update week 11

Author: mhjensen

doc/pub/week11/html/week11-bs.html

@@ -0,0 +1,91 @@
+\documentclass{beamer}
+
+\usepackage{amsmath, amssymb, braket, tikz}
+\usetikzlibrary{quantikz}
+
+\title{Quantum Fourier Transform (QFT) on Two Qubits}
+\author{MHJ}
+\date{\today}
+
+\begin{document}
+
+\begin{frame}
+   \titlepage
+\end{frame}
+
+% Slide 1: Introduction
+\begin{frame}{Introduction}
+   \textbf{Quantum Fourier Transform (QFT)}
+   \begin{itemize}
+       \item Quantum analogue of the Discrete Fourier Transform (DFT).
+       \item Used in quantum algorithms like Shor's algorithm and phase estimation.
+       \item Converts computational basis states into a superposition of phases.
+   \end{itemize}
+\end{frame}
+
+% Slide 2: Definition
+\begin{frame}{Definition of QFT}
+   The QFT on an \( n \)-qubit system is defined as:
+   \[
+   \text{QFT} \ket{x} = \frac{1}{\sqrt{2^n}} \sum_{y=0}^{2^n-1} e^{2\pi i xy / 2^n} \ket{y}
+   \]
+   where \( x \) is the input integer in binary representation.
+\end{frame}
+
+% Slide 3: QFT on Two Qubits
+\begin{frame}{QFT on Two Qubits}
+   For a two-qubit system (\( n=2 \)), the QFT transformation is:
+   \[
+   \text{QFT} \ket{x_1 x_0} = \frac{1}{2} \sum_{y=0}^{3} e^{2\pi i x y / 4} \ket{y}
+   \]
+   where \( x = x_1 x_0 \) (binary representation).
+\end{frame}
+
+
+
+% Slide 5: Matrix Representation
+\begin{frame}{Matrix Representation}
+   The QFT on two qubits has the matrix:
+   \[
+   U_{\text{QFT}} = \frac{1}{2}
+   \begin{bmatrix}
+       1 & 1 & 1 & 1 \\
+       1 & i & -1 & -i \\
+       1 & -1 & 1 & -1 \\
+       1 & -i & -1 & i
+   \end{bmatrix}
+   \]
+\end{frame}
+
+% Slide 6: Example Calculation
+\begin{frame}{Example Calculation}
+   Applying QFT to \( \ket{10} \) (binary 2):
+   \[
+   \text{QFT} \ket{10} = \frac{1}{2} (\ket{0} + i\ket{1} - \ket{2} - i\ket{3})
+   \]
+\end{frame}
+
+% Slide 7: Conclusion
+\begin{frame}{Conclusion}
+   \begin{itemize}
+       \item The QFT transforms basis states into Fourier superpositions.
+       \item Two-qubit QFT involves Hadamards, phase gates, and a SWAP.
+       \item It is essential for quantum algorithms like phase estimation.
+   \end{itemize}
+\end{frame}
+
+\end{document}
+% Slide 4: Quantum Circuit
+\begin{frame}{Quantum Circuit for QFT (2 Qubits)}
+   \begin{center}
+       \begin{quantikz}
+           \lstick{\ket{x_1}} & \gate{H} & \ctrl{1} & \qw      & \swap{1} & \qw \\
+           \lstick{\ket{x_0}} & \gate{H} & \gate{R_2} & \qw & \targX{} & \qw
+       \end{quantikz}
+   \end{center}
+   \begin{itemize}
+       \item Hadamard $H$ applies equal superposition.
+%       \item $R_2$ is the controlled phase gate: $R_2 = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/2} \end{bmatrix}$.
+%       \item The final SWAP reorders the qubits.
+   \end{itemize}
+\end{frame}