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{
"nbformat": 4,
"nbformat_minor": 0,
"metadata": {
"colab": {
"provenance": []
},
"kernelspec": {
"name": "python3",
"display_name": "Python 3"
},
"language_info": {
"name": "python"
}
},
"cells": [
{
"cell_type": "markdown",
"source": [
"QOSF Screening tasks submission for Unitary fund Hackathon Tasks descriptions."
],
"metadata": {
"id": "Er7RFLDDO8T5"
}
},
{
"cell_type": "markdown",
"source": [
"\n",
"\n",
"QOSF Screening Tasks Submission for Unitary Fund Hackathon\n",
"\n",
"Quantum-Inspired Function Optimization\n",
"\n",
"The task involves optimizing a mathematical function using a quantum-inspired algorithm. This optimization aims to find either the minimum or maximum value of the objective function over a specified range of input variables.\n",
"\n",
"\n",
"\n",
"Select a mathematical function to be optimized. This function should be parameterized by one or more variables.\n",
"\n",
" A simple quadratic function:\n",
"\\[\n",
"f(x) = x^2 - 4x + 4\n",
"\\]\n",
"\n",
"The goal is to find the minimum value of this function over a specified range of input variables \\( x \\).\n",
"\n",
"\n",
"\n",
"Design a quantum circuit ansatz that parameterizes the quantum state preparation. This ansatz should depend on the parameters to be optimized and the input variables of the objective function.\n",
"\n",
" For a simple one-qubit rotation ansatz, we can use rotation gates such as \\( R_y \\) and \\( R_z \\) with parameters \\( \\theta \\) and \\( \\phi \\):\n",
"\\[\n",
"|\\psi(\\theta, \\phi)\\rangle = R_y(\\theta) R_z(\\phi) |0\\rangle\n",
"\\]\n",
"\n",
"\n",
"\n",
"Define a quantum cost function that evaluates the expectation value of an observable operator on the quantum circuit output. This cost function will be optimized to minimize or maximize the objective function.\n",
"\n",
" If we are using the Pauli-Z operator as the observable, the cost function \\( C(\\theta, \\phi) \\) can be expressed as:\n",
"\\[\n",
"C(\\theta, \\phi) = \\langle \\psi(\\theta, \\phi) | Z | \\psi(\\theta, \\phi) \\rangle\n",
"\\]\n",
"\n",
"\n",
"\n",
"Choose a classical optimization algorithm to optimize the parameters of the quantum circuit ansatz based on the quantum cost function's evaluation.\n",
"\n",
" Gradient Descent, Nelder-Mead, BFGS, etc.\n",
"\n",
"\n",
"\n",
"Implement a routine to optimize the quantum circuit parameters iteratively. This routine should update the parameters using the classical optimization algorithm until convergence is reached.\n",
"\n",
" Use the Gradient Descent method to iteratively adjust the parameters \\( \\theta \\) and \\( \\phi \\) to minimize the cost function \\( C(\\theta, \\phi) \\).\n",
"\n",
"\n",
"\n",
"Test the quantum-inspired optimization algorithm with different functions and analyze its performance. Compare the optimized function values obtained using the quantum-inspired algorithm with classical optimization methods.\n",
"\n",
" Evaluate the minimum value of \\( f(x) = x^2 - 4x + 4 \\) and compare it with the known analytical minimum.\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"Conduct a comparative analysis of the quantum-inspired optimization algorithm with classical optimization algorithms in terms of convergence speed, accuracy, and scalability.\n",
"\n",
" Compare the performance of the quantum-inspired algorithm with classical methods like Gradient Descent and Newton's Method.\n",
"\n",
"\n",
"\n",
"Extend the optimization algorithm to handle noise and errors in quantum devices. Implement error mitigation techniques to enhance the optimization algorithm's robustness.\n",
"\n",
" Zero Noise Extrapolation (ZNE), Measurement Error Mitigation.\n",
"\n",
"\n",
"\n",
"Explore different strategies for initializing the quantum circuit parameters and evaluate their impact on optimization performance.\n",
"\n",
" Random initialization, Heuristic-based initialization, Grid search.\n",
"\n",
"\n",
"\n",
"Investigate hybrid quantum-classical optimization techniques, where classical optimization algorithms are combined with quantum circuits to achieve better optimization results.\n",
"\n",
"Use a classical optimizer to refine the parameters obtained from a quantum circuit.\n",
"\n",
"\n",
"\n",
"Generalize the optimization algorithm to handle functions with multiple variables. Design quantum circuit ansatz and cost functions suitable for multi-variable optimization tasks.\n",
"\n",
"Extend the ansatz to a multi-qubit system to handle a function \\( f(x, y) = x^2 + y^2 - 4x - 4y + 8 \\).\n",
"\n",
"\n",
"\n",
"Given the function \\( f(x) = x^2 - 4x + 4 \\), we can find its minimum value by following these steps:\n",
"\n",
"\n",
"\n",
"\\[\n",
"f(x) = x^2 - 4x + 4\n",
"\\]\n",
"\n",
"\n",
"\n",
"Use a one-qubit rotation ansatz:\n",
"\\[\n",
"|\\psi(\\theta)\\rangle = R_y(\\theta) |0\\rangle\n",
"\\]\n",
"\n",
"\n",
"\n",
"Define the cost function as the expectation value of the Pauli-Z operator:\n",
"\\[\n",
"C(\\theta) = \\langle \\psi(\\theta) | Z | \\psi(\\theta) \\rangle\n",
"\\]\n",
"\n",
"\n",
"\n",
"Use a classical optimizer like Gradient Descent to minimize \\( C(\\theta) \\).\n",
"\n",
"\n",
"\n",
"Initialize \\( \\theta \\) and iteratively update it to minimize the cost function.\n",
"\n",
"\n",
"\n",
"Compare the result with the analytical minimum of \\( f(x) \\), which is \\( x = 2 \\).\n",
"\n",
"\n"
],
"metadata": {
"id": "g01ycumc9Wdk"
}
},
{
"cell_type": "markdown",
"source": [
"Quantum-Inspired Function Optimization\n",
"\n",
"Problem Statement: The task involves optimizing a mathematical function using a quantum-inspired algorithm. This optimization aims to find either the minimum or maximum value of the objective function over a specified range of input variables.\n",
"\n",
"Requirements:\n",
"\n",
"Choose a Function: Select a mathematical function that serves as the objective function to be optimized. This function should be parameterized by one or more variables.\n",
"\n",
"Quantum Circuit Ansatz: Design a quantum circuit ansatz that parameterizes the quantum state preparation. This ansatz should depend on the parameters to be optimized and the input variables of the objective function.\n",
"\n",
"Quantum Cost Function: Define a quantum cost function that evaluates the expectation value of an observable operator on the quantum circuit output. This cost function will be optimized to minimize or maximize the objective function.\n",
"\n",
"Classical Optimization: Choose a classical optimization algorithm, such as gradient descent, to optimize the parameters of the quantum circuit ansatz based on the quantum cost function's evaluation.\n",
"\n",
"Optimization Routine: Implement a routine to optimize the quantum circuit parameters iteratively. This routine should update the parameters using the classical optimization algorithm until convergence is reached.\n",
"\n",
"Evaluate Results: Test the quantum-inspired optimization algorithm with different functions and analyze its performance. Compare the optimized function values obtained using the quantum-inspired algorithm with classical optimization methods.\n",
"\n",
"Bonus Tasks:\n",
"\n",
"Performance Analysis: Conduct a comparative analysis of the quantum-inspired optimization algorithm with classical optimization algorithms in terms of convergence speed, accuracy, and scalability.\n",
"\n",
"Noise Handling: Extend the optimization algorithm to handle noise and errors in quantum devices. Implement error mitigation techniques to enhance the optimization algorithm's robustness.\n",
"\n",
"Parameter Initialization: Explore different strategies for initializing the quantum circuit parameters and evaluate their impact on optimization performance.\n",
"\n",
"Hybrid Approaches: Investigate hybrid quantum-classical optimization techniques, where classical optimization algorithms are combined with quantum circuits to achieve better optimization results.\n",
"\n",
"Extension to Multi-variable Functions: Generalize the optimization algorithm to handle functions with multiple variables. Design quantum circuit ansatz and cost functions suitable for multi-variable optimization tasks."
],
"metadata": {
"id": "LTIB2rGWPBMX"
}
},
{
"cell_type": "markdown",
"source": [
"You are provided with a simple quadratic function \\( f(x) = x^2 - 4x + 4 \\) as the objective function to optimize. Your task is to find the minimum value of this function over a specified range of input variables \\( x \\)."
],
"metadata": {
"id": "iPTisIX4QnRc"
}
},
{
"cell_type": "code",
"source": [
"import pennylane as qml\n",
"from pennylane import numpy as np\n",
"\n",
"# Define the function to optimize\n",
"# your code here#\n",
" # your code here#\n",
"\n",
"# Define the quantum device\n",
"dev = qml.device(\"default.qubit\", wires=1)"
],
"metadata": {
"id": "kJUpNqTM8HCq"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"Design a quantum circuit ansatz that parameterizes the quantum state preparation. The ansatz should depend on the parameters to be optimized and the input variables of the objective function."
],
"metadata": {
"id": "zeb8kWS-Qqw0"
}
},
{
"cell_type": "code",
"source": [
"# Define the quantum circuit ansatz\n",
"# your code here#\n",
" # your code here#\n",
" # your code here#"
],
"metadata": {
"id": "jAD_z7DFQv0m"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"Define a quantum cost function that evaluates the expectation value of an observable operator on the quantum circuit output. This cost function will be optimized to minimize the objective function."
],
"metadata": {
"id": "2WL9fgRwRAvW"
}
},
{
"cell_type": "code",
"source": [
"# Define the quantum cost function\n",
"@qml.qnode(dev)\n",
"# your code here#\n",
" # your code here#\n",
" # your code here#"
],
"metadata": {
"id": "ylZMRFezRD0-"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"Define the VQE optimzer and the function optimization route"
],
"metadata": {
"id": "Vi2-QqI1RQ6u"
}
},
{
"cell_type": "code",
"source": [
"# Define the VQE optimizer\n",
"# your code here#\n",
"\n",
"# Define the function optimization routine\n",
"def optimize_function(x):\n",
" # your code here#\n",
" # your code here#\n",
"\n",
" for _ in range(num_steps):\n",
" # Optimize parameters\n",
" params = optimizer.step(lambda v: (cost(v, x),), params)\n",
"\n",
" return params"
],
"metadata": {
"id": "U49MjBgC8R75"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"Define the range of values for x and optimize the function for each value"
],
"metadata": {
"id": "jDA2nIyyS0PN"
}
},
{
"cell_type": "code",
"source": [
"# Define the range of values for the variable x\n",
"# your code here#\n",
"\n",
"# Optimize the function for each value of x in the range\n",
"# your code here#\n",
"# your code here#\n",
"\n",
"for x_val in x_range:\n",
" params = optimize_function(x_val)\n",
" optimized_value = my_function(x_val)\n",
" if optimized_value < minimum_value:\n",
" minimum_value = optimized_value\n",
" optimal_x = x_val"
],
"metadata": {
"id": "W7ci9cn48Ypp"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"Print and get the optimal value"
],
"metadata": {
"id": "nXmd8tiETL3E"
}
},
{
"cell_type": "code",
"source": [
"print(\"Optimal value of x:\", optimal_x)\n",
"print(\"Minimum value of the function:\", minimum_value)"
],
"metadata": {
"id": "W7wdGZAM8rR6"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"References and resources\n",
"https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Raji)/02%3A_Prime_Numbers/2.06%3A_The_function_x._the_symbols_O_o_and\n",
"\n",
"https://github.com/iQuHACK/2024_Classiq/blob/main/Classiq%E2%80%99s%20Generalized%20Arithmetic%20Challenge.pdf"
],
"metadata": {
"id": "dttmj6ggTY6E"
}
}
]
}