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‎doc/src/week12/Latexslides/qpe_lecture.tex‎

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\documentclass[11pt]{article}
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\usepackage[utf8]{inputenc}
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\usepackage{amsmath, amssymb}
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\usepackage{braket}
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\usepackage{quantikz}
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\usepackage{geometry}
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\usepackage{graphicx}
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\geometry{a4paper, margin=1in}
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\title{Quantum Phase Estimation (QPE)}
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\author{Morten HJ}
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\date{\today}
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\begin{document}
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\maketitle
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\tableofcontents
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\newpage
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% Introduction
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\section{Introduction}
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Quantum Phase Estimation (QPE) is a fundamental quantum algorithm that estimates the eigenphase $\phi$ of a unitary operator $U$. Given an eigenstate $\ket{\psi}$ satisfying:
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\begin{equation}
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U\ket{\psi} = e^{2\pi i \phi} \ket{\psi},
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\end{equation}
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QPE provides an estimate of $\phi$ with high probability. It is a crucial subroutine for algorithms like Shor's factoring algorithm and quantum simulations.
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% Mathematical Background
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\section{Mathematical Background}
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The goal of QPE is to estimate the phase $\phi \in [0,1)$ where:
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\begin{equation}
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U\ket{\psi} = e^{2\pi i \phi} \ket{\psi}.
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\end{equation}
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\subsection{Quantum Fourier Transform (QFT)}
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The Quantum Fourier Transform (QFT) on $n$ qubits is defined as:
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\begin{equation}
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\ket{x} \rightarrow \frac{1}{\sqrt{2^n}}\sum_{y=0}^{2^n-1} e^{2\pi i xy / 2^n}\ket{y}.
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\end{equation}
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QFT is essential for extracting phase information in QPE.
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\newpage
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% Quantum Circuit and Working
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\section{Quantum Circuit and Working}
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The quantum circuit for QPE consists of two quantum registers:
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\begin{itemize}
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\item The first register with $n$ qubits is initialized to $\ket{0}^{\otimes n}$.
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\item The second register is initialized to the eigenstate $\ket{\psi}$ of $U$.
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\end{itemize}
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\begin{figure}[h!]
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\centering
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\begin{quantikz}
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\lstick{$\ket{0}^{\otimes n}$} & \gate{H} & \ctrl{1} & \gate{QFT^\dagger} & \meter{} \\
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\lstick{$\ket{\psi}$} & \qw & \gate{U^{2^j}} & \qw & \qw
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\end{quantikz}
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\caption{Quantum Phase Estimation Circuit}
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\end{figure}
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% Derivation of the Algorithm
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\section{Derivation of the Algorithm}
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\subsection{Step 1: Initialization}
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The system starts in the state:
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\begin{equation}
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\ket{0}^{\otimes n} \otimes \ket{\psi}.
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\end{equation}
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Applying Hadamard gates to the first register creates a superposition:
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\begin{equation}
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\frac{1}{2^{n/2}}\sum_{k=0}^{2^n -1} \ket{k}\otimes \ket{\psi}.
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\end{equation}
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\subsection{Step 2: Controlled Unitary Operations}
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Controlled-$U^{2^j}$ gates apply the operation $U$ with exponential powers:
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\begin{equation}
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\frac{1}{2^{n/2}}\sum_{k=0}^{2^n -1} \ket{k} \otimes U^k\ket{\psi}.
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\end{equation}
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For eigenstate $\ket{\psi}$, this becomes:
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\begin{equation}
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\frac{1}{2^{n/2}}\sum_{k=0}^{2^n -1} e^{2\pi i \phi k}\ket{k} \otimes \ket{\psi}.
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\end{equation}
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\subsection{Step 3: Quantum Fourier Transform (QFT)}
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Applying QFT to the first register transforms the state to:
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\begin{equation}
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\sum_{k=0}^{2^n -1} c_k \ket{k},
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\end{equation}
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where coefficients $c_k$ are peaked near $k \approx 2^n \phi$.
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\subsection{Step 4: Measurement}
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Measuring the first register gives an $n$-bit estimate of $\phi$ with high probability.
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\newpage
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% Applications
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\section{Applications of QPE}
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QPE is a foundational algorithm with several key applications:
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\begin{itemize}
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\item \textbf{Factoring and Cryptography:} Integral to Shor's algorithm for integer factoring.
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\item \textbf{Quantum Simulations:} Estimating eigenvalues of Hamiltonians in quantum chemistry.
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\item \textbf{Amplitude Estimation:} Enhances algorithms like Grover's search.
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\end{itemize}
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% Challenges and Practical Considerations
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\section{Challenges and Practical Considerations}
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\begin{itemize}
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\item \textbf{Qubit Requirements:} Requires a large number of qubits for high precision.
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\item \textbf{Gate Depth:} Controlled-$U^{2^j}$ operations increase gate complexity.
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\item \textbf{Noise Sensitivity:} Errors in QFT or controlled gates impact accuracy.
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\end{itemize}
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% Conclusion
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\section{Conclusion}
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Quantum Phase Estimation is a powerful quantum algorithm for extracting phase information of unitary operators. It forms the foundation for many advanced quantum algorithms and demonstrates the power of quantum Fourier transforms in computational tasks.
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% References
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\section{References}
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\begin{thebibliography}{9}
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\bibitem{nielsen} M. A. Nielsen and I. L. Chuang, \textit{Quantum Computation and Quantum Information}, Cambridge University Press, 2000.
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\bibitem{preskill} J. Preskill, \textit{Lecture Notes on Quantum Computation}, Caltech.
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\end{thebibliography}
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\end{document}

‎doc/src/week12/Latexslides/shor_beamer.tex‎

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\documentclass{beamer}
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\usepackage[utf8]{inputenc}
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\usepackage{quantikz}
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\usepackage{amsmath}
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\usepackage{braket}
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\title{Shor's Algorithm}
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\author{Morten HJ}
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\date{2006}
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\begin{document}
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% Title Slide
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\begin{frame}
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\titlepage
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\end{frame}
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% Outline Slide
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\begin{frame}{Outline}
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\tableofcontents
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\end{frame}
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% Introduction Slide
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\section{Introduction}
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\begin{frame}{What is Shor's Algorithm?}
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\begin{itemize}
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\item Shor's algorithm is a quantum algorithm for integer factorization.
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\item It efficiently factors large integers by leveraging quantum period finding.
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\item Exponential speedup over the best-known classical algorithms.
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\item Breaks RSA encryption, which relies on the difficulty of factoring.
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\end{itemize}
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\end{frame}
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% Mathematical Background
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\section{Mathematical Background}
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\begin{frame}{Number Theory Basics}
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\begin{itemize}
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\item The problem: Factor an integer $N$ into its prime factors.
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\item Reduce factoring to period finding:
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\[ f(x) = a^x \mod N, \]
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where $a$ is randomly chosen such that $\gcd(a, N) = 1$.
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\item The period $r$ satisfies:
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\[ a^r \equiv 1 \mod N. \]
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\item Once $r$ is known, factors are given by:
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\[ \gcd(a^{r/2} \pm 1, N). \]
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\end{itemize}
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\end{frame}
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% Quantum Period Finding
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\section{Quantum Period Finding}
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\begin{frame}{Quantum Period Finding: Key to Shor's Algorithm}
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\begin{itemize}
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\item Quantum subroutine for finding the period $r$ of $f(x)$.
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\item Similar to Quantum Phase Estimation (QPE).
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\item Uses two quantum registers:
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\begin{itemize}
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\item First register: Superposition of all possible inputs $x$.
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\item Second register: Computes $f(x) = a^x \mod N$.
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\end{itemize}
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\end{itemize}
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\end{frame}
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% Quantum Circuit Design
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\section{Quantum Circuit Design}
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\begin{frame}{Quantum Circuit for Shor's Algorithm}
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\begin{figure}
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\centering
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\begin{quantikz}
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\lstick{$\ket{0}^{\otimes n}$} & \qwbundle{n} & \gate{H} & \ctrl{1} & \gate{QFT^\dagger} & \meter{} \\
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\lstick{$\ket{1}$} & \qw & \qw & \gate{U_f} & \qw & \qw
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\end{quantikz}
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\caption{Quantum circuit for period finding in Shor's algorithm.}
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\end{figure}
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\end{frame}
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\begin{frame}{Applying Quantum Fourier Transform (QFT)}
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\begin{itemize}
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\item The QFT is applied to the first register after the controlled unitary operations.
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\item Peaks in the measurement correspond to multiples of $1/r$.
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\item Post-processing classically determines $r$ by continued fractions.
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\end{itemize}
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\end{frame}
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% Classical Post-Processing
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\section{Classical Post-Processing}
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\begin{frame}{Classical Post-Processing}
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\begin{itemize}
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\item Once $r$ is found, factors of $N$ are computed by:
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\[ \gcd(a^{r/2} \pm 1, N). \]
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\item If $r$ is odd or $a^{r/2} \equiv -1 \mod N$, try a different $a$.
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\item With high probability, this yields a non-trivial factor of $N$.
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\end{itemize}
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\end{frame}
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% Complexity and Practical Considerations
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\section{Complexity and Practical Considerations}
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\begin{frame}{Complexity and Practical Considerations}
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\begin{itemize}
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\item Quantum Complexity: Polynomial in $\log N$.
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\item Classical Best Known: Sub-exponential (Number Field Sieve).
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\item Quantum Advantage: Exponential speedup over classical algorithms.
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\item Practical Challenges: Quantum error correction, large qubit counts.
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\end{itemize}
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\end{frame}
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% Conclusion
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\section{Conclusion}
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\begin{frame}{Conclusion}
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\begin{itemize}
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\item Shor's algorithm is a groundbreaking quantum algorithm for factoring.
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\item Combines quantum period finding with classical number theory.
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\item Highlights the potential of quantum computing to break RSA.
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\end{itemize}
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\end{frame}
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% References
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\begin{frame}{References}
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\begin{itemize}
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\item P. Shor, "Algorithms for Quantum Computation: Discrete Logarithms and Factoring".
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\item M. A. Nielsen and I. L. Chuang, \textit{Quantum Computation and Quantum Information}.
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\end{itemize}
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\end{frame}
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\end{document}

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