Create qpe.tex

Author: mhjensen

doc/src/week12/Latexslides/qpe.tex

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+\documentclass{beamer}
+\usepackage[utf8]{inputenc}
+\usepackage{quantikz}
+\usepackage{amsmath}
+\usepackage{braket}
+
+\title{Quantum Phase Estimation (QPE)}
+\author{Morten Hjorth-Jensen}
+\date{Slides from 2007}
+
+\begin{document}
+
+% Title Slide
+\begin{frame}
+    \titlepage
+\end{frame}
+
+% Outline Slide
+\begin{frame}{Outline}
+    \tableofcontents
+\end{frame}
+
+% Introduction Slide
+\section{Introduction}
+\begin{frame}{What is Quantum Phase Estimation?}
+    \begin{itemize}
+        \item Quantum Phase Estimation (QPE) is a fundamental quantum algorithm to estimate the eigenphase $\phi$ of a unitary operator $U$.
+        \item Given an eigenstate $\ket{\psi}$ such that $U\ket{\psi} = e^{2\pi i \phi}\ket{\psi}$, QPE estimates $\phi$ with high probability.
+        \item It is a crucial subroutine in algorithms like Shor's and quantum simulations.
+    \end{itemize}
+\end{frame}
+
+% Mathematical Background
+\section{Mathematical Background}
+\begin{frame}{Mathematical Background}
+    \begin{itemize}
+        \item Consider a unitary operator $U$ with eigenstate $\ket{\psi}$:
+        \[ U\ket{\psi} = e^{2\pi i \phi} \ket{\psi}, \quad \phi \in [0,1). \]
+        \item Goal: Estimate $\phi$ to $n$ bits of precision.
+        \item Quantum Fourier Transform (QFT) plays a key role in extracting phase information.
+    \end{itemize}
+\end{frame}
+
+% Circuit Structure
+\section{Circuit Structure}
+\begin{frame}{Quantum Circuit for QPE}
+    \begin{figure}
+        \centering
+        \begin{quantikz}
+            \lstick{$\ket{0}^{\otimes n}$} & \qwbundle{n} & \gate[2]{QFT^\dagger} & \meter{} \\
+            \lstick{$\ket{\psi}$} & \gate{U^{2^0}} & \gate{U^{2^1}} & \qw & \qw
+        \end{quantikz}
+        \caption{Quantum Phase Estimation Circuit with $n$ qubits.}
+    \end{figure}
+\end{frame}
+
+% Derivation of the Algorithm
+\section{Derivation of the Algorithm}
+\begin{frame}{Step 1: Initialization}
+    \begin{itemize}
+        \item The system starts in $\ket{0}^{\otimes n} \otimes \ket{\psi}$.
+        \item Apply Hadamard gates to the first $n$ qubits:
+        \[ \frac{1}{2^{n/2}} \sum_{k=0}^{2^n -1} \ket{k} \otimes \ket{\psi}. \]
+    \end{itemize}
+\end{frame}
+
+\begin{frame}{Step 2: Controlled Unitary Operations}
+    \begin{itemize}
+        \item Apply controlled-$U^{2^j}$ operations to create phase kickbacks:
+        \[ \frac{1}{2^{n/2}} \sum_{k=0}^{2^n -1} \ket{k} \otimes U^k\ket{\psi}. \]
+        \item For eigenstate $\ket{\psi}$ with eigenvalue $e^{2\pi i \phi}$:
+        \[ U^k\ket{\psi} = e^{2\pi i \phi k} \ket{\psi}. \]
+    \end{itemize}
+\end{frame}
+
+\begin{frame}{Step 3: Quantum Fourier Transform (QFT)}
+    \begin{itemize}
+        \item QFT on the first $n$ qubits transforms the state:
+        \[ \frac{1}{2^{n/2}} \sum_{k=0}^{2^n -1} e^{2\pi i \phi k} \ket{k}. \]
+        \item After QFT, we measure the most probable state representing $\phi$.
+    \end{itemize}
+\end{frame}
+
+% Applications
+\section{Applications}
+\begin{frame}{Applications of QPE}
+    \begin{itemize}
+        \item Estimating eigenvalues of Hermitian operators.
+        \item Integral part of Shor's algorithm for integer factorization.
+        \item Quantum simulation for molecular energy estimation.
+    \end{itemize}
+\end{frame}
+
+% Conclusion
+\section{Conclusion}
+\begin{frame}{Conclusion}
+    \begin{itemize}
+        \item QPE is a powerful quantum algorithm for phase estimation.
+        \item Leverages QFT and controlled unitary operations.
+        \item Forms a foundational block for many quantum algorithms.
+    \end{itemize}
+\end{frame}
+
+% References
+\begin{frame}{References}
+    \begin{itemize}
+        \item M. A. Nielsen and I. L. Chuang, \textit{Quantum Computation and Quantum Information}.
+        \item Quantum algorithms lecture notes.
+    \end{itemize}
+\end{frame}
+
+\end{document}