mirror of
https://github.com/quantumjim/Quantum-Computation-course-Basel.git
synced 2025-11-24 01:14:30 +01:00
update for 2023
This commit is contained in:
James Wootton
20187
exercises_2022/Exercise1.ipynb
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20187
exercises_2022/Exercise1.ipynb
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exercises_2022/Exercise10.ipynb
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exercises_2022/Exercise10.ipynb
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {
|
||||
"colab_type": "text",
|
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"id": "pmm5uV8cQQN6"
|
||||
},
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"source": [
|
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"# Exercise 10"
|
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]
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},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## 1. Shor Code\n",
|
||||
"\n",
|
||||
"The Shor (or 9 Qubit) code is a method for quantum error correction based on the classical repetition code. Here we will consider the version where a logical qubit is first stored in three physical qubits using the encoding\n",
|
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"$$\n",
|
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"|+\\rangle_3 = |+++\\rangle, \\,\\,\\, |-\\rangle_3 = |---\\rangle.\n",
|
||||
"$$\n",
|
||||
"This protects against errors that try to flip $|+\\rangle_3$ to $|-\\rangle_3$, and vice-versa. But the corresponding errors for the $Z$ basis states\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"|0\\rangle_3 = \\frac{1}{\\sqrt{2}} (|+++\\rangle+|---\\rangle), \\,\\,\\, |1\\rangle_3 = \\frac{1}{\\sqrt{2}} (|+++\\rangle-|---\\rangle)\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"become more likely. To deal with this we take three of these logical qubits and use them (like the original physical qubits) to encode a single logical qubit. This uses the encoding:\n",
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||||
"\n",
|
||||
"$$\n",
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||||
"|0\\rangle_9 = |0\\rangle_3 \\otimes |0\\rangle_3 \\otimes |0\\rangle_3, \\,\\,\\, |1\\rangle_9 = |1\\rangle_3 \\otimes |1\\rangle_3 \\otimes |1\\rangle_3.\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"The end result is then a code that stores one logical qubit in 9 physical qubits, with stabilizer states\n",
|
||||
"\n",
|
||||
"$$\n",
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||||
"|0\\rangle_9 = \\left\\lbrack \\frac{1}{\\sqrt{2}} \\left(|+++\\rangle +|---\\rangle \\right) \\right\\rbrack^{\\otimes 3}, \\,\\,\\, |1\\rangle_9 = \\left\\lbrack\\frac{1}{\\sqrt{2}}\\left(|+++\\rangle-|---\\rangle\\right) \\right\\rbrack^{\\otimes 3}\n",
|
||||
"$$\n",
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"\n",
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"a) Find operators that act as $X$ and $Z$ on the logical qubit. What are the minimal number of qubits these act on?\n",
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"\n",
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||||
"b) Suppose $\\sigma_x$ errors occur independently on each qubit with probability $p_x$. What is the probability $P_x$ that a logical $X$ occurs after syndrome measurement and error correction? For simplicity you can determine this only up to lowest order in $p_x$.\n",
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"\n",
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||||
"c) Similarly, what is the probability $P_z$ of $Z$ errors, given that $\\sigma_z$ errors occur with probability $p_z$? For simplicity you can determine this only up to lowest order in $p_z$."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"colab_type": "text",
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||||
"id": "CJyxxSDUQQN9"
|
||||
},
|
||||
"source": [
|
||||
"## 2. Concatenated Shor Code\n",
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"\n",
|
||||
"In order to increase the performance of a code we can use the concept of concatenation. We will now consider this process for the Shor code.\n",
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||||
"\n",
|
||||
"Let us describe physical qubits as level-0 qubits, and suppose we have $n$ of them. We can use these to encode $n/9$ logical qubits, which we call level-1 qubits. These will have lower probabilities for noise than the level-0 qubits, but maybe not as low as we require. We can then use the level-1 qubits as if they were physical qubits, using them to encode $n/9^2$ level-2 qubits. This procedure can then be continued as many times as required, with the level-$(l-1)$ qubits always used as the physical qubits of the Shor codes that encode the level-$l$ qubits.\n",
|
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"\n",
|
||||
"a) In order to encode a single level-$l$ qubit, for arbitrary $l$, what is the number $n(l)$ of level-0 qubits required?\n",
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"\n",
|
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"b) The standard Shor code has distance $d=3$. What is the distance of a level-$l$ concatenated Shor code?\n",
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"\n",
|
||||
"c) We will use $p_x^{(l)}$ to denote the bit flip error probility at level-$l$. The level-0 error rate is that of the phyical qubit errors: $p_x^{(0)}=p_x$. The level-1 error rate is that of the logical qubits of the 9 qubit code: $p_x^{(1)}=P_x$. Use your result from question 1(b) to show that $p_x^{(l-1)}$ and $p_x^{(l)}$ are related by\n",
|
||||
"$$\n",
|
||||
"p_x^{(l)} < 27 \\left( p_x^{(l-1)} \\right)^2.\n",
|
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"$$\n",
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"\n",
|
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"d) Show that $p_x^{(l)}$ decays exponentially with $n(l)^\\beta$ when $p_x < 1/27$, and find $\\beta$.\n",
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"\n",
|
||||
"This is a proof of the `threshold theorem' for this code and error model. As long as the physical noise rate $p_x$ is below the threshold value of $1/27$ ( and $p_z$ is below its threshold of $1/9$), concatenation of the Shor code can achieve arbitrarily low error rates.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
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||||
"execution_count": null,
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||||
"metadata": {},
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||||
"outputs": [],
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||||
"source": []
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||||
}
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||||
],
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||||
"metadata": {
|
||||
"colab": {
|
||||
"collapsed_sections": [],
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||||
"name": "Exercise_4.ipynb",
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"provenance": [],
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"version": "0.3.2"
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},
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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||||
"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.10.4"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 1
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}
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631
exercises_2022/Exercise2.ipynb
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exercises_2022/Exercise2.ipynb
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||||
{
|
||||
"cells": [
|
||||
{
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||||
"cell_type": "markdown",
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"metadata": {
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||||
"tags": [
|
||||
"remove_cell"
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||||
]
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},
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"source": [
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"<h1 style=\"font-size:35px;\n",
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" color:black;\n",
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" \">Exercise 2: Quantum Logic Gates</h1>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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||||
"source": [
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||||
"Prerequisite\n",
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||||
"- [Qiskit basics](https://qiskit.org/documentation/tutorials/circuits/1_getting_started_with_qiskit.html)\n",
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||||
"- [Ch.1.2 The Atoms of Computation](https://qiskit.org/textbook/ch-states/atoms-computation.html)\n",
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||||
"\n",
|
||||
"Other relevant materials\n",
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||||
"- [Access IBM Quantum Systems](https://qiskit.org/documentation/install.html#access-ibm-quantum-systems)\n",
|
||||
"- [IBM Quantum Systems Configuration](https://quantum-computing.ibm.com/docs/manage/backends/configuration)\n",
|
||||
"- [Transpile](https://qiskit.org/documentation/apidoc/transpiler.html)\n",
|
||||
"- [IBM Quantum account](https://quantum-computing.ibm.com/docs/manage/account/ibmq)\n",
|
||||
"- [Quantum Circuits](https://qiskit.org/documentation/apidoc/circuit.html) "
|
||||
]
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||||
},
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||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
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||||
"metadata": {},
|
||||
"outputs": [],
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||||
"source": [
|
||||
"from qiskit import *\n",
|
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"from qiskit.visualization import plot_histogram\n",
|
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"import numpy as np"
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||||
]
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||||
},
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||||
{
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||||
"cell_type": "markdown",
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||||
"metadata": {},
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||||
"source": [
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||||
"## **Exercise 1**\n",
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"\n",
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"See 'Part 1' below, and find the circuits required for the:\n",
|
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"* (a) `XOR` gate;\n",
|
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"* (b) `AND` gate;\n",
|
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"* (c) `NAND` gate;\n",
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"* (d) `OR` gate.\n",
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"\n",
|
||||
"## **Exercise 2**\n",
|
||||
"\n",
|
||||
"See 'Part 2' below, and find a `layout` for which the AND gate compiles to 6 non-local gates for `ibmqx2`. Note that there is some randomness in the compiling process. So you might need to try a few times."
|
||||
]
|
||||
},
|
||||
{
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||||
"cell_type": "markdown",
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||||
"metadata": {},
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||||
"source": [
|
||||
"<h2 style=\"font-size:24px;\">Part 1: Classical logic gates with quantum circuits</h2>\n",
|
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"\n",
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||||
"<br>\n",
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"\n",
|
||||
"An implementation of the `NOT` gate is provided as an example."
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||||
]
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||||
},
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||||
{
|
||||
"cell_type": "code",
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||||
"execution_count": null,
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||||
"metadata": {},
|
||||
"outputs": [],
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||||
"source": [
|
||||
"def NOT(inp):\n",
|
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" \"\"\"An NOT gate.\n",
|
||||
" \n",
|
||||
" Parameters:\n",
|
||||
" inp (str): Input, encoded in qubit 0.\n",
|
||||
" \n",
|
||||
" Returns:\n",
|
||||
" QuantumCircuit: Output NOT circuit.\n",
|
||||
" str: Output value measured from qubit 0.\n",
|
||||
" \"\"\"\n",
|
||||
"\n",
|
||||
" qc = QuantumCircuit(1, 1) # A quantum circuit with a single qubit and a single classical bit\n",
|
||||
" qc.reset(0)\n",
|
||||
" \n",
|
||||
" # We encode '0' as the qubit state |0⟩, and '1' as |1⟩\n",
|
||||
" # Since the qubit is initially |0⟩, we don't need to do anything for an input of '0'\n",
|
||||
" # For an input of '1', we do an x to rotate the |0⟩ to |1⟩\n",
|
||||
" if inp=='1':\n",
|
||||
" qc.x(0)\n",
|
||||
" \n",
|
||||
" # barrier between input state and gate operation \n",
|
||||
" qc.barrier()\n",
|
||||
" \n",
|
||||
" # Now we've encoded the input, we can do a NOT on it using x\n",
|
||||
" qc.x(0)\n",
|
||||
" \n",
|
||||
" #barrier between gate operation and measurement\n",
|
||||
" qc.barrier()\n",
|
||||
" \n",
|
||||
" # Finally, we extract the |0⟩/|1⟩ output of the qubit and encode it in the bit c[0]\n",
|
||||
" qc.measure(0,0)\n",
|
||||
" qc.draw('mpl')\n",
|
||||
" \n",
|
||||
" # We'll run the program on a simulator\n",
|
||||
" backend = Aer.get_backend('qasm_simulator')\n",
|
||||
" # Since the output will be deterministic, we can use just a single shot to get it\n",
|
||||
" job = execute(qc, backend, shots=1, memory=True)\n",
|
||||
" output = job.result().get_memory()[0]\n",
|
||||
" \n",
|
||||
" return qc, output"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"## Test the function\n",
|
||||
"for inp in ['0', '1']:\n",
|
||||
" qc, out = NOT(inp)\n",
|
||||
" print('NOT with input',inp,'gives output',out)\n",
|
||||
" display(qc.draw())\n",
|
||||
" print('\\n')"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<h3 style=\"font-size: 20px\">📓 XOR gate</h3>\n",
|
||||
"\n",
|
||||
"Takes two binary strings as input and gives one as output.\n",
|
||||
"\n",
|
||||
"The output is '0' when the inputs are equal and '1' otherwise."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def XOR(inp1,inp2):\n",
|
||||
" \"\"\"An XOR gate.\n",
|
||||
" \n",
|
||||
" Parameters:\n",
|
||||
" inpt1 (str): Input 1, encoded in qubit 0.\n",
|
||||
" inpt2 (str): Input 2, encoded in qubit 1.\n",
|
||||
" \n",
|
||||
" Returns:\n",
|
||||
" QuantumCircuit: Output XOR circuit.\n",
|
||||
" str: Output value measured from qubit 1.\n",
|
||||
" \"\"\"\n",
|
||||
" \n",
|
||||
" qc = QuantumCircuit(2, 1) \n",
|
||||
" qc.reset(range(2))\n",
|
||||
" \n",
|
||||
" if inp1=='1':\n",
|
||||
" qc.x(0)\n",
|
||||
" if inp2=='1':\n",
|
||||
" qc.x(1)\n",
|
||||
" \n",
|
||||
" # barrier between input state and gate operation \n",
|
||||
" qc.barrier()\n",
|
||||
" \n",
|
||||
" # this is where your program for quantum XOR gate goes\n",
|
||||
" \n",
|
||||
" \n",
|
||||
" \n",
|
||||
" \n",
|
||||
" \n",
|
||||
" \n",
|
||||
" \n",
|
||||
" \n",
|
||||
" # barrier between input state and gate operation \n",
|
||||
" qc.barrier()\n",
|
||||
" \n",
|
||||
" qc.measure(1,0) # output from qubit 1 is measured\n",
|
||||
" \n",
|
||||
" #We'll run the program on a simulator\n",
|
||||
" backend = Aer.get_backend('qasm_simulator')\n",
|
||||
" #Since the output will be deterministic, we can use just a single shot to get it\n",
|
||||
" job = execute(qc, backend, shots=1, memory=True)\n",
|
||||
" output = job.result().get_memory()[0]\n",
|
||||
" \n",
|
||||
" return qc, output"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"## Test the function\n",
|
||||
"for inp1 in ['0', '1']:\n",
|
||||
" for inp2 in ['0', '1']:\n",
|
||||
" qc, output = XOR(inp1, inp2)\n",
|
||||
" print('XOR with inputs',inp1,inp2,'gives output',output)\n",
|
||||
" display(qc.draw())\n",
|
||||
" print('\\n')"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<h3 style=\"font-size: 20px\">📓 AND gate</h3>\n",
|
||||
"\n",
|
||||
"Takes two binary strings as input and gives one as output.\n",
|
||||
"\n",
|
||||
"The output is `'1'` only when both the inputs are `'1'`."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def AND(inp1,inp2):\n",
|
||||
" \"\"\"An AND gate.\n",
|
||||
" \n",
|
||||
" Parameters:\n",
|
||||
" inpt1 (str): Input 1, encoded in qubit 0.\n",
|
||||
" inpt2 (str): Input 2, encoded in qubit 1.\n",
|
||||
" \n",
|
||||
" Returns:\n",
|
||||
" QuantumCircuit: Output XOR circuit.\n",
|
||||
" str: Output value measured from qubit 2.\n",
|
||||
" \"\"\"\n",
|
||||
" qc = QuantumCircuit(3, 1) \n",
|
||||
" qc.reset(range(2))\n",
|
||||
" \n",
|
||||
" if inp1=='1':\n",
|
||||
" qc.x(0)\n",
|
||||
" if inp2=='1':\n",
|
||||
" qc.x(1)\n",
|
||||
" \n",
|
||||
" qc.barrier()\n",
|
||||
"\n",
|
||||
" # this is where your program for quantum AND gate goes\n",
|
||||
"\n",
|
||||
" \n",
|
||||
" \n",
|
||||
" \n",
|
||||
" \n",
|
||||
" \n",
|
||||
"\n",
|
||||
" qc.barrier()\n",
|
||||
" qc.measure(2, 0) # output from qubit 2 is measured\n",
|
||||
" \n",
|
||||
" # We'll run the program on a simulator\n",
|
||||
" backend = Aer.get_backend('qasm_simulator')\n",
|
||||
" # Since the output will be deterministic, we can use just a single shot to get it\n",
|
||||
" job = execute(qc, backend, shots=1, memory=True)\n",
|
||||
" output = job.result().get_memory()[0]\n",
|
||||
" \n",
|
||||
" return qc, output"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"## Test the function\n",
|
||||
"for inp1 in ['0', '1']:\n",
|
||||
" for inp2 in ['0', '1']:\n",
|
||||
" qc, output = AND(inp1, inp2)\n",
|
||||
" print('AND with inputs',inp1,inp2,'gives output',output)\n",
|
||||
" display(qc.draw())\n",
|
||||
" print('\\n')"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<h3 style=\"font-size: 20px\">📓 NAND gate</h3>\n",
|
||||
"\n",
|
||||
"Takes two binary strings as input and gives one as output.\n",
|
||||
"\n",
|
||||
"The output is `'0'` only when both the inputs are `'1'`."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def NAND(inp1,inp2):\n",
|
||||
" \"\"\"An NAND gate.\n",
|
||||
" \n",
|
||||
" Parameters:\n",
|
||||
" inpt1 (str): Input 1, encoded in qubit 0.\n",
|
||||
" inpt2 (str): Input 2, encoded in qubit 1.\n",
|
||||
" \n",
|
||||
" Returns:\n",
|
||||
" QuantumCircuit: Output NAND circuit.\n",
|
||||
" str: Output value measured from qubit 2.\n",
|
||||
" \"\"\"\n",
|
||||
" qc = QuantumCircuit(3, 1) \n",
|
||||
" qc.reset(range(3))\n",
|
||||
" \n",
|
||||
" if inp1=='1':\n",
|
||||
" qc.x(0)\n",
|
||||
" if inp2=='1':\n",
|
||||
" qc.x(1)\n",
|
||||
" \n",
|
||||
" qc.barrier()\n",
|
||||
" \n",
|
||||
" # this is where your program for quantum NAND gate goes\n",
|
||||
"\n",
|
||||
"\n",
|
||||
" \n",
|
||||
" \n",
|
||||
" \n",
|
||||
" \n",
|
||||
" \n",
|
||||
" qc.barrier()\n",
|
||||
" qc.measure(2, 0) # output from qubit 2 is measured\n",
|
||||
" \n",
|
||||
" # We'll run the program on a simulator\n",
|
||||
" backend = Aer.get_backend('qasm_simulator')\n",
|
||||
" # Since the output will be deterministic, we can use just a single shot to get it\n",
|
||||
" job = execute(qc,backend,shots=1,memory=True)\n",
|
||||
" output = job.result().get_memory()[0]\n",
|
||||
" \n",
|
||||
" return qc, output"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"## Test the function\n",
|
||||
"for inp1 in ['0', '1']:\n",
|
||||
" for inp2 in ['0', '1']:\n",
|
||||
" qc, output = NAND(inp1, inp2)\n",
|
||||
" print('NAND with inputs',inp1,inp2,'gives output',output)\n",
|
||||
" display(qc.draw())\n",
|
||||
" print('\\n')"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<h3 style=\"font-size: 20px\">📓 OR gate</h3>\n",
|
||||
"\n",
|
||||
"Takes two binary strings as input and gives one as output.\n",
|
||||
"\n",
|
||||
"The output is '1' if either input is '1'."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def OR(inp1,inp2):\n",
|
||||
" \"\"\"An OR gate.\n",
|
||||
" \n",
|
||||
" Parameters:\n",
|
||||
" inpt1 (str): Input 1, encoded in qubit 0.\n",
|
||||
" inpt2 (str): Input 2, encoded in qubit 1.\n",
|
||||
" \n",
|
||||
" Returns:\n",
|
||||
" QuantumCircuit: Output XOR circuit.\n",
|
||||
" str: Output value measured from qubit 2.\n",
|
||||
" \"\"\"\n",
|
||||
"\n",
|
||||
" qc = QuantumCircuit(3, 1) \n",
|
||||
" qc.reset(range(3))\n",
|
||||
" \n",
|
||||
" if inp1=='1':\n",
|
||||
" qc.x(0)\n",
|
||||
" if inp2=='1':\n",
|
||||
" qc.x(1)\n",
|
||||
" \n",
|
||||
" qc.barrier()\n",
|
||||
" \n",
|
||||
" # this is where your program for quantum OR gate goes\n",
|
||||
"\n",
|
||||
"\n",
|
||||
" \n",
|
||||
" \n",
|
||||
" \n",
|
||||
" \n",
|
||||
" \n",
|
||||
" qc.barrier()\n",
|
||||
" qc.measure(2, 0) # output from qubit 2 is measured\n",
|
||||
" \n",
|
||||
" # We'll run the program on a simulator\n",
|
||||
" backend = Aer.get_backend('qasm_simulator')\n",
|
||||
" # Since the output will be deterministic, we can use just a single shot to get it\n",
|
||||
" job = execute(qc,backend,shots=1,memory=True)\n",
|
||||
" output = job.result().get_memory()[0]\n",
|
||||
" \n",
|
||||
" return qc, output"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"## Test the function\n",
|
||||
"for inp1 in ['0', '1']:\n",
|
||||
" for inp2 in ['0', '1']:\n",
|
||||
" qc, output = OR(inp1, inp2)\n",
|
||||
" print('OR with inputs',inp1,inp2,'gives output',output)\n",
|
||||
" display(qc.draw())\n",
|
||||
" print('\\n')"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<h2 style=\"font-size:24px;\">Part 2: AND gate on Quantum Computer</h2>\n",
|
||||
"<br>\n",
|
||||
"\n",
|
||||
"Real quantum computers are not able to implement arbitary gates directly. Instead, everything needs to be compiled (or 'transpiled') to the set of basic gates that the device can use. This usually consists of a set of single qubit rotations, as well as two qubit gates like `cx`.\n",
|
||||
"\n",
|
||||
"There are also limits on which `cx` gates can be used directly: only some pairs of control and target qubits are possible. To implement other `cx` gates, tricks such as using `swap` gates to effectively move information around must be used. The possible pairs of qubits on which `cx` gates can be applied is known as the 'connectivity' of the device.\n",
|
||||
"\n",
|
||||
"We'll now look at some examples. To make sure you don't end up in a queue for a busy device, we'll be using mock backends. These are designed to act exactly like real backends."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from qiskit.test.mock import FakeYorktown\n",
|
||||
"backend = FakeYorktown()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Upon executing the following cell you will be presented with a widget that displays all of the information about your choice of the backend. You can obtain information that you need by clicking on the tabs. For example, backend status, number of qubits and the connectivity are under `configuration` tab, where as the `Error Map` tab will reveal the latest noise information for the system. "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import qiskit.tools.jupyter\n",
|
||||
"\n",
|
||||
"backend"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The two system we are using (or at least pretending to) is `ibmqx2` (also known as `ibmq_yorktown`).\n",
|
||||
"\n",
|
||||
"Here's a circuit that applies an `AND` gate, compiled into single and two qubit gates (assuming full connectivity)."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"qc_and = QuantumCircuit(3)\n",
|
||||
"qc_and.ccx(0,1,2)\n",
|
||||
"print('AND gate')\n",
|
||||
"display(qc_and.draw())\n",
|
||||
"print('\\n\\nTranspiled AND gate with all the required connectiviy')\n",
|
||||
"qc_and.decompose().draw()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"This ideal transpilation requires 6 `cx` gates.\n",
|
||||
"\n",
|
||||
"There are often optimizations that the transpiler can perform that reduce the overall gate count, and thus total length of the input circuits. Note that the addition of swaps to match the device topology, and optimizations for reducing the length of a circuit are at odds with each other. In what follows we will make use of `initial_layout` that allows us to pick the qubits on a device used for the computation and `optimization_level`, an argument that allows selecting from internal defaults for circuit swap mapping and optimization methods to perform.\n",
|
||||
"\n",
|
||||
"You can learn more about transpile function in depth [here](https://qiskit.org/documentation/apidoc/transpiler.html)."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Rather than actually running the AND function, let's just look at the transpiled circuits. The following function does this for a given set of inputs."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# run the cell to define AND gate for real quantum system\n",
|
||||
"\n",
|
||||
"def AND(inp1, inp2, backend, layout):\n",
|
||||
" \n",
|
||||
" qc = QuantumCircuit(3, 1) \n",
|
||||
" qc.reset(range(3))\n",
|
||||
" \n",
|
||||
" if inp1=='1':\n",
|
||||
" qc.x(0)\n",
|
||||
" if inp2=='1':\n",
|
||||
" qc.x(1)\n",
|
||||
" \n",
|
||||
" qc.barrier()\n",
|
||||
" qc.ccx(0, 1, 2) \n",
|
||||
" qc.barrier()\n",
|
||||
" qc.measure(2, 0) \n",
|
||||
" \n",
|
||||
" qc_trans = transpile(qc, backend, initial_layout=layout, optimization_level=3)\n",
|
||||
" \n",
|
||||
" return qc_trans"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<h4 style=\"font-size: 16px\">Three qubits on <code>ibmqx2</code> with the triangle connectivity</h4>\n",
|
||||
" \n",
|
||||
"First, examine `ibmqx2` using the widget introduced earlier. Find the best set of three qubits to use for the `AND` gate, making best use of the connectivity."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# run this cell for the widget\n",
|
||||
"backend"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<p>📓 Assign your choice of layout to the list variable <code>layout</code> in the cell below</p>"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# Assign your choice of the initial_layout to the variable layout1 as a list \n",
|
||||
"# ex) layout = [0,2,4]\n",
|
||||
"layout = "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Compile the `AND` gate on `ibmqx2` by running the cell below."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {
|
||||
"scrolled": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"for input1 in ['0','1']:\n",
|
||||
" for input2 in ['0','1']:\n",
|
||||
" qc_trans1 = AND(input1, input2, backend, layout)\n",
|
||||
" \n",
|
||||
" print('For input '+input1+input2)\n",
|
||||
" print('# of nonlocal gates =',qc_trans1.num_nonlocal_gates())"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"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
|
||||
}
|
||||
87
exercises_2022/Exercise3.ipynb
Normal file
87
exercises_2022/Exercise3.ipynb
Normal file
@@ -0,0 +1,87 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "9ae539e2",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Exercise 3"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "fe84364c",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## 1. Mutually unbiased bases\n",
|
||||
"Show that the $X$, $Y$ and $Z$ bases are all unbiased with respect to each other: each\n",
|
||||
"state for one basis results in a completely random result for the other two bases."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "7e5f84c8",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## 2. Shifting certainty\n",
|
||||
"\n",
|
||||
"The state of a single qubit is characterized by three numbers, $\\langle \\sigma^x \\rangle$, $\\langle \\sigma^y \\rangle$ and $\\langle \\sigma^z \\rangle$,\n",
|
||||
"defined as\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\langle \\sigma^\\alpha \\rangle = p^\\alpha_0 - p^\\alpha_1,\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"where $p^z_0$ is the probability of the outcome \\texttt{0} for a $Z$ measurement, and so on.\n",
|
||||
"\n",
|
||||
"For any single qubit superposition $|\\psi\\rangle = c_0 |0\\rangle + c_1 |1\\rangle$,\n",
|
||||
"\n",
|
||||
"$\n",
|
||||
"\\langle \\sigma^x \\rangle^2 + \\langle \\sigma^y \\rangle^2 + \\langle \\sigma^z \\rangle^2 = 1\n",
|
||||
"$\n",
|
||||
"\n",
|
||||
"Verify this for the following states.\n",
|
||||
"\n",
|
||||
"* (a) $|\\psi\\rangle = \\cos \\theta \\, |0\\rangle + \\sin \\theta \\, |1\\rangle$\n",
|
||||
"* (b) $|\\psi\\rangle = \\frac{1}{\\sqrt{2}} \\, \\left( |0\\rangle \\,+\\, e^{i \\phi} |1\\rangle \\right)$\n",
|
||||
"\n",
|
||||
"## 3. A useful matrix\n",
|
||||
"\n",
|
||||
"Find the $2\\times2$ matrix $M$ such that\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"p^z_0 - p^z_1 = \\langle \\psi | M | \\psi \\rangle \\,\\,\\, \\forall |\\psi\\rangle\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "ac918ed2",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"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": 5
|
||||
}
|
||||
136
exercises_2022/Exercise4.ipynb
Normal file
136
exercises_2022/Exercise4.ipynb
Normal file
@@ -0,0 +1,136 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Exercise 4\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## 1. Alternative Pauli Basis States\n",
|
||||
"\n",
|
||||
"There are an infinite number of possible single qubit states. From a theoretical stand-point, the one we choose to label $| 0 \\rangle$ is arbitrary. So let’s consider the following alternative.\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"| \\bar 0 \\rangle = \\cos(\\theta) \\, | 0 \\rangle + \\sin(\\theta) \\, | 1 \\rangle.\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"For this $| \\bar 0 \\rangle$:\n",
|
||||
"\n",
|
||||
"(a) Find a corresponding orthogonal state $| \\bar 1 \\rangle$;\n",
|
||||
"\n",
|
||||
"(b) For this basis $| \\bar 0 \\rangle$, $| \\bar 1 \\rangle$, find mutually unbiased basis states $| \\bar + \\rangle$ and $| \\bar - \\rangle$;\n",
|
||||
"\n",
|
||||
"## 2. Properties of the Pauli Matrices\n",
|
||||
"\n",
|
||||
"Note: Sometimes the Pauli matrices are written as $X$, $Y$ and $Z$, and sometimes as $\\sigma_x$, $\\sigma_y$ and $\\sigma_z$. For the most part, the convention is an arbitrary choice. Once you’ve used them enough, you’ll hardly notice the difference (to the great annoyance of your students!).\n",
|
||||
"\n",
|
||||
"The Pauli matrices are defined\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"X = \n",
|
||||
"\\begin{pmatrix} \n",
|
||||
"0 & 1 \\\\\n",
|
||||
"1 & 0 \\\\\n",
|
||||
"\\end{pmatrix}, \\,\\,\n",
|
||||
"Y = \n",
|
||||
"\\begin{pmatrix} \n",
|
||||
"0 & -i \\\\\n",
|
||||
"i & 0 \\\\\n",
|
||||
"\\end{pmatrix}, \\,\\,\n",
|
||||
"Z = \n",
|
||||
"\\begin{pmatrix} \n",
|
||||
"1 & 0 \\\\\n",
|
||||
"0 & -1 \\\\\n",
|
||||
"\\end{pmatrix}, \\,\\,\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"(a) Show that each squares to the identity matrix.\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"I = \n",
|
||||
"\\begin{pmatrix} \n",
|
||||
"1 & 0 \\\\\n",
|
||||
"0 & 1 \\\\\n",
|
||||
"\\end{pmatrix}\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"(b) Show that $P_1 P_2 = - P_2 P_1$ for any pair of Paulis $P_1$ and $P_2$.\n",
|
||||
"\n",
|
||||
"(c) Show that $P_1 P_2 \\sim P_3$ for any pair of Paulis $P_1$ and $P_2$, where $P_3$ is the remaining Pauli.\n",
|
||||
"\n",
|
||||
"(d) Find the eigenvectors and eigenvalues of each Pauli.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## 3. The Hadamard\n",
|
||||
"\n",
|
||||
"The Hadamard matrix can be expressed\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"H = \\frac{1}{\\sqrt{2}}\n",
|
||||
"\\begin{pmatrix} \n",
|
||||
"1 & 1 \\\\\n",
|
||||
"1 & -1 \\\\\n",
|
||||
"\\end{pmatrix}\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"(a) Find the eigenvectors and eigenvalues of this matrix.\n",
|
||||
"\n",
|
||||
"(b) Show that $H$ also squares to identity.\n",
|
||||
"\n",
|
||||
"(c) Show that $H P_1 H^\\dagger \\sim P_2$ for Paulis $P_1$ and $P_2$. \n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## 4. Two-qubit Paulis\n",
|
||||
"\n",
|
||||
"For two qubits we can define a set of matrices $X_0$, $Y_0$ and $Z_0$ that behave as Paulis (i.e. they have the same properties as in 3a, 3b and 3c above). We can also define another separate set of matrices $X_1$, $Y_1$ and $Z_1$ that also behave as Paulis. Furthermore, any matrix from one of these sets will commute with any matrix from the other\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"P_0 P_1 = P_1 P_0, \\,\\, \\forall \\,\\, P_j \\, \\in \\, \\{X_j, Y_j, Z_j\\}.\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"Usually we define these sets in a very simple way, using the Paulis of one qubit for one set, and the Paulis of the other qubit for the other set,\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"X_0 = X \\otimes I, \\, X_1 = I \\otimes X, \\, \\rm{etc}.\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"But since this is an exercise, let's do something a bit more interesting! Consider the following choice of the first set,\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"X_0 = X \\otimes X, \\, Y_0 = Y \\otimes X,\\, Z_0 = Z \\otimes I\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"Find a corresponding $X_1$, $Y_1$ and $Z_1$."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"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
|
||||
}
|
||||
87
exercises_2022/Exercise5.ipynb
Normal file
87
exercises_2022/Exercise5.ipynb
Normal file
@@ -0,0 +1,87 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Exercise 5\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## 1. Clifford Gates and Paulis\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"(a) For $n$ qubits, there are $4^n$ possible tensor products of Paulis (one of which is the $n$-qubit identity). Show that (up to a phase) these can be expressed as a product of $2n$ $n$-qubit Paulis.\n",
|
||||
"\n",
|
||||
"(b) If $U$ is a Clifford gate, the following property holds\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"U P U^\\dagger \\sim P' \\,\\,\\,\\,\\, \\forall P,\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"where $P$ and $P'$ are Paulis and $\\sim$ denotes equality up to a factor of $\\pm 1$ or $\\pm i$. If this relation holds for the $2n$ Pauli generators of part (a), show that it also holds for all $n$-qubit Paulis.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## 2. Single-Qubit Clifford Gates\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"(a) Show that the Paulis are Cliffords themselves.\n",
|
||||
"\n",
|
||||
"(b) Show that $H$, $S$ and $S^\\dagger$ are Clifford gates.\n",
|
||||
"\n",
|
||||
"(c) Show that $T=S^{1/2}$ is not a Clifford gate.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## 3. Two-Qubit Clifford Gates\n",
|
||||
"\n",
|
||||
"For more than one qubit, Clifford gates map between tensor products of Pauli operators.\n",
|
||||
"\n",
|
||||
"For two qubits\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"U \\,( P \\otimes Q )\\, U^\\dagger \\sim P' \\otimes Q' \\,\\,\\,\\,\\, \\forall P,Q\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"where $P$, $Q$, $P'$ and $Q'$ are all Paulis and $\\sim$ denotes equality up to a factor of $\\pm 1$ or $\\pm i$.\n",
|
||||
"\n",
|
||||
"(a) Show that the controlled-NOT is a Clifford gate.\n",
|
||||
"\n",
|
||||
"(b) Show that the controlled-Hadamard is not a Clifford gate.\n",
|
||||
"\n",
|
||||
"## 4. Three-Qubit Clifford Gates\n",
|
||||
"\n",
|
||||
"(a) Provide an example of a three-qubit Clifford gate, and show that it is indeed a Clifford. This should be a truly three qubit gate, and therefore not one that can be expressed purely as a tensor product of single- and two-qubit gates.\n",
|
||||
"\n",
|
||||
"(b) Show that the Toffoli gate is not Clifford."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"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
|
||||
}
|
||||
243
exercises_2022/Exercise6.ipynb
Normal file
243
exercises_2022/Exercise6.ipynb
Normal file
File diff suppressed because one or more lines are too long
106
exercises_2022/Exercise7.ipynb
Normal file
106
exercises_2022/Exercise7.ipynb
Normal file
@@ -0,0 +1,106 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"colab_type": "text",
|
||||
"id": "pmm5uV8cQQN6"
|
||||
},
|
||||
"source": [
|
||||
"# Exercise 7\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## 1\n",
|
||||
"\n",
|
||||
"Given the ability to perform `cx` and `h` gates, and given a supply of auxiliary qubits in state $|0\\rangle$, show that it is possible to perform `x` gates with arbitrarily high probability of success."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"colab_type": "text",
|
||||
"id": "CJyxxSDUQQN9"
|
||||
},
|
||||
"source": [
|
||||
"## 2\n",
|
||||
"\n",
|
||||
"A general $n$-qubit state can be written\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"| \\psi \\rangle = \\sum_x c_x |x\\rangle = \\sum_x \\Re (c_x) |x\\rangle + i \\Im (c_x) |x\\rangle,\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"where the $|x\\rangle$ here denote the $n$-qubit Z basis states.\n",
|
||||
"\n",
|
||||
"For each such state we can write an equivalent state for which all amplitudes are real. Since the complex nature of the amplitudes effectively adds an extra degree of superposition to the state, we need to add an extra qubit to the system to encode the same information. The $n+1$ qubit state $| \\tilde \\psi \\rangle$ equivalent to $| \\psi \\rangle$ is then\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"| \\tilde \\psi \\rangle = \\sum_x \\Re (c_x) |x\\rangle \\otimes |0\\rangle + \\Im (c_x) |x\\rangle \\otimes |1\\rangle.\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"Note that the $i$ in $| \\psi \\rangle$ is replaced by the $|1\\rangle$ state on the extra qubit in $| \\tilde \\psi \\rangle$.\n",
|
||||
"\n",
|
||||
"a) For each $n$ qubit unitary $U$ we can define an equivalent $n+1$ qubit unitary $\\tilde U$, such that\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\tilde U (|x\\rangle \\otimes |0\\rangle) = \\widetilde{U |x\\rangle}, \\,\\, \\forall x\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"Write the effects of $\\tilde U (|x\\rangle \\otimes |0\\rangle)$ and $\\tilde U (|x\\rangle \\otimes |1\\rangle)$ in terms of $U$.\n",
|
||||
"\n",
|
||||
"b) Show that any two gates $\\tilde U$ and $\\tilde V$ will combine equivalently to their counterparts $U$ and $V$, i.e.,\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\tilde U \\tilde V = \\widetilde{U V}.\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"c) The controlled-$S$ gate is a two qubit gate which applies a phase of $i$ when the two qubits are in the state $|11\\rangle$ and acts trivially otherwise. Find the equivalent gate acting on real states, and express it in terms of gates that we have seen already during the course."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## 3\n",
|
||||
"\n",
|
||||
"a) Given all Clifford gates and any an additional non-Clifford, universal quantum computation can be perfomed. Given this fact, show that the controlled-$S$ and Hadamard are a universal gate set.\n",
|
||||
"\n",
|
||||
"b) Show that the Toffoli and Hadamard can simulate universal quantum computation on real states."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"collapsed_sections": [],
|
||||
"name": "Exercise_4.ipynb",
|
||||
"provenance": [],
|
||||
"version": "0.3.2"
|
||||
},
|
||||
"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": 1
|
||||
}
|
||||
95
exercises_2022/Exercise8.ipynb
Normal file
95
exercises_2022/Exercise8.ipynb
Normal file
@@ -0,0 +1,95 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Exercise 8\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## 1. Delayed measurement with Clifford only circuits\n",
|
||||
"\n",
|
||||
"In general, we can consider quantum circuits with classically-controlled operations: this is when measurements are made throught the circuit, with the outcomes being used to determine what gates are subsequently applied.\n",
|
||||
"\n",
|
||||
"Nevertheless, when reasoning about what kind of computations we can do with quantum circuits, it is simplest to consider only circuits of the form represented below.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"Here all measurements are deferred to the end, and so there are no classically-controlled operations.\n",
|
||||
"\n",
|
||||
"Remarkably, this comes with no loss of generality. This is because all classically-controlled operations (with measurement) can be replaced by fully quantum controlled gates (unitary and without measurement). For example\n",
|
||||
"\n",
|
||||
"<div>\n",
|
||||
"<img src=\"delayed.png\" width=\"500\"/>\n",
|
||||
"</div>\n",
|
||||
"\n",
|
||||
"Any circuit with classically-controlled gates can therefore be replaced by an equivalent one with only fully quantum gates and with all measurements deferred to the end. However, the gate set required for the latter will typically need to be more powerful that the set of unitary gates used in the former."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"To show this, consider circuits for which classically-control is allowed, but for which the unitary part of all gates must be Clifford. Show that, in general, the equivalent circuit without classically-controlled gates requires non-Clifford gates."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## 2. Unitarity of the order-finding operator\n",
|
||||
"\n",
|
||||
"For integers $x$, $N$ and $L$ with $x < N \\leq 2^L-1$ and $\\rm { gcd} (x,N) = 1$, consider the following operation,\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"U = \\sum_{y=0}^{2^L-1} \\left| \\, f(y) \\, \\right\\rangle \\left\\langle \\,y \\, \\right|,\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"Where $f(y) = x \\times y \\,\\,\\, {\\rm mod}\\, N$ for $0 \\leq y <N$ and $f(y)=y$ otherwise. Show that $U$ is unitary."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## 3. Eigenstates of the order-finding operator\n",
|
||||
"\n",
|
||||
"* (a) Show that the following states are eigenstates of $U$,\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\left| u_s \\right\\rangle = \\frac{1}{\\sqrt{r}} \\sum_{k=0}^{r-1} \\exp \\left[ \\frac{-2 \\pi i s k}{r} \\right] \\left| x^k \\,\\,\\, {\\rm mod}\\, N \\right\\rangle.\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"Here $0 \\leq s \\leq r-1$, where $r$ is the smallest integer such that $x^r = 1 \\,\\,\\, {\\rm mod}\\, N$. Show also that the corresponding eigenvalues are $u_s = \\exp ( {2 \\pi i s}/{r} )$.\n",
|
||||
"\n",
|
||||
"* (b) There are also many states with eigenvalue $1$. What are these?\n",
|
||||
"\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"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
|
||||
}
|
||||
61
exercises_2022/Exercise9.ipynb
Normal file
61
exercises_2022/Exercise9.ipynb
Normal file
@@ -0,0 +1,61 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Exercise 9\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## 1. Grover's algorithm for two qubits\n",
|
||||
"\n",
|
||||
"(a) Implement an oracle and diffusion operator to search all strings of two bits to find `01`.\n",
|
||||
"\n",
|
||||
"(b) Run the algorithm, and determine how many iterations are required to output `01` with high probability."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## 2. Grover's algorithm for three qubits\n",
|
||||
"\n",
|
||||
"(a) Implement an oracle and diffusion operator to search all strings of three bits to find `100`.\n",
|
||||
"\n",
|
||||
"(b) Run the algorithm, and determine how many iterations are required to output `100` with high probability."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"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
|
||||
}
|
||||
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exercises_2022/circuit_model.png
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exercises_2022/circuit_model.png
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exercises_2022/circuits.png
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exercises_2022/circuits.png
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exercises_2022/delayed.png
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exercises_2022/delayed.png
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Reference in New Issue
Block a user