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| { | ||
| "cells": [ | ||
| { | ||
| "cell_type": "markdown", | ||
| "id": "e4f44eb7", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "<h1 style=\"text-align: center;\"><b>TMQS Workshop 2024</b> @ Zuse Institute Berlin</h1>\n", | ||
| "<h2 style=\"text-align: center;\">Summer School on Tensor Methods for Quantum Simulation</h2>\n", | ||
| "<h2 style=\"text-align: center;\">June 3 - 5, 2024</h2>\n", | ||
| "$\\def\\tcoreleft{\\textcolor{MidnightBlue}{\\Huge⦸}}$\n", | ||
| "$\\def\\tcorecenter{\\textcolor{RedOrange}{\\Huge⦿}}$\n", | ||
| "$\\def\\tcoreright{\\textcolor{MidnightBlue}{\\Huge\\oslash}}$\n", | ||
| "<div style=\"text-align: center; background-color:#D6EAF8;padding-top:20px;padding-bottom:5px\">\n", | ||
| "<table style=\"font-size:16pt\">\n", | ||
| " <tr style=\"background-color:#D6EAF8;\">\n", | ||
| " <td>$\\tcoreleft$</td>\n", | ||
| " <td>$-$</td>\n", | ||
| " <td>$\\tcoreleft$</td>\n", | ||
| " <td>$-$</td>\n", | ||
| " <td>$\\tcoreleft$</td>\n", | ||
| " <td>$-$</td>\n", | ||
| " <td>$\\cdots$</td>\n", | ||
| " <td>$-$</td>\n", | ||
| " <td>$\\tcorecenter$</td>\n", | ||
| " <td>$-$</td>\n", | ||
| " <td>$\\cdots$</td>\n", | ||
| " <td>$-$</td>\n", | ||
| " <td>$\\tcoreright$</td>\n", | ||
| " <td>$-$</td>\n", | ||
| " <td>$\\tcoreright$</td>\n", | ||
| " </tr>\n", | ||
| " <tr style=\"background-color:#D6EAF8;\">\n", | ||
| " <td style=\"text-align: center;\">$\\tiny\\mid$</td>\n", | ||
| " <td></td>\n", | ||
| " <td style=\"text-align: center;\">$\\tiny\\mid$</td>\n", | ||
| " <td></td>\n", | ||
| " <td style=\"text-align: center;\">$\\tiny\\mid$</td>\n", | ||
| " <td></td>\n", | ||
| " <td></td>\n", | ||
| " <td></td>\n", | ||
| " <td style=\"text-align: center;\">$\\tiny\\mid$</td>\n", | ||
| " <td></td>\n", | ||
| " <td></td>\n", | ||
| " <td></td>\n", | ||
| " <td style=\"text-align: center;\">$\\tiny\\mid$</td>\n", | ||
| " <td></td>\n", | ||
| " <td style=\"text-align: center;\">$\\tiny\\mid$</td>\n", | ||
| " </tr>\n", | ||
| "</table>\n", | ||
| "</div>" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "id": "6f6cc702", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "***" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "id": "74ce99d5-268a-4ad8-980b-de19d60b6be6", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## **Session 4 - Quantum Information**" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "id": "e8441c54-ffd4-45b4-ab04-ff8cf85e6474", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "***" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "id": "a781c1c8", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Exercise 4.1\n", | ||
| "\n", | ||
| "Which of the following are valid quantum states?\n", | ||
| "\n", | ||
| "$\\hspace{1cm}$$\\begin{pmatrix} 0 \\\\ 1\\end{pmatrix}$, $\\quad \\begin{pmatrix} 1 \\\\ 1\\end{pmatrix}$, $\\quad \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0 \\\\ -i\\end{pmatrix}$, $\\quad \\frac{1}{\\sqrt{3}}\\begin{pmatrix} 1 \\\\ 2\\end{pmatrix}$, $\\quad \\begin{pmatrix} \\sqrt{2/3} \\\\ i/\\sqrt{3}\\end{pmatrix}$\n", | ||
| "\n", | ||
| "What is the probability to measure $0$ and $1$ for the valid quantum states?" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "id": "6e1b9898-10a5-422d-8cdb-a8f42bdc01c4", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "***" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "id": "0e8243a2-a54e-4aaf-b460-f1f7f9e369b0", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Exercise 4.2\n", | ||
| "\n", | ||
| "**a)**$\\quad$Write down the state vector of two quantum states \n", | ||
| "\n", | ||
| "$\\hspace{1cm}$$\\displaystyle|\\Psi_1\\rangle = \\alpha_1 |0\\rangle + \\beta_1 |1\\rangle \\quad $ and $\\quad |\\Psi_2\\rangle = \\alpha_2 |0\\rangle + \\beta_2 |1\\rangle$, \n", | ||
| "\n", | ||
| "$\\hspace{0.35cm}$$\\quad$i.e. the tensor product, in the computational basis. Write down the basis vectors of the composite system.\n", | ||
| "\n", | ||
| "**b)**$\\quad$Consider the $2$-qubit state \n", | ||
| "\n", | ||
| "$\\hspace{1cm}$$\\displaystyle|\\Psi\\rangle = \\frac{1}{\\sqrt{2}} |00\\rangle + \\frac{1}{2}|01\\rangle + \\frac{1}{2} |11\\rangle$. \n", | ||
| "\n", | ||
| "$\\hspace{0.35cm}$$\\quad$What is the state after a measurement of the first qubit where you obtain $|0\\rangle$?\n", | ||
| "\n", | ||
| "$\\hspace{0.35cm}$$\\quad$Is this an entangled state?\n", | ||
| "\n", | ||
| "$\\hspace{0.35cm}$$\\quad$*Hint:* Quantum states are normalized!" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "id": "818bf57f-be0c-408a-9776-414578b7763a", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "***" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "id": "bed85201-1bfb-45a2-a57a-44332f370276", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Exercise 4.3\n", | ||
| "\n", | ||
| "Suppose you have $n + 1$ qubits. We will write $|\\vec{x}\\rangle$ to mean the $n$-qubit classical state given by the number $x$ in binary. For instance, if $n = 2$ then:\n", | ||
| "\n", | ||
| "$\\hspace{0.5cm}$$|\\vec{0}\\rangle = |00\\rangle , \\quad |\\vec{1}\\rangle = |01\\rangle, \\quad |\\vec{2}\\rangle = |10\\rangle, \\quad |\\vec{3}\\rangle = |11\\rangle$.\n", | ||
| "\n", | ||
| "Assume the qubits are in the state\n", | ||
| "\n", | ||
| "$\\hspace{0.5cm}$$\\displaystyle |\\Psi\\rangle = \\frac{1}{\\sqrt{2^n}} \\sum_{x=0}^{2^n -1} | \\vec{x} \\rangle \\otimes | x~\\text{mod}~2 \\rangle$.\n", | ||
| "\n", | ||
| "**a)**$\\quad$What is the resulting state if we measure the last qubit and obtain $|0\\rangle$?\n", | ||
| "\n", | ||
| "**b)**$\\quad$What is the resulting state if we measure the last qubit and obtain $|1\\rangle$?" | ||
| ] | ||
| } | ||
| ], | ||
| "metadata": { | ||
| "kernelspec": { | ||
| "display_name": "Python 3 (ipykernel)", | ||
| "language": "python", | ||
| "name": "python3" | ||
| }, | ||
| "language_info": { | ||
| "codemirror_mode": { | ||
| "name": "ipython", | ||
| "version": 3 | ||
| }, | ||
| "file_extension": ".py", | ||
| "mimetype": "text/x-python", | ||
| "name": "python", | ||
| "nbconvert_exporter": "python", | ||
| "pygments_lexer": "ipython3", | ||
| "version": "3.11.8" | ||
| } | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 5 | ||
| } |
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