al., ``Designing fault-tolerant
|
|
circuits using detector error models,'' \emph{Quantum}, 2025.
|
|
}
|
|
\stopreferencesmanual
|
|
\end{frame}
|
|
|
|
\begin{frame}
|
|
\frametitle{Decoding using Detector Error Models}
|
|
|
|
% Related interesting stuff
|
|
% - Roffe et al. use the min-sum variant of BP.
|
|
% - Babar et al. use the SPA
|
|
% - A lot of publications use Roffe's ldpc package -> min-sum
|
|
|
|
\vspace*{-10mm}
|
|
|
|
\begin{itemize}
|
|
\item A \schlagwort{\acl{dem}} (DEM) combines a detector error
|
|
matrix and a noise model
|
|
\visible<2->{
|
|
\item Tanner graph of detector error matrix of \ac{bb} code
|
|
\citereferencemanual{KSW$^+$25}
|
|
}
|
|
\end{itemize}
|
|
|
|
\vspace*{5mm}
|
|
|
|
\visible<2->{
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\includegraphics[scale=5,angle=90]{res/stergios_tanner_graph}
|
|
\end{figure}
|
|
}
|
|
|
|
\visible<3->{
|
|
\begin{itemize}
|
|
\item Challenges
|
|
\begin{itemize}
|
|
\item Repeated syndrome measurements lead to
|
|
increased decoding complexity
|
|
\citereferencemanual{GCR24}
|
|
\item Degeneracy and short cycles lead to degraded
|
|
performance of \ac{bp}
|
|
\citereferencemanual{BBA$^+$15}
|
|
\end{itemize}
|
|
\end{itemize}
|
|
}
|
|
|
|
\vspace*{15mm}
|
|
|
|
\addreferencesmanual
|
|
{KSW$^+$25}{
|
|
S. Koutsioumpas et al., ``Automorphism Ensemble Decoding of
|
|
Quantum LDPC Codes,'' \emph{arXiv:2503.01738}, 2025.
|
|
}
|
|
{GCR24}{
|
|
A. Gong, S. Cammerer, and J. M. Renes, ``Toward
|
|
Low-latency Iterative Decoding of QLDPC Codes Under
|
|
Circuit-Level Noise,'' 2024.
|
|
}
|
|
{BBA$^+$15}{
|
|
Z. Babar et al., ``Fifteen Years of
|
|
Quantum LDPC Coding and Improved Decoding Strategies,''
|
|
\emph{IEEE Access}, 2015.
|
|
}
|
|
\stopreferencesmanual
|
|
\end{frame}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Research Gap}
|
|
\label{sec:Research Gap}
|
|
|
|
%%%%%%%%%%%%%%%%
|
|
\subsection{State of the Art}
|
|
\label{subsec:State of the Art}
|
|
|
|
% TODO: Mention somewhere that we are particularly interested in QLDPC codes
|
|
\begin{frame}
|
|
\frametitle{Addressing the Challenges}
|
|
|
|
\vspace*{-14mm}
|
|
|
|
\begin{itemize}
|
|
\item Decoding complexity addressed with window-based approaches
|
|
\begin{itemize}
|
|
\item Parallel decoding \citereferencemanual{SBB$^+$23}
|
|
\item Sliding windows
|
|
\citereferencemanual{HP23}
|
|
\citereferencemanual{GCR24}
|
|
\end{itemize}
|
|
\visible<2>{
|
|
\item Degraded \ac{bp} performance addressed with
|
|
modification or extension
|
|
\begin{itemize}
|
|
\item \Ac{osd} post-processing
|
|
\citereferencemanual{RWB$^+$20}
|
|
\item Guided decimation \citereferencemanual{GCR24}
|
|
\item Neural approaches
|
|
\citereferencemanual{KL22}
|
|
\citereferencemanual{MSL$^+$25}
|
|
\item Ensemble decoding
|
|
\citereferencemanual{KSW$^+$25}
|
|
\end{itemize}
|
|
}
|
|
\end{itemize}
|
|
|
|
\vspace*{15mm}
|
|
|
|
\addreferencesmanual
|
|
{SBB$^+$23}{
|
|
L. Skoric et al., ``Parallel window decoding enables scalable
|
|
fault tolerant quantum computation,'' \emph{Nature
|
|
Communications}, 2023.
|
|
}
|
|
{HP23}{
|
|
S. Huang and S. Puri, ``Improved Noisy Syndrome Decoding of
|
|
Quantum LDPC Codes with Sliding Window,'' \emph{arXiv:2311.03307}, 2023.
|
|
}
|
|
{GCR24}{
|
|
A. Gong, S. Cammerer, and J. M. Renes, ``Toward
|
|
Low-latency Iterative Decoding of QLDPC Codes Under
|
|
Circuit-Level Noise,'' 2024.
|
|
}
|
|
{RWB$^+$20}{
|
|
J. Roffe, et al., ``Decoding across the quantum low-density
|
|
parity-check code landscape,'' \emph{Physical Review}, 2020.
|
|
}
|
|
{KL22}{
|
|
K.- Y. Kuo and C.- Y. Lai, ``Exploiting degeneracy in belief
|
|
propagation decoding of quantum codes,'' \emph{npj Quantum
|
|
Information}, 2022.
|
|
}
|
|
{MSL$^+$25}{
|
|
S. Miao et al., ``Quaternary Neural Belief Propagation
|
|
Decoding of Quantum LDPC Codes with Overcomplete
|
|
Check Matrices'', \emph{IEEE Access}, 2025.
|
|
}
|
|
{KSW$^+$25}{
|
|
S. Koutsioumpas et al., ``Automorphism Ensemble Decoding of
|
|
Quantum LDPC Codes,'' \emph{arXiv:2503.01738}, 2025.
|
|
}
|
|
\stopreferencesmanual
|
|
\end{frame}
|
|
|
|
% TODO: Understand update equation for s_2'
|
|
\begin{frame}[fragile]
|
|
\frametitle{Sliding-Window Decoding}
|
|
|
|
\vspace*{-12mm}
|
|
|
|
\begin{itemize}
|
|
\item Approach taken in \citereferencemanual{GCR24}
|
|
resembles \acf{scldpc} code
|
|
\item They try \ac{bp} + \ac{osd} and a modification of
|
|
\ac{bp} with guided decimation
|
|
\end{itemize}
|
|
|
|
\vspace*{5mm}
|
|
|
|
\begin{figure}
|
|
\begin{subfigure}[b]{0.5\textwidth}
|
|
\newcommand{\pz}{\phantom{\bm{0}}}
|
|
\[
|
|
\left(
|
|
\hspace*{-4mm}
|
|
\begin{tikzpicture}[baseline=(m.center)]
|
|
\matrix (m)[
|
|
matrix of math nodes,
|
|
nodes in empty cells,
|
|
column sep={14mm,between origins},
|
|
row sep={10mm,between origins},
|
|
] {
|
|
% tex-fmt: off
|
|
\bm{H}_0 & \bm{H}_1 & \pz & \pz & \pz & \pz & \pz & \pz \\
|
|
\pz & \bm{H}_2 & \bm{H}_0 & \bm{H}_1 & \pz & \pz & \pz & \pz & \pz \\
|
|
\pz & \pz & \pz & \bm{H}_2 & \bm{H}_0 & \bm{H}_1 & \pz & \pz & \pz \\
|
|
\pz & \pz & \pz & \pz & \pz & \bm{H}_2 & \bm{H}_0 & \bm{H}_1 & \pz \\
|
|
\pz & \pz & \pz & \pz & \pz & \pz & \pz & \pz & \ddots \\
|
|
% tex-fmt: on
|
|
} ;
|
|
|
|
\draw[kit-red, line width=2pt]
|
|
($(m-3-1.south west) + (-0.1,-0.1)$)
|
|
rectangle
|
|
($(m-1-6.north east) + (0.1,0.1)$);
|
|
|
|
\draw[kit-orange, line width=2pt]
|
|
($(m-4-3.south west) + (-0.1,-0.1)$)
|
|
rectangle
|
|
($(m-2-8.north east) + (0.1,0.1)$);
|
|
|
|
\draw[kit-blue, line width=2pt]
|
|
($(m-5-5.south west) + (-0.1,-0.1)$)
|
|
rectangle
|
|
($(m-3-9.north east) + (0.1,0.1)$);
|
|
|
|
\draw[-{Latex},line width=2pt]
|
|
($(m-1-6.north) + (0,0.8)$)
|
|
--
|
|
($(m-3-9.north) + (0.5,0.8)$);
|
|
\end{tikzpicture}
|
|
\hspace*{-2mm}
|
|
\right)
|
|
\]
|
|
|
|
\caption{Visualization of sliding window procedure}
|
|
\end{subfigure}%
|
|
\begin{subfigure}[b]{0.5\textwidth}
|
|
\begin{gather*}
|
|
\begin{pmatrix}
|
|
\bm{H}_0 & \bm{H}_1 & \bm{0} & \bm{0} &
|
|
\bm{0} & \bm{0} \\
|
|
\bm{0} & \bm{H}_2 & \bm{H}_0 & \bm{H}_1 &
|
|
\bm{0} & \bm{0} \\
|
|
\bm{0} & \bm{0} & \bm{0} & \bm{H}_2 &
|
|
\bm{H}_0 & \bm{H}_1
|
|
\end{pmatrix}
|
|
\begin{pmatrix}
|
|
\hat{\bm{e}}_0 \\
|
|
\vdots \\
|
|
\hat{\bm{e}}_5
|
|
\end{pmatrix}
|
|
=
|
|
\begin{pmatrix}
|
|
\bm{s}_1 \\
|
|
\bm{s}_2 \\
|
|
\bm{s}_3
|
|
\end{pmatrix} \\[5mm]
|
|
\bm{s}_2' = \bm{s}_2 + \bm{H}_2 \hat{\bm{e}}_1
|
|
\end{gather*}
|
|
|
|
\vspace*{2mm}
|
|
|
|
\caption{Equations for the decoding of the first window}
|
|
\end{subfigure}%
|
|
\end{figure}
|
|
|
|
\visible<2->{
|
|
\begin{itemize}
|
|
\item However, no passing of soft information between windows
|
|
\end{itemize}
|
|
}
|
|
|
|
\vspace*{10mm}
|
|
|
|
\addreferencesmanual
|
|
{GCR24}{
|
|
A. Gong, S. Cammerer, and J. M. Renes, ``Toward
|
|
Low-latency Iterative Decoding of QLDPC Codes Under
|
|
Circuit-Level Noise,'' 2024.
|
|
}
|
|
\stopreferencesmanual
|
|
\end{frame}
|
|
|
|
%%%%%%%%%%%%%%%%
|
|
\subsection{Future Work}
|
|
\label{subsec:Future Work}
|
|
|
|
\begin{frame}
|
|
\frametitle{Future Work}
|
|
|
|
\vspace*{-15mm}
|
|
|
|
\begin{itemize}
|
|
\item Completed work
|
|
\begin{itemize}
|
|
\item Review literature on fault-tolerant \ac{qec}
|
|
using \acp{dem}
|
|
\item Identify research gap
|
|
\item Familiarize with software toolboxes
|
|
\end{itemize}
|
|
\vspace*{7mm}
|
|
\item Research gap
|
|
\begin{itemize}
|
|
\item Existing literature into circuit-level noise
|
|
fails to properly consider \ac{scldpc}-like
|
|
structure
|
|
\end{itemize}
|
|
\vspace*{7mm}
|
|
\visible<2->{
|
|
\item Future directions
|
|
\begin{itemize}
|
|
\item Adapt modified guided decimation decoder from
|
|
\citereferencemanual{GCR24} to pass soft
|
|
information
|
|
\item Investigate performance of different
|
|
modifications of \ac{bp} for "inner decoder"
|
|
(e.g., quaternary neural \ac{bp}
|
|
\citereferencemanual{MSL$^+$25})
|
|
\item \ldots
|
|
\end{itemize}
|
|
}
|
|
\end{itemize}
|
|
|
|
\vspace*{10mm}
|
|
|
|
\addreferencesmanual
|
|
{GCR24}{
|
|
A. Gong, S. Cammerer, and J. M. Renes, ``Toward
|
|
Low-latency Iterative Decoding of QLDPC Codes Under
|
|
Circuit-Level Noise,'' 2024.
|
|
}
|
|
{MSL$^+$25}{
|
|
S. Miao et al., ``Quaternary Neural Belief Propagation
|
|
Decoding of Quantum LDPC Codes with Overcomplete
|
|
Check Matrices'', \emph{IEEE Access}, 2025.
|
|
}
|
|
\stopreferencesmanual
|
|
\end{frame}
|
|
|
|
% TODO: Organize sections properly
|
|
%%%%%%%%%%%%%%%%
|
|
\section{Remarks on Evaluation}
|
|
\label{sec:Remarks on Evaluation}
|
|
|
|
%%%%%%%%%%%%%%%%
|
|
\subsection{Figures of Merit}
|
|
\label{subsec:Figures of Merit}
|
|
|
|
\begin{frame}
|
|
\frametitle{Performance Evaluation}
|
|
|
|
% - Gong et al. don't actually analyze the latency ->
|
|
% Benchmarking against other methods would be interesting
|
|
|
|
% \item For circuit-level noise, use same
|
|
% \schlagwort{physical error rate} for all error
|
|
% locations \citereferencemanual{FSG09}
|
|
|
|
\vspace*{-15mm}
|
|
|
|
\begin{minipage}{0.35\textwidth}
|
|
\only<1>{
|
|
\begin{itemize}
|
|
\item Independent variables
|
|
\begin{itemize}
|
|
\item Physical error rate
|
|
\item CNOT infidelity
|
|
\item Total qubit count
|
|
\item \ldots
|
|
\end{itemize}
|
|
\end{itemize}
|
|
}
|
|
\only<2->{
|
|
\begin{itemize}
|
|
\item Independent variables
|
|
\begin{itemize}
|
|
\item \textbf{Physical error rate}
|
|
\item CNOT infidelity
|
|
\item \textbf{Total qubit count}
|
|
\item \ldots
|
|
\end{itemize}
|
|
\end{itemize}
|
|
}
|
|
\end{minipage}%
|
|
\begin{minipage}{0.65\textwidth}
|
|
\begin{itemize}
|
|
\only<1>{
|
|
\item Noise models
|
|
\begin{itemize}
|
|
\item Standard circuit-based depolarizing noise
|
|
\citereferencemanual{FSG09}
|
|
\item Superconductor inspired (SI1000)
|
|
\citereferencemanual{GNF$^+$21}
|
|
\item Entangling Measurements (EM3)
|
|
\citereferencemanual{GNF$^+$21}
|
|
\item \ldots
|
|
\end{itemize}
|
|
}
|
|
\only<2->{
|
|
\item Noise models
|
|
\begin{itemize}
|
|
\item \textbf{Standard circuit-based depolarizing noise}
|
|
\citereferencemanual{FSG09}
|
|
\item Superconductor inspired (SI1000)
|
|
\citereferencemanual{GNF$^+$21}
|
|
\item Entangling Measurements (EM3)
|
|
\citereferencemanual{GNF$^+$21}
|
|
\item \ldots
|
|
\end{itemize}
|
|
}
|
|
\end{itemize}
|
|
\end{minipage}
|
|
|
|
\vspace{5mm}
|
|
|
|
\visible<3->{
|
|
\begin{itemize}
|
|
\item Degeneracy, information stored in correlations
|
|
$\rightarrow$ Consider \schlagwort{\acl{ler}} (LER)
|
|
\end{itemize}
|
|
}
|
|
\visible<4->{
|
|
\begin{itemize}
|
|
\item Types of benchmarking plots
|
|
\end{itemize}
|
|
|
|
\vspace*{5mm}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
\begin{subfigure}{0.35\textwidth}
|
|
\centering
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
domain=-5:5,
|
|
width=7cm,
|
|
height=5.5cm,
|
|
xticklabels=\empty,
|
|
yticklabels=\empty,
|
|
xlabel={Physical error rate},
|
|
xlabel style={yshift=5mm},
|
|
ylabel={LER},
|
|
ylabel style={yshift=-5mm},
|
|
grid,
|
|
]
|
|
|
|
\addplot+[
|
|
mark=none,
|
|
kit-red,
|
|
line width=2pt,
|
|
]
|
|
table[row sep=crcr] {
|
|
x y \\
|
|
1.134800559068837 0.5575221183357257 \\
|
|
2.0632737437615223 0.9764009116710485 \\
|
|
2.861072612292603 1.7787608707489788 \\
|
|
3.7551580964997053 2.8407080379684153 \\
|
|
4.264098875196703 3.513274267363004 \\
|
|
4.573589936760932 3.9911505302955272 \\
|
|
4.903713970055305 4.268436552233389 \\
|
|
};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\end{subfigure}%
|
|
\begin{subfigure}{0.35\textwidth}
|
|
\centering
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
domain=-5:5,
|
|
width=7cm,
|
|
height=5.5cm,
|
|
xticklabels=\empty,
|
|
yticklabels=\empty,
|
|
xlabel={Total qubit count},
|
|
xlabel style={yshift=5mm},
|
|
ylabel={LER},
|
|
ylabel style={yshift=-5mm},
|
|
grid,
|
|
]
|
|
|
|
\addplot+[
|
|
mark=none,
|
|
kit-blue,
|
|
line width=2pt,
|
|
]
|
|
table[row sep=crcr] {
|
|
x y \\
|
|
1.147643096789246 3.8430493581808607 \\
|
|
1.7245658892318043 2.762331811591747 \\
|
|
2.3573205843145306 2.3587443650766753 \\
|
|
2.9156332708646624 1.560537992857378 \\
|
|
3.6352360073136527 1.0403588210329737 \\
|
|
4.392060012189421 0.7130042787942606 \\
|
|
};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\end{subfigure}
|
|
\end{figure}
|
|
}
|
|
|
|
\vspace*{4mm}
|
|
|
|
\addreferencesmanual
|
|
{FSG09}{
|
|
A. G. Fowler, A. M. Stephens, and P. Groszkowski,
|
|
``High-threshold universal quantum computation on the surface
|
|
code,'' \emph{Physical Review}, 2009.
|
|
}
|
|
{GNF$^+$21}{
|
|
C. Gidney et al., ``A Fault-Tolerant Honeycomb Memory'',
|
|
\emph{Quantum}, 2021.
|
|
}
|
|
\stopreferencesmanual
|
|
\end{frame}
|
|
|
|
%%%%%%%%%%%%%%%%
|
|
\subsection{Conclusion and Outlook}
|
|
\label{subsec:Conclusion and Outlook}
|
|
|
|
\begin{frame}
|
|
\frametitle{Conclusion and Outlook}
|
|
|
|
\vspace*{-10mm}
|
|
|
|
\begin{minipage}[c]{0.65\textwidth}
|
|
\begin{itemize}
|
|
\item Problem setting
|
|
\begin{itemize}
|
|
\item Research area: Decoder design for \acp{dem}
|
|
under circuit-level noise
|
|
\item Research gap: Consideration of \acp{dem} as
|
|
\ac{scldpc} codes
|
|
\end{itemize}
|
|
\vspace*{5mm}
|
|
\item Future work
|
|
\begin{itemize}
|
|
\item Modify existing decoder to pass soft information
|
|
\item Test different \ac{bp} variations
|
|
\item \ldots
|
|
\end{itemize}
|
|
\vspace*{5mm}
|
|
\item Parameters
|
|
\begin{itemize}
|
|
\item Use standard depolarizing noise for comparability
|
|
\item Compare performance with other \ac{bb} code decoders
|
|
\end{itemize}
|
|
\end{itemize}
|
|
\end{minipage}%
|
|
\begin{minipage}[c]{0.35\textwidth}
|
|
\centering
|
|
\pause
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\vspace*{-25mm}
|
|
\begin{tikzpicture}
|
|
\node[scale=10] at (0, 0)
|
|
{\textcolor{kit-blue}{{\fontfamily{phv}\selectfont ?}}};
|
|
|
|
\node[align=center] at (0,-5) {Thank you for your
|
|
attention! \\ Any questions?};
|
|
\end{tikzpicture}
|
|
\end{figure}
|
|
\end{minipage}
|
|
\end{frame}
|
|
|
|
\appendix
|
|
\beginbackup
|
|
|
|
% TODO: Move arrow into syndrome extraction lower (branch from other
|
|
% arrow) and change caption to "modified from [MSLS25]"
|
|
\begin{frame}
|
|
\frametitle{System Level Overview}
|
|
|
|
\vspace*{-10mm}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
\begin{subfigure}[t]{0.5\textwidth}
|
|
\centering
|
|
|
|
\includegraphics[scale=1.1]{res/architecture}
|
|
|
|
\vspace*{5mm}
|
|
|
|
\caption{Schematic workflow of surface code quantum
|
|
computation \citereferencemanual{ZZC$^+$23}.}
|
|
\end{subfigure}%
|
|
\begin{subfigure}[t]{0.5\textwidth}
|
|
\centering
|
|
|
|
\tikzset{
|
|
block/.style={
|
|
draw, rectangle,
|
|
fill = kit-blue!25,
|
|
minimum width=75mm, minimum height=15mm,
|
|
}
|
|
}
|
|
|
|
\scalebox{0.7}{
|
|
\begin{tikzpicture}[node distance=15mm and 20mm]
|
|
\node[block] (encoding) {Encoding};
|
|
\node[block, below=of encoding] (channel)
|
|
{Quantum Channel};
|
|
\node[block, below=of channel] (reverse-op)
|
|
{Reverse Operation};
|
|
|
|
\node[block, right=of channel] (syn-extr)
|
|
{Syndrome Extraction};
|
|
\node[block, below=of syn-extr] (syn-dec)
|
|
{Syndrome Decoder};
|
|
|
|
\node[above=of encoding] (input) {$\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}}$};
|
|
|
|
\draw [-{Latex}] (input) -- (encoding);
|
|
\draw [-{Latex}] (reverse-op) -- (output);
|
|
\end{tikzpicture}
|
|
}
|
|
|
|
\vspace*{5mm}
|
|
|
|
\caption{Block diagram of QEC using stabilizer codes
|
|
\citereferencemanual{MSL$^+$25}.}
|
|
\end{subfigure}
|
|
\end{figure}
|
|
|
|
\vspace*{5mm}
|
|
|
|
\addreferencesmanual
|
|
{ZZC$^+$23}{
|
|
F. Zhang et al., ``A Classical Architecture for Digital
|
|
Quantum Computers,'' \emph{ACM Transactions on Quantum
|
|
Computing}, 2023.
|
|
}
|
|
{MSL$^+$25}{
|
|
S. Miao et al., ``Quaternary Neural Belief Propagation
|
|
Decoding of Quantum LDPC Codes with Overcomplete
|
|
Check Matrices'', \emph{IEEE Access}, 2025.
|
|
}
|
|
\stopreferencesmanual
|
|
\end{frame}
|
|
|
|
\begin{frame}
|
|
\frametitle{Guided Decimation Guessing Decoding}
|
|
|
|
\begin{minipage}{0.57\textwidth}
|
|
\begin{itemize}
|
|
\item BP guided decimation (BPGD) \\
|
|
$\rightarrow$ Iteratively fix most reliable variable node (VN)
|
|
\vspace*{10mm}
|
|
\item \schlagwort{Guided decimation guessing} (GDG)
|
|
\citereferencemanual{GCR24}
|
|
\begin{itemize}
|
|
\item Choose VN with \schlagwort{lowest}
|
|
log-likelihood ratio
|
|
\item Choose VN to fix based on \schlagwort{LLR history}
|
|
\item Explore both VN values in parallel
|
|
(\schlagwort{guessing})
|
|
\end{itemize}
|
|
\end{itemize}
|
|
\end{minipage}%
|
|
\begin{minipage}{0.43\textwidth}
|
|
\begin{figure}[H]
|
|
\centering
|
|
\includegraphics[scale=1.3]{res/gdg.pdf}
|
|
\end{figure}
|
|
\end{minipage}%
|
|
|
|
\vspace*{30mm}
|
|
|
|
\addreferencesmanual
|
|
{GCR24}{
|
|
A. Gong, S. Cammerer, and J. M. Renes, ``Toward
|
|
Low-latency Iterative Decoding of QLDPC Codes Under
|
|
Circuit-Level Noise,'' 2024.
|
|
}
|
|
\stopreferencesmanual
|
|
\end{frame}
|
|
|
|
\begin{frame}
|
|
\frametitle{The Quantum Error Correcting Landscape}
|
|
|
|
\vspace*{-10mm}
|
|
|
|
\begin{itemize}
|
|
\item Taxonomy of main QEC code families \citereferencemanual{SPG$^+$25}
|
|
\end{itemize}
|
|
|
|
\vspace*{2mm}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
\includegraphics[scale=2.5]{res/taxonomy.pdf}
|
|
\end{figure}
|
|
|
|
\begin{itemize}
|
|
\item Surface code is the industry standard for
|
|
experimental implementations, but has poor encoding
|
|
efficiency \citereferencemanual{BCG$^+$24}
|
|
\item \Ac{qldpc} codes particularly interesting because of
|
|
\begin{itemize}
|
|
\item Constant overhead scaling \citereferencemanual{Got14}
|
|
\item Linear distance scaling \citereferencemanual{BCG$^+$24}
|
|
\end{itemize}
|
|
\end{itemize}
|
|
|
|
\vspace*{15mm}
|
|
|
|
\addreferencesmanual
|
|
{SPG$^+$25}{
|
|
A. Swierkowska et al., ``ECCentric: An Empirical
|
|
Analysis of Quantum Error Correction Codes'',
|
|
\emph{arXiv:2511.01062v1}, 2025.
|
|
}
|
|
{BCG$^+$24}{
|
|
S. Bravyi et al., ``High-threshold and low-overhead
|
|
fault-tolerant quantum memory,'' \emph{Nature}, 2024.
|
|
}
|
|
{Got14}{
|
|
D. Gottesman, ``Fault-Tolerant Quantum Computation with
|
|
Constant Overhead'', \emph{arXiv:1310.2984}, 2014.
|
|
}
|
|
\stopreferencesendmanual
|
|
\end{frame}
|
|
|
|
% TODO: Is this really necessary?
|
|
% \begin{frame}
|
|
% \frametitle{The Quantum Error Correction Landscape}
|
|
%
|
|
% \begin{itemize}
|
|
% \item \red{Give basic overview of most promising avenues of
|
|
% research (as in \citereference{swierkowska_eccentric_2025})}
|
|
% \end{itemize}
|
|
%
|
|
% \vspace*{15mm}
|
|
%
|
|
% \addreferences
|
|
% {swierkowska_eccentric_2025}
|
|
% \stopreferences
|
|
% \end{frame}
|
|
|
|
\backupend
|
|
|
|
\end{document}
|
|
|