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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"
+      }
+    }
+  ]
+}
\ No newline at end of file