Create qpe1.tex

Author: mhjensen

doc/src/week12/Latexslides/qpe1.tex

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+\documentclass{beamer}
+\usepackage{graphicx}
+\usepackage{amsmath, amssymb}
+\usepackage{qcircuit} % For quantum circuits
+\usepackage{tikz}
+\usetikzlibrary{quantikz} % Quantum circuit diagrams
+
+\title{Quantum Phase Estimation}
+\author{Morten HJ}
+\date{Modified slides from 2009}
+
+\begin{document}
+
+\begin{frame}
+    \titlepage
+\end{frame}
+
+% Slide: Introduction
+\begin{frame}{Introduction to Quantum Phase Estimation (QPE)}
+    \begin{itemize}
+        \item QPE is a fundamental quantum algorithm used to estimate the phase $\phi$ in eigenvalue equations of unitary operators.
+        \item Given a unitary operator $U$ and an eigenvector $| \psi \rangle$ such that:
+        \[
+        U | \psi \rangle = e^{2\pi i \phi} | \psi \rangle
+        \]
+        the goal of QPE is to estimate $\phi$.
+        \item Applications:
+        \begin{itemize}
+            \item Shor's factoring algorithm
+            \item Hamiltonian simulation
+            \item Quantum chemistry
+        \end{itemize}
+    \end{itemize}
+\end{frame}
+
+% Slide: Quantum Circuit
+\begin{frame}{Quantum Circuit for QPE}
+    \centering
+    \begin{quantikz}
+        \lstick[wires=1]{$n$ qubits} & \gate{H} & \qwbundle{n} & \ctrl{1} & \qw & \qw & \gate{\text{QFT}^{\dagger}} & \meter{} \\
+        \lstick[wires=1]{1 qubit} & \qw & \qw & \gate{U^{2^j}} & \qw & \qw & \qw & \qw
+    \end{quantikz}
+    \begin{itemize}
+        \item The first register contains $n$ qubits initialized to $|0\rangle$.
+        \item Hadamard gates create a superposition state.
+        \item Controlled-$U^{2^j}$ operations apply phase shifts.
+        \item The inverse Quantum Fourier Transform (QFT) extracts phase information.
+    \end{itemize}
+\end{frame}
+
+% Slide: State Evolution
+\begin{frame}{State Evolution in QPE}
+    \begin{itemize}
+        \item Initial state:
+        \[
+        | \psi_0 \rangle = | 0 \rangle^{\otimes n} \otimes | \psi \rangle
+        \]
+        \item After Hadamard transform:
+        \[
+        \frac{1}{\sqrt{2^n}} \sum_{k=0}^{2^n - 1} | k \rangle | \psi \rangle
+        \]
+        \item Applying controlled-$U$ gates:
+        \[
+        \frac{1}{\sqrt{2^n}} \sum_{k=0}^{2^n - 1} | k \rangle U^k | \psi \rangle
+        \]
+        Since $U | \psi \rangle = e^{2\pi i \phi} | \psi \rangle$, this becomes:
+        \[
+        \frac{1}{\sqrt{2^n}} \sum_{k=0}^{2^n - 1} e^{2\pi i k \phi} | k \rangle | \psi \rangle
+        \]
+    \end{itemize}
+\end{frame}
+
+% Slide: Inverse QFT
+\begin{frame}{Inverse Quantum Fourier Transform (QFT)}
+    \begin{itemize}
+        \item The QFT maps computational basis states to phase-encoded states:
+        \[
+        | k \rangle \to \frac{1}{\sqrt{2^n}} \sum_{j=0}^{2^n - 1} e^{2\pi i k j / 2^n} | j \rangle
+        \]
+        \item Applying inverse QFT reveals the phase $\phi$ as a binary fraction.
+    \end{itemize}
+    \centering
+    \begin{tikzpicture}
+%        \node (qft) at (0,0) {\includegraphics[width=0.7\textwidth]{qft_diagram.png}};
+    \end{tikzpicture}
+\end{frame}
+
+% Slide: Example Computation
+\begin{frame}{Example: Phase of a Simple Unitary}
+    \begin{itemize}
+        \item Consider $U = \begin{bmatrix} 1 & 0 \\ 0 & e^{2\pi i / 8} \end{bmatrix}$, which has eigenvalues $e^{2\pi i \phi}$ with $\phi = 1/8$.
+        \item Using 3 qubits for precision:
+        \[
+        \phi = 0.001_2 = \frac{1}{8}
+        \]
+        \item Measurement should give result $\phi = 0.001$ (binary).
+    \end{itemize}
+\end{frame}
+
+% Slide: Applications of QPE
+\begin{frame}{Applications of QPE}
+    \begin{itemize}
+        \item \textbf{Shor's Algorithm}: Finds the period of a modular function.
+        \item \textbf{Hamiltonian Simulation}: Extracts energy eigenvalues for molecular systems.
+        \item \textbf{Quantum Metrology}: Improves phase measurement precision.
+    \end{itemize}
+\end{frame}
+
+% Slide: Error Analysis
+\begin{frame}{Error Analysis and Precision}
+    \begin{itemize}
+        \item The precision of QPE depends on the number of qubits used.
+        \item Probability of failure decreases exponentially with $n$ qubits.
+        \[
+        P_{\text{error}} \approx \frac{1}{2^{2n}}
+        \]
+        \item Noise and decoherence can impact accuracy, requiring error correction.
+    \end{itemize}
+\end{frame}
+
+% Slide: Conclusion
+\begin{frame}{Conclusion}
+    \begin{itemize}
+        \item QPE is a key quantum subroutine for extracting eigenphases.
+        \item Involves controlled unitary operations and inverse QFT.
+        \item Essential for algorithms like Shor's and Hamiltonian simulation.
+    \end{itemize}
+\end{frame}
+
+\end{document}