Finish first version of intro

src/midterm_presentation/main.tex

@@ -47,6 +47,7 @@
 \usepackage{amssymb}
 \usepackage{acro}
 \usepackage{braket}
+\usepackage{qcircuit}
 
 \title{Fault Tolerant Quantum Error Correction}
 \subtitle{Master's Thesis Midterm Presentation}
@@ -67,6 +68,7 @@
 %
 %
 
+\newcommand{\red}[1]{\textcolor{red}{#1}}
 \newcommand{\res}{src/midterm_presentation/res}
 
 %
@@ -80,6 +82,11 @@
     long=quantum error correction
 }
 
+\DeclareAcronym{css}{
+    short=CSS,
+    long=Calderbank Shor Steane
+}
+
 %
 %
 % Document body
@@ -111,7 +118,7 @@
     %   [How to factor 2048 bit RSA integers with less than a million
     %   noisy qubits]
 
-    \vspace*{-19mm}
+    \vspace*{-15mm}
 
     \begin{itemize}
         \item Simulating quantum systems on classical hardware
@@ -121,7 +128,6 @@
         \item Some problems that are ``hard'' to solve on classical
             computers we can ``easily'' solve on quantum computers
             \citereference{preskill_quantum_2018}
-        \item We are still in the early days of quantum computing
     \end{itemize}
 
     \vspace*{-5mm}
@@ -138,6 +144,8 @@
         }
     \end{figure}
 
+    \vspace*{3mm}
+
     \addreferences
     {feynman_simulating_1982}
     {preskill_quantum_2018}
@@ -150,7 +158,7 @@
 \begin{frame}
     \frametitle{The Need for Quantum Error Correction}
 
-    \vspace*{-10mm}
+    \vspace*{-17mm}
 
     % Related interesting stuff
     % - Qubits differ from bits in that they can be in superpositions
@@ -163,6 +171,9 @@
     %   2048 bit RSA integer
     %   [How to factor 2048 bit RSA integers with less than a million
     %   noisy qubits]
+    % - The "physical error rate" is really the value all error rates
+    %   in the system are set to for circuit level noise simulations
+    %   [High-threshold universal quantum computation on the surface code]
     % - The backlog problem is the fact that an increasing backlog of
     %   syndrome data will lead to an exponential slowdown during the
     %   computation
@@ -172,31 +183,37 @@
             correlations of qubits, not their values \\
             directly \citereference{preskill_quantum_2018}
         \item We want to not disturb the quantum state but need to
-            interact with the system \\
+            interact with the system $\rightarrow$ Protect the state
-            $\rightarrow$ Protect the state with \ac{qec}
+            with \ac{qec}
         \item We employ more physical qubits to introduce
             redundancy and use the resulting \emph{physical} state to
             represent the \emph{logical} state
             \citereference{roffe_quantum_2019}
-        \item IBM recently introduced a scheme encoding 12 logical
+            \vspace*{8mm}
-            qubits in 288 physical ones
+        \item Typical scales
-            \citereference{bravyi_high-threshold_2024}
+            \begin{itemize}
-        \item The physical error rate is typically assumed to be $10^{-3}$ for
+                \item IBM recently introduced a scheme encoding $12$ logical
-            simulations (e.g., \citereference{bravyi_high-threshold_2024})
+                    qubits in $288$ physical ones
-        \item Decoding has to happen with ultra-low latency to avoid
+                    \citereference{bravyi_high-threshold_2024}
-            the backlog problem (about $\SI{1}{us}$ per data
+                \item The physical error rate is typically assumed to
-            extraction round) \citereference{caune_demonstrating_2024}
+                    be $10^{-3}$ for
-            % \citereference{terhal_quantum_2015}
+                    simulations (e.g.,
+                    \citereference{bravyi_high-threshold_2024})
+                \item Decoding has to happen with ultra-low latency to avoid
+                    the backlog problem (about $\SI{1}{us}$ per data
+                    extraction round) \citereference{caune_demonstrating_2024}
+                    % \citereference{terhal_quantum_2015}
+            \end{itemize}
     \end{itemize}
 
-    \vspace*{12mm}
+    \vspace*{7mm}
 
     \addreferences
     % {terhal_quantum_2015}
-    {caune_demonstrating_2024}
     {preskill_quantum_2018}
     {roffe_quantum_2019}
     {bravyi_high-threshold_2024}
+    {caune_demonstrating_2024}
     \stopreferences
 \end{frame}
 
@@ -208,6 +225,8 @@
 \begin{frame}
     \frametitle{Peculiarities of the Quantum Setting}
 
+    \vspace*{-5mm}
+
     % Related interesting stuff
     % - No cloning theorem -> Not replication of state, protection
     %   through further entanglement
@@ -215,53 +234,164 @@
     %   to correct infinitely many different types of errors. Luckily,
     %   it turns out that in actual fact we only really need to correct
     %   two [Gottesman's Thesis]
+    % - Mention that kets are just vectors, used here to represent the state
+    % - There are actually infinitely many different errors that can
+    %   happen, but we can digitize them and only need to consider X and Z
+    % - Not only do we only care about the coset, we specifically
+    %   don't want to know more than the syndrome can tell us because
+    %   that would mean that "we collapse the quantum mechanical state too
+    %   much"
 
     \begin{itemize}
-        \item Measuring the system collapses the quantum state
+        \item As mentioned earlier, \ac{qec} is actually able to
-            $\rightarrow$ Loss of benefit of quantum system \\
+            protect the quantum state with all its correlations
-            $\rightarrow$ For BP, we work with the syndrome and not
+        \item We have to consider phase flip errors in addition to
-            the variable nodes \textcolor{red}{This can't be here,
+            bit flip errors \citereference{roffe_quantum_2019}
-            it's before introducing how QEC works}
+            \vspace*{-10mm}
-        \item X and Z errors
+            \begin{figure}[H]
-        \item With QEC we are able to restore the quantum state, not
+                \centering
-            "just the bits"
+                \begin{subfigure}{0.5\textwidth}
-        \item We don't care about the specific error, only the coset
+                    \centering
-            its in $\rightarrow$ We only really care about the syndrome
+
+                    \begin{align*}
+                        \ket{0} &\rightarrow \ket{1} \\
+                        \ket{1} &\rightarrow \ket{0}
+                    \end{align*}
+
+                    \caption{Bit flip (X) error}
+                \end{subfigure}%
+                \begin{subfigure}{0.5\textwidth}
+                    \centering
+
+                    \begin{align*}
+                        \ket{0} &\rightarrow \phantom{-}\ket{0} \\
+                        \ket{1} &\rightarrow -\ket{1}
+                    \end{align*}
+
+                    \caption{Phase flip (Z) error}
+                \end{subfigure}
+            \end{figure}
+        \item Measuring the qubits directly destroys superpositions
+            and entanglement \\
+            $\rightarrow$ We generally only work with the syndrome,
+            which we can measure \citereference{nielsen_quantum_2010}
+        \item We don't care about restoring the specific codeword,
+            only finding the coset it's in
     \end{itemize}
+
+    \vspace*{15mm}
+
+    \addreferences
+    {nielsen_quantum_2010}
+    {roffe_quantum_2019}
+    \stopreferences
 \end{frame}
 
 \begin{frame}
-    \frametitle{Fundamentals of Quantum Error Correction}
+    \frametitle{Stabilizer and Calderbank Shor Steane Codes}
+
+    \vspace*{-5mm}
+
+    % Related interesting stuff
+    % - Using stabilizers to describe quantum codes is a bit like
+    %   using parity check equations to describe classical codes
+    %   -> stabilizer codes are the quantum analog of binary linear codes
+    % - For CSS codes, "the parity checks for the X errors and the
+    %   parity checks for the Z errors can be represented independently
+    %   of one another"
 
     \begin{itemize}
-        \item Stabilizer codes: the quantum equivalent of binary linear codes
+        \item Stabilizer codes \citereference{nielsen_quantum_2010}
-        \item CSS codes: separate corection of X and Z errors
+            \begin{itemize}
-            $\rightarrow$ simpler circuitry
+                \item The code space can implicitly be defined using
-        \item Construction of CSS codes from binary linear codes
+                    \emph{stabilizer generators}
-            \textcolor{red}{Is this really necessary?}
+                \item We can represent them using parity
+                    check matrices
+                \item Quantum analog of linear codes
+            \end{itemize}
+            \vspace*{10mm}
+        \item \Ac{css} codes \citereference{nielsen_quantum_2010}
+            \begin{itemize}
+                \item Subset of stabilizer codes
+                \item Can correct X and Z errors independently
+                \item Described using two separate parity check
+                    matrices $\bm{H}_\text{X}$ and $\bm{H}_\text{Z}$
+                \item Can be constructed from two binary linear codes
+                    $\mathcal{C}_1 \left[ n, k_1 \right]$ and
+                    $\mathcal{C}_2 \left[ n, k_2 \right]$ with
+                    $\mathcal{C}_2 \subset \mathcal{C}_1$
+            \end{itemize}
+
+            \vspace*{10mm}
+
+        \item \red{Do I need to go more in depth for either
+            stabilizer codes or CSS codes?}
     \end{itemize}
+
+    \vspace*{10mm}
+
+    \addreferences
+    {nielsen_quantum_2010}
+    \stopreferences
 \end{frame}
 
-% TODO: Is this really necessary?
+% TODO: Do I need to show what the syndrome extraction circuitry for
+% Z errors looks like?
 \begin{frame}
-    \frametitle{The Quantum Error Correction Landscape}
+    \frametitle{Syndrome Extraction Circuits}
+
+    \vspace*{-16mm}
 
     \begin{itemize}
-        \item Give basic overview of most promising avenues of
+        \item We entangle the state with \emph{ancilla qubits} to
-            research (as in ECCentric paper)
+            perform syndrome measurements \citereference{nielsen_quantum_2010}
+        \item Example: The 3-qubit repetition code%
+            \footnote {
+                Note that, for simplicity, this chosen example is a
+                code that is not only able to correct X errors (bit flips)
+            } %
+            \red{Do I need to show what the syndrome extraction
+            circuitry for Z errors looks like?}
     \end{itemize}
-\end{frame}
 
-\begin{frame}
+    \vspace*{-10mm}
-    \frametitle{An Example: The Steane Code}
+
+    \begin{align*}
+        \bm{H} =
+        \begin{pmatrix}
+            1 & 1 & 0 \\
+            0 & 1 & 1
+        \end{pmatrix}
+    \end{align*}
+
+    \vspace*{5mm}
+
+    \begin{figure}[H]
+        \centering
+        \mbox{
+            % tex-fmt: off
+            \Qcircuit @C=1em @R=.7em {
+                                     & & \ctrl{3} & \qw      & \qw      & \qw      & \qw    & \qw \\
+                \ket{\psi}_\text{L}  & & \qw      & \ctrl{2} & \ctrl{3} & \qw      & \qw    & \qw \\
+                                     & & \qw      & \qw      & \qw      & \ctrl{2} & \qw    & \qw \\
+                \ket{0}_{\text{A}_1} & & \targ    & \targ    & \qw      & \qw      & \meter &     \\
+                \ket{0}_{\text{A}_2} & & \qw      & \qw      & \targ    & \targ    & \meter &
+            }
+            % tex-fmt: on
+        }
+
+        \vspace*{5mm}
+
+        \caption{Syndrome extraction circuit for the 3-qubit repetition code}
+    \end{figure}
+
+    % \vspace*{5mm}
+    \vspace*{-2mm}
+
+    \addreferences
+    {nielsen_quantum_2010}
+    \stopreferences
 
-    \begin{itemize}
-        \item \textcolor{red}{Give example slides grey background or something?}
-        \item The Steane code is the quantum equivalent of the
-            [7,4]-Hamming code
-        \item Construction from Hamming code
-        \item Syndrome Extraction Circuitry
-    \end{itemize}
 \end{frame}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -287,10 +417,29 @@
         \item The Threshold theorem
             \setcounter{footnote}{0}
         \item Definition of fault tolerance \footnotemark
-        \item \textcolor{red}{Different approaches to fault tolerance?}
+        \item \red{Different approaches to fault tolerance?}
     \end{itemize}
 \end{frame}
 
+% TODO: Where to we introduces the different kinds of noise models?
+\begin{frame}
+    \frametitle{Noise models}
+
+    \begin{itemize}
+        \item The depolarizing channel
+        \item Phenomenological noise
+        \item Circuit-level noise (we generally have all error
+                probabilities equal the same value \\
+            for simulations \citereference{fowler_high-threshold_2009})
+    \end{itemize}
+
+    \vspace*{15mm}
+
+    \addreferences
+    {fowler_high-threshold_2009}
+    \stopreferences
+\end{frame}
+
 %%%%%%%%%%%%%%%%
 \subsection{Detector Error Models}
 \label{subsec:Detector Error Models}
@@ -305,10 +454,11 @@
 \end{frame}
 
 \begin{frame}
-    \frametitle{An Example: Steane Code Detector Error Model}
+    \frametitle{Example: 3-Qubit Repetition Code Detector\\ Error
+    Model for Circuit Level Noise}
 
     \begin{itemize}
-        \item New Syndrome Extraction Circuitry \textcolor{red}{Is a
+        \item New Syndrome Extraction Circuitry \red{Is a
             repetition of the old circuitry needed?}
         \item New parity check matrix
         \item Highlighting of the SC-LDPC-code-like structure
@@ -319,16 +469,16 @@
     \frametitle{Challenges}
 
     \begin{itemize}
-        \item Multiple different errors are summarized $\rightarrow$
+        \item \red{Multiple different errors are summarized
-            short cycles \& degeneracy
+            $\rightarrow$ short cycles \& degeneracy}
             \footnote{
                 \texttt{
-                    https://www.math.cit.tum.de/fileadmin/w00ccg/math/\_my\_direct\_uploads/Dan\_Browne.pdf
+                    \red{https://www.math.cit.tum.de/fileadmin/w00ccg/math/\_my\_direct\_uploads/Dan\_Browne.pdf}
                 }
             }
             \\
-            $\rightarrow$ We generally don't use "normal BP" (BP +
+            \red{$\rightarrow$ We generally don't use "normal BP" (BP
-            OSD, BPGD, etc.)
+            + OSD, BPGD, etc.)}
     \end{itemize}
 \end{frame}
 
@@ -344,6 +494,8 @@
     \frametitle{Sliding Window Decoding}
 
     \begin{itemize}
+            % TODO: Do I have to explain BP?
+        \item \red{Do I have to explain BP}
         \item Give overview of existing research
         \item Explain exactly what they do in the main paper I am
             basing my work on
@@ -376,8 +528,14 @@
     \begin{itemize}
         \item Top level overview of entire system: X and Z syndrome
             extraction, logical operator measurement, where decoding
-            takes place, etc.
+            takes place, etc. (fig. 3 of \citereference{derks_designing_2025})
     \end{itemize}
+
+    \vspace*{15mm}
+
+    \addreferences
+    {derks_designing_2025}
+    \stopreferences
 \end{frame}
 
 \begin{frame}
@@ -400,7 +558,7 @@
     \frametitle{The lack of a Standard Evaluation System}
 
     \begin{itemize}
-        \item \textcolor{red}{Look into ECCentric}
+        \item \red{Look into ECCentric}
         \item There is not even a standard figure of merit (e.g.,
             FER/BER over SNR in classical case) $\rightarrow$
             Multiple different kinds of plots (e.g., footprint)
@@ -417,9 +575,33 @@
     \end{itemize}
 \end{frame}
 
+\begin{frame}[t]
+    \frametitle{Questions}
+
+    \begin{minipage}[c]{0.65\textwidth}
+        \centering
+
+        \LARGE Thank you for your attention!\\ Any questions?
+    \end{minipage}%
+    \begin{minipage}[c]{0.35\textwidth}
+        \centering
+        \begin{figure}[H]
+            \centering
+
+            \begin{tikzpicture}[every node/.style={scale=10}]
+                \node at (0, 0)
+                {\textcolor{kit-blue}{{\fontfamily{phv}\selectfont ?}}};
+            \end{tikzpicture}
+        \end{figure}
+    \end{minipage}
+
+\end{frame}
+
 \appendix
 \beginbackup
 
+% TODO: Move arrow into syndrome extraction lower (branch from other
+% arrow) and change caption to "modified from [MSLS25]"
 \begin{frame}
     \frametitle{System Level Overview}
 
@@ -495,6 +677,16 @@
     \stopreferences
 \end{frame}
 
+% TODO: Is this really necessary?
+\begin{frame}
+    \frametitle{The Quantum Error Correction Landscape}
+
+    \begin{itemize}
+        \item Give basic overview of most promising avenues of
+            research (as in ECCentric paper)
+    \end{itemize}
+\end{frame}
+
 \backupend
 
 \end{document}