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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", | ||
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" <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 $|1\\rangle$?" | ||
] | ||
} | ||
], | ||
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"display_name": "Python 3 (ipykernel)", | ||
"language": "python", | ||
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