personal_interest_mirrors/Quantum-Computation-course-Basel
1
0
Fork 0
mirror of https://github.com/quantumjim/Quantum-Computation-course-Basel.git synced 2025-11-08 22:14:25 +01:00
Code Issues Actions Packages Projects Releases Wiki Activity

add ex 3

Browse Source
This commit is contained in:
quantumjim 2024-10-29 09:31:42 +01:00
parent 53920df68e
commit 0f01a5d8ab
2 changed files with 78 additions and 2 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

6
README.md
View File

@ -51,8 +51,10 @@ For some lectures there is also some extra content. We'll cover it if there is t
Four sets of take-home exercises set throughout the course. There will be hints sessions at 16:15 on the dates that these are set. Solutions will be presented after the exercises have been graded. Four sets of take-home exercises set throughout the course. There will be hints sessions at 16:15 on the dates that these are set. Solutions will be presented after the exercises have been graded.
* [Exercise 1](exercises/Exercise1.ipynb): Set 01/10/2024, due 15/10/2024. * [Exercise 1](exercises/Exercise1.ipynb): Set 01/10/2024, due 15/10/2024, Anatolii as TA.
* [Exercise 2](exercises/Exercise2.ipynb): Set 15/10/2024, due 29/10/2024. * [Exercise 2](exercises/Exercise2.ipynb): Set 15/10/2024, due 29/10/2024, Anatolii as TA.
* [Exercise 3](exercises/Exercise3.ipynb): Set 29/10/2024, due 12/11/2024, Kacper as TA.
**Note: These exercises form 50% of your final grade** **Note: These exercises form 50% of your final grade**

74
exercises/Exercise3.ipynb Normal file
View File

@ -0,0 +1,74 @@
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Exercise 3\n",
"\n",
"This sheet is about [density matrices](../QI_course/2_The_Qubit.pdf), [the partial trace](../QI_course/3_Quantum_Information.pdf) and [the Schmidt decomposition](../QI_course/6_Quantum_Correlations_part_1.pdf). Take a look at the linked materials to find out more."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 1\n",
"\n",
"Given an abitrary single qubit state $a|0\\rangle +b|1\\rangle$:\n",
"\n",
"(a) Write down the correponding density matrix.\n",
"\n",
"(b) Write down the density matrix representing the effect of applying an $X$ with probability $q_x$, $Y$ with probability $q_y$ and $Z$ with probability $q_z$.\n",
"\n",
"(c) Write down the density matrix for representing the effect of replacing the state with $I/2$ with probability $p$.\n",
"\n",
"(d) For what values of $q_x$, $q_y$ and $q_z$ is the effect of (b) equivalent to that of (c)?\n",
"\n",
"## 2\n",
"\n",
"(a) Write down the density matrix of a single qubit which is in state $|0\\rangle$ with probability $2/3$ and $|+\\rangle$ with probability $1/3$.\n",
"\n",
"(b) Find a two qubit state such that one of the reduced density matrices is equal to the density matrix in (a).\n",
"\n",
"(c) Write down a density matrix for a state that is not entangled, but which has the same reduced density matrices as your answer for (b).\n",
"\n",
"## 3\n",
"\n",
"Consider the three qubit state $(|001\\rangle + |010\\rangle)+ |100\\rangle/\\sqrt{3}$.\n",
"\n",
"(a) What is the reduced density matrix for a single qubit in this state?\n",
"\n",
"(b) What is the reduced density matrix for a pair of qubits in this state?\n",
"\n",
"(c) Use the Schmidt decomposition to rewrite the state, with one qubit on one side of the bipartition and two qubits on the other."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"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": "3.9.1"
}
},
"nbformat": 4,
"nbformat_minor": 4
}