From 57d2ce2dd6dd24d3cead8c2ffc4ff4de0b32a05c Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Wed, 4 Feb 2026 13:56:33 +0100 Subject: [PATCH] Add changes up to slide 11 --- src/midterm_presentation/main.tex | 542 ++++++++++++++++++++---------- 1 file changed, 358 insertions(+), 184 deletions(-) diff --git a/src/midterm_presentation/main.tex b/src/midterm_presentation/main.tex index 1eacca5..84f629e 100644 --- a/src/midterm_presentation/main.tex +++ b/src/midterm_presentation/main.tex @@ -1,4 +1,4 @@ -\documentclass[overviewatsection, showsubsectionsatfirstoverview]{CELbeamer} +\documentclass[overviewatsection]{CELbeamer} % % @@ -53,7 +53,7 @@ \title{Fault Tolerant Quantum Error Correction} \subtitle{Master's Thesis Midterm Presentation} \author[Tsouchlos]{Andreas Tsouchlos} -\date[]{February 5th, 2026} +\date[]{} \DeclareFieldFormat{note}{} \DeclareFieldFormat{issn}{} @@ -72,6 +72,11 @@ \newcommand{\red}[1]{\textcolor{red}{#1}} \newcommand{\res}{src/midterm_presentation/res} +\newcommand{\X}{\textcolor{kit-blue}{\bm{X}}} +\newcommand{\Z}{\textcolor{kit-orange}{\bm{Z}}} +\newcommand{\Y}{\textcolor{kit-red}{\bm{Y}}} +\newcommand{\I}{\bm{I}} + % % % Acronyms @@ -85,7 +90,7 @@ \DeclareAcronym{css}{ short=CSS, - long=Calderbank Shor Steane + long=Calderbank -- Shor -- Steane } \DeclareAcronym{bb}{ @@ -110,12 +115,12 @@ \DeclareAcronym{qldpc}{ short=QLDPC, - long=quantum low density parity check, + long=quantum low - density parity - check, } \DeclareAcronym{scldpc}{ short=SC-LDPC, - long=spatially coupled low density parity check + long=spatially - coupled low - density parity - check } % @@ -154,25 +159,20 @@ \begin{itemize} \item Simulating quantum systems on classical hardware is exponentially complex \\ - $\rightarrow$ Can't we use quantum hardware to simulate - quantum systems? \citereference{feynman_simulating_1982} - \item Some problems that are ``hard'' to solve on classical - computers we can ``easily'' solve on quantum computers + $\rightarrow$ Use quantum hardware to simulate quantum + systems \citereference{feynman_simulating_1982} + \item ``Hard'' to solve problems on classical computers can + be ``easy'' on quantum computers \citereference{preskill_quantum_2018} + \item Google Quantum AI's quantum computing roadmap + \citereference{google_quantum_ai_quantum_nodate} \end{itemize} - \vspace*{-5mm} + \vspace*{3mm} \begin{figure}[H] \centering - \includegraphics[scale=0.43]{res/google_roadmap.png} - - \vspace*{-3mm} - \caption{ - Google Quantum AI's quantum computing roadmap - \citereference{google_quantum_ai_quantum_nodate}. - } \end{figure} \vspace*{3mm} @@ -189,7 +189,7 @@ \begin{frame} \frametitle{The Need for Quantum Error Correction} - \vspace*{-17mm} + \vspace*{-15mm} % Related interesting stuff % - Qubits differ from bits in that they can be in superpositions @@ -210,38 +210,61 @@ % computation \begin{itemize} - \item Quantum computers represent information through - correlations of qubits, not their values \\ - directly \citereference{preskill_quantum_2018} - \item Errors during quantum computation are inevitable - because quantum systems are fragile - \item We want to interact with the quantum state but not disturb it - \item We employ more physical qubits to introduce - redundancy and use the resulting \emph{physical} state to - represent the \emph{logical} state - \citereference{roffe_quantum_2019} - \vspace*{8mm} - \item Typical scales - \begin{itemize} - \item IBM recently introduced a scheme encoding $12$ logical - qubits in $288$ physical ones - \citereference{bravyi_high-threshold_2024} - \item The physical error rate is typically assumed to - be $10^{-3}$ for - simulations (e.g., - \citereference{bravyi_high-threshold_2024}) - \item Decoding has to happen with ultra-low latency to avoid - the backlog problem (about $\SI{1}{us}$ per data - extraction round) \citereference{caune_demonstrating_2024} - % \citereference{terhal_quantum_2015} - \end{itemize} + % \item Quantum computers represent information through + % correlations of qubits, not their values \\ + % directly \citereference{preskill_quantum_2018} + \item Quantum systems are inherently fragile + \item Interacting with the quantum state disturbs it + \item Idea: Represent \schlagwort{logical qubits} using more + \schlagwort{physical qubits} \citereference{roffe_quantum_2019} + + \vspace*{2mm} + + \begin{figure}[H] + \centering + \begin{tikzpicture} + \node[ + rectangle, + draw, fill=kit-blue!25, + minimum height=15mm, + ] + (enc) {Three-qubit encoder}; + + \node[left=of enc] (in) + {$\ket{\psi} = \alpha\ket{0} + \beta\ket{1}$}; + + \node[right=of enc,yshift=6mm] (out) + {$\alpha\overbrace{\ket{000}}^{\ket{0}_\text{L}} + +\; \beta\overbrace{\ket{111}}^{\ket{1}_\text{L}} = + \ket{\psi}_\text{L}$}; + + \draw[-{Latex}] (in) -- (enc); + \draw[-{Latex}] (enc) -- (enc -| out.west); + \end{tikzpicture} + \end{figure} + + \vspace*{5mm} + \visible<2>{ + \item Typical scales + \begin{itemize} + \item Recent scheme by IBM encodes $12$ logical + qubits in $288$ physical ones + \citereference{bravyi_high-threshold_2024} + \item Physical error rate typically set to $10^{-3}$ + for simulations (e.g., + \citereference{bravyi_high-threshold_2024}) + \item Decode with ultra-low latency to avoid + \schlagwort{backlog problem} (about + $\SI{1}{\micro s}$ per data \\ + extraction round) + \citereference{caune_demonstrating_2024} + \end{itemize} + } \end{itemize} - \vspace*{7mm} + \vspace*{10mm} \addreferences - % {terhal_quantum_2015} - {preskill_quantum_2018} {roffe_quantum_2019} {bravyi_high-threshold_2024} {caune_demonstrating_2024} @@ -256,7 +279,7 @@ \begin{frame} \frametitle{Peculiarities of the Quantum Setting} - \vspace*{-18mm} + \vspace*{-13mm} % Related interesting stuff % - No cloning theorem -> Not replication of state, protection @@ -274,9 +297,10 @@ % much" \begin{itemize} - \item \Ac{qec} is actually able to protect the actual quantum state - \item Similar to bits and gates, quantum systems are built on - top of qubits and quantum gates + % \item \Ac{qec} is actually able to protect the actual + % quantum state + \item Classical systems built with bits and gates, quantum + systems with qubits and quantum gates \item We have to consider phase flip errors in addition to bit flip errors \citereference{roffe_quantum_2019} \end{itemize} @@ -289,47 +313,49 @@ \centering \begin{align*} - \ket{0} &\rightarrow \ket{1} \\ - \ket{1} &\rightarrow \ket{0} + \ket{0} &\mapsto \ket{1} \\ + \ket{1} &\mapsto \ket{0} \end{align*} - \caption{Bit flip (X) error} + \caption{Bit flip ($\X$) error} \end{subfigure}% \begin{subfigure}{0.32\textwidth} \centering \begin{align*} - \ket{0} &\rightarrow \phantom{-}\ket{0} \\ - \ket{1} &\rightarrow -\ket{1} + \ket{0} &\mapsto \phantom{-}\ket{0} \\ + \ket{1} &\mapsto -\ket{1} \end{align*} - \caption{Phase flip (Z) error} + \caption{Phase flip ($\Z$) error} \end{subfigure}% \begin{subfigure}{0.32\textwidth} \centering \begin{align*} - \ket{0} &\rightarrow \phantom{-j}\ket{1} \\ - \ket{1} &\rightarrow -j\ket{0} + \ket{0} &\mapsto \phantom{-j}\ket{1} \\ + \ket{1} &\mapsto -j\ket{0} \end{align*} - \caption{Y error: Combination of X and Z} + \caption{$\Y$ error} \end{subfigure} \end{figure} - \vspace*{-3mm} - \begin{itemize} - \item Measuring the qubits directly destroys superpositions - and entanglement \\ - $\rightarrow$ We generally only work with the syndrome, - which we can measure \citereference{nielsen_quantum_2010} - \item Sometimes superposition permits multiple equivalent - solutions to the decoding problem (\emph{quantum - degeneracy}) \citereference{roffe_decoding_2020} + \visible<2->{ + \item Measuring the qubits directly destroys superpositions + and entanglement \\ + $\rightarrow$ Use syndrome for decoding + \citereference{nielsen_quantum_2010} + } + \visible<3>{ + \item Superposition $\rightarrow$ multiple solutions to the + decoding problem + (\schlagwort{quantum degeneracy}) + \citereference{roffe_decoding_2020}} \end{itemize} - \vspace*{7mm} + \vspace*{12mm} \addreferences {nielsen_quantum_2010} @@ -354,24 +380,25 @@ \begin{itemize} \item Stabilizer codes \citereference{nielsen_quantum_2010} \begin{itemize} - \item The code space can implicitly be defined using - \emph{stabilizer generators} - \item We can represent them using parity - check matrices - \item Quantum analog of linear codes + \item Implicitly defined using \schlagwort{stabilizer + generators} + \item Can be represented using parity check matrices + \item Quantum analog of linear block codes \end{itemize} \vspace*{10mm} - \item \Ac{css} codes \citereference{nielsen_quantum_2010} - \begin{itemize} - \item Subset of stabilizer codes - \item Can correct X and Z errors independently - \item Described using two separate parity check - matrices $\bm{H}_\text{X}$ and $\bm{H}_\text{Z}$ - \item Can be constructed from two binary linear codes - $\mathcal{C}_1 \left[ n, k_1 \right]$ and - $\mathcal{C}_2 \left[ n, k_2 \right]$ with - $\mathcal{C}_2 \subset \mathcal{C}_1$ - \end{itemize} + \visible<2->{ + \item \Acf{css} codes \citereference{nielsen_quantum_2010} + \begin{itemize} + \item Subset of stabilizer codes + \item Able to correct $\X$ and $\Z$ errors independently + \item Described using two separate parity check + matrices $\bm{H}_X$ and $\bm{H}_Z$ + \item Can be constructed from two binary linear codes + $\mathcal{C}_1 \left[ n, k_1 \right]$ and + $\mathcal{C}_2 \left[ n, k_2 \right]$ with + $\mathcal{C}_2 \subset \mathcal{C}_1$ + \end{itemize} + } \end{itemize} \vspace*{20mm} @@ -386,21 +413,16 @@ \begin{frame} \frametitle{Syndrome Extraction Circuits} - \vspace*{-16mm} + \vspace*{-10mm} \begin{itemize} - \item We entangle the state with \emph{ancilla qubits} to - perform syndrome measurements \citereference{nielsen_quantum_2010} - % \item \red{Do I need to show what the syndrome extraction - % circuitry for Z errors looks like?} - \item Example: The 3-qubit repetition code% - \footnote { - Note that, for simplicity, this chosen example is a - code that is only able to correct X errors (bit flips) - } % + \item Entangle the state $\ket{\psi}$ with + \schlagwort{ancilla qubits} to perform syndrome + measurements \citereference{nielsen_quantum_2010} + \item Example: The 3-qubit repetition code for $\X$ errors \end{itemize} - \vspace*{-10mm} + \vspace*{-5mm} \begin{align*} \bm{H} = @@ -410,8 +432,6 @@ \end{pmatrix} \end{align*} - \vspace*{5mm} - \begin{figure}[H] % \newcommand{\anyerrgate}{\gate[style={fill=red!20}]{\mathcal{E}_\text{XYZ}}} \newcommand{\preperr}{\gate[style={fill=orange!20}]{\phantom{1}}} @@ -421,18 +441,17 @@ \centering % tex-fmt: off \begin{quantikz}%[row sep=4mm, column sep=4mm] - & \ctrl{3} & & & & & \\ - \lstick{$\ket{\psi}$} & & \ctrl{2} & \ctrl{3} & & & \\ - & & & & \ctrl{2} & & \\ - \lstick{$\ket{0}_{\text{A}_1}$} & \targ{} & \targ{} & & & \meter{} \\ - \lstick{$\ket{0}_{\text{A}_2}$} & & & \targ{} & \targ{} & \meter{} + & \ctrl{3} & & & & & \\ + \lstick{$\ket{\psi}$} & & \ctrl{2} & \ctrl{3} & & & \\ + & & & & \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 % \caption{Circuit-level noise model for the 3-qubit repetition code} \end{figure} - % \vspace*{5mm} - \vspace*{-2mm} + \vspace*{5mm} \addreferences {nielsen_quantum_2010} @@ -451,19 +470,18 @@ \begin{frame} \frametitle{Fault Tolerance} - \vspace*{-18mm} + \vspace*{-10mm} \begin{itemize} - \item The quantum gates we use for syndrome extraction are + \item Quantum gates used for syndrome extraction are faulty themselves \\ - $\rightarrow$ We need \emph{fault-tolerant} \ac{qec} - \item A \ac{qec} procedure is said to be fault tolerant if, - in addition to correcting \emph{input errors}, the spread - of \emph{internal errors} is sufficiently limited + $\rightarrow$ Need for \schlagwort{fault-tolerant} \acf{qec} + \item In addition to correcting \schlagwort{input errors}, + limit spread of \schlagwort{internal errors} \citereference{derks_designing_2025} \end{itemize} - % \vspace*{3mm} + \vspace*{3mm} \begin{figure}[H] \centering @@ -487,22 +505,20 @@ \node[above] at (internal.north) {\small QEC}; \node[above] at (output.north) {\small Output State}; \end{tikzpicture} - - \caption{Overview of the flow of errors in a \ac{qec} system. - Adapted from \citereference{derks_designing_2025}.} \end{figure} - % \vspace*{3mm} + \vspace*{3mm} \begin{itemize} - \item We have to modify the syndrome extraction circuitry to - be fault tolerant (e.g., by using specially prepared - multi-qubit states for each ancilla - \citereference{shor_fault-tolerant_1997}) - \item We generally perform multiple rounds of syndrome extraction + \visible<2->{ + \item Modify syndrome extraction circuitry (e.g., multi-qubit + states for each ancilla + \citereference{shor_fault-tolerant_1997}) + \item Multiple rounds of syndrome extraction + } \end{itemize} - \vspace*{8mm} + \vspace*{15mm} \addreferences {shor_fault-tolerant_1997} @@ -520,43 +536,108 @@ \vspace*{-18mm} \begin{itemize} - \item Each column of the \emph{measurement syndrome matrix} - $\bm{\Omega}$ corresponds to a measurement pattern an - error produces \citereference{derks_designing_2025} + \item \schlagwort{Measurement syndrome matrix} $\bm{\Omega}$ \\ + contains error patterns \citereference{derks_designing_2025} \item Example: 3-qubit repetition code \\ (Only bit flips on data qubits) \end{itemize} - \vspace*{-28mm} + \vspace*{-35mm} \centering \only<1>{ \begin{minipage}{0.4\textwidth} \centering - \begin{align*} - \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) - \end{align*} + + \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} + ] + (2.4,1.2) -- (2.5,2.85) + node[midway,right,xshift=10mm]{$\text{SE}_1$}; + + \draw [ + line width=1pt, + decorate, + decoration={brace,mirror,amplitude=3mm,raise=5mm} + ] + (2.4,-0.75) -- (2.5,0.9) + node[midway,right,xshift=10mm]{$\text{SE}_2$}; + + \draw [ + line width=1pt, + decorate, + decoration={brace,mirror,amplitude=3mm,raise=5mm} + ] + (2.4,-2.7) -- (2.5,-1.1) + node[midway,right,xshift=10mm]{$\text{SE}_3$}; + \end{tikzpicture} + + \vspace*{-10mm} + + \begin{gather*} + \bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\} + \end{gather*} \end{minipage}% \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}}} \centering - % tex-fmt: off - \begin{quantikz}[row sep=4mm, column sep=4mm, wire types={q,q,q,q,q,n,n,n,n}] + \begin{quantikz}[ + row sep=4mm, column sep=4mm, + wire types={q,q,q,q,q,n,n,n,n}, + execute at end picture={ + \draw [ + line width=1pt, + decorate, + decoration={brace,amplitude=3mm,raise=5mm} + ] + (\tikzcdmatrixname-4-19.north east) + -- + (\tikzcdmatrixname-5-19.south east) + node[midway,right,xshift=10mm]{$\text{SE}_1$}; + \draw [ + line width=1pt, + decorate, + decoration={brace,amplitude=3mm,raise=5mm} + ] + (\tikzcdmatrixname-6-19.north east) + -- + (\tikzcdmatrixname-7-19.south east) + node[midway,right,xshift=10mm]{$\text{SE}_2$}; + \draw [ + line width=1pt, + decorate, + decoration={brace,amplitude=3mm,raise=5mm} + ] + (\tikzcdmatrixname-8-19.north east) + -- + (\tikzcdmatrixname-9-19.south east) + node[midway,right,xshift=10mm]{$\text{SE}_3$}; + } + ] + % tex-fmt: off & \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} & \\ @@ -566,26 +647,61 @@ & & & & & & \lstick{$\ket{0}_{\text{A}_4}$} & \setwiretype{q} & & & \targ{} & \targ{} & & & & & & & \meter{} \\ & & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_5}$} & \setwiretype{q} & \targ{} & \targ{} & & & \meter{} \\ & & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_6}$} & \setwiretype{q} & & & \targ{} & \targ{} & \meter{} + % tex-fmt: on \end{quantikz} - % tex-fmt: on \end{figure} \end{minipage} } \only<2>{ \begin{minipage}{0.4\textwidth} \centering - \begin{align*} - \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) - \end{align*} + + \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} + ] + (2.4,1.2) -- (2.5,2.85) + node[midway,right,xshift=10mm]{$\text{SE}_1$}; + + \draw [ + line width=1pt, + decorate, + decoration={brace,mirror,amplitude=3mm,raise=5mm} + ] + (2.4,-0.75) -- (2.5,0.9) + node[midway,right,xshift=10mm]{$\text{SE}_2$}; + + \draw [ + line width=1pt, + decorate, + decoration={brace,mirror,amplitude=3mm,raise=5mm} + ] + (2.4,-2.7) -- (2.5,-1.1) + node[midway,right,xshift=10mm]{$\text{SE}_3$}; + \end{tikzpicture} + + \vspace*{-10mm} + + \begin{gather*} + \bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\} + \end{gather*} \end{minipage}% \begin{minipage}{0.6\textwidth} \begin{figure}[H] @@ -615,7 +731,7 @@ \tikzset{ noisy/.style={ starburst, - starburst point height=2.5mm, + starburst point height=2mm, fill=red!25, draw=red!85!black, line width=2pt, inner xsep=-2pt, inner ysep=-2pt @@ -624,8 +740,40 @@ \centering - % tex-fmt: off - \begin{quantikz}[row sep=4mm, column sep=4mm, wire types={q,q,q,q,q,n,n,n,n}] + \begin{quantikz}[ + row sep=4mm, column sep=4mm, + wire types={q,q,q,q,q,n,n,n,n}, + execute at end picture={ + \draw [ + line width=1pt, + decorate, + decoration={brace,amplitude=3mm,raise=5mm} + ] + (\tikzcdmatrixname-4-19.north east) + -- + (\tikzcdmatrixname-5-19.south east) + node[midway,right,xshift=10mm]{$\text{SE}_1$}; + \draw [ + line width=1pt, + decorate, + decoration={brace,amplitude=3mm,raise=5mm} + ] + (\tikzcdmatrixname-6-19.north east) + -- + (\tikzcdmatrixname-7-19.south east) + node[midway,right,xshift=10mm]{$\text{SE}_2$}; + \draw [ + line width=1pt, + decorate, + decoration={brace,amplitude=3mm,raise=5mm} + ] + (\tikzcdmatrixname-8-19.north east) + -- + (\tikzcdmatrixname-9-19.south east) + node[midway,right,xshift=10mm]{$\text{SE}_3$}; + } + ] + % tex-fmt: off & \noise\redwire{17} & \redctrl{3} & & & & & & \redctrl{5} & & & & & & \redctrl{7} & & & & \\ \lstick{$\ket{\psi}$} & \preperr{E_1} & & \ctrl{2} & \ctrl{3} & & & & & \ctrl{4} & \ctrl{5} & & & & & \ctrl{6} & \ctrl{7} & & \\ & \preperr{E_2} & & & & \ctrl{2} & & & & & & \ctrl{4} & & & & & & \ctrl{6} & \\ @@ -635,13 +783,13 @@ & & & & & & \lstick{$\ket{0}_{\text{A}_4}$} & \setwiretype{q} & & & \targ{} & \targ{} & & & & & & & \meter{} \\ & & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_5}$} & \setwiretype{q} & \redtarg\redwire{4} & \targ{} & & & \redmeter \\ & & & & & & & & & & & & \lstick{$\ket{0}_{\text{A}_6}$} & \setwiretype{q} & & & \targ{} & \targ{} & \meter{} + % tex-fmt: on \end{quantikz} - % tex-fmt: on \end{figure} \end{minipage} } - \vspace*{2mm} + \vspace*{8mm} \addreferences {derks_designing_2025} @@ -649,29 +797,29 @@ \end{frame} \begin{frame}[fragile] - \frametitle{The Measurement Syndromemani Matrix II} + \frametitle{The Measurement Syndrome Matrix II} \vspace*{-18mm} \begin{itemize} - \item Each column of the \emph{measurement syndrome matrix} - $\bm{\Omega}$ corresponds to a measurement pattern an - error produces \citereference{derks_designing_2025} - \item - Example: 3-qubit repetition code \\ + \item \schlagwort{Measurement syndrome matrix} $\bm{\Omega}$ \\ + contains error patterns \citereference{derks_designing_2025} + \item Example: 3-qubit repetition code \\ (Phenomenological noise \citereference{derks_designing_2025}) \end{itemize} - \vspace*{-28mm} + \vspace*{-40mm} \centering \only<1>{ \begin{minipage}{0.4\textwidth} \centering + \vspace*{61mm} + \hspace*{-75mm} \scalebox{0.85}{ \parbox{.5\linewidth}{% - \begin{align*} + \begin{gather*} \bm{\Omega} = \left( \begin{array}{ccccccccccccccc} @@ -687,8 +835,11 @@ & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 - \end{array}\right) - \end{align*} + \end{array} + \right) \\[10mm] + \hspace*{50mm} % + \bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\} + \end{gather*} } } \end{minipage}% @@ -749,11 +900,11 @@ \begin{minipage}{0.4\textwidth} \centering - \newcommand{\pz}{\phantom{0}} - + \vspace*{61mm} + \hspace*{-75mm} \scalebox{0.85}{ \parbox{.5\linewidth}{% - \begin{align*} + \begin{gather*} \bm{\Omega} = \left( \begin{array}{ @@ -773,8 +924,11 @@ & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 - \end{array}\right) - \end{align*} + \end{array} + \right) \\[10mm] + \hspace*{50mm} % + \bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\} + \end{gather*} } } \end{minipage}% @@ -835,11 +989,11 @@ \begin{minipage}{0.4\textwidth} \centering - \newcommand{\pz}{\phantom{0}} - + \vspace*{61mm} + \hspace*{-75mm} \scalebox{0.85}{ \parbox{.5\linewidth}{% - \begin{align*} + \begin{gather*} \bm{\Omega} = \left( \begin{array}{ @@ -859,8 +1013,11 @@ & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 - \end{array}\right) - \end{align*} + \end{array} + \right) \\[10mm] + \hspace*{50mm} % + \bm{s} \in \text{span} \mleft\{ \bm{\Omega} \mright\} + \end{gather*} } } \end{minipage}% @@ -918,7 +1075,7 @@ \end{minipage} } - \vspace*{2mm} + \vspace*{3mm} \addreferences {derks_designing_2025} @@ -934,7 +1091,7 @@ \begin{itemize} \item A detector is a parity constraint on a set of measurement outcomes \citereference{derks_designing_2025} - \item Each column of the \emph{detector error matrix} $\bm{H}$ + \item Each column of the \schlagwort{detector error matrix} $\bm{H}$ corresponds to a detector pattern an error produces \item We can mitigate the propagation of errors into subsequent rounds by XORing the measurements, i.e., @@ -1056,6 +1213,9 @@ % - 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*{-15mm} @@ -1068,19 +1228,19 @@ \begin{minipage}{0.60\textwidth} \begin{itemize} - \item The \emph{depolarizing channel} considers + \item The \schlagwort{depolarizing channel} considers \citereference{nielsen_quantum_2010} \begin{itemize} \item X, Y or Z errors on the data qubits \end{itemize} - \item \emph{Phenomenological noise} considers + \item \schlagwort{Phenomenological noise} considers \citereference{derks_designing_2025} \begin{itemize} \item X errors on data qubits before each \\ measurement round \item X errors on measurement outcomes \end{itemize} - \item \emph{Circuit-level noise} considers + \item \schlagwort{Circuit-level noise} considers \citereference{derks_designing_2025} \begin{itemize} \item \colorbox{orange!20}{X, Y or Z errors after @@ -1523,6 +1683,20 @@ \stopreferences \end{frame} +\begin{frame} + \frametitle{Guided Decimation Guessing Decoding} + + \begin{itemize} + \item \red{Explain paper} + \end{itemize} + + \vspace*{25mm} + + \addreferences + {gong_toward_2024} + \stopreferences +\end{frame} + % TODO: Is this really necessary? % \begin{frame} % \frametitle{The Quantum Error Correction Landscape}