|
| 1 | +{ |
| 2 | + "cells": [ |
| 3 | + { |
| 4 | + "cell_type": "markdown", |
| 5 | + "id": "7aa9cc8f-4e42-401f-a1fd-665e5cda19c7", |
| 6 | + "metadata": {}, |
| 7 | + "source": [ |
| 8 | + "# The Deutsch-Jozsa Algorithm\n", |
| 9 | + "\n", |
| 10 | + "Here is the link to the original paper: [Deutsch-Jozsa algorithm](http://rspa.royalsocietypublishing.org/content/439/1907/553). This algorithm is an earlier demonstration of the computational advantage of quantum algorithm over classical one. It addresses the problem of identifying the nature of a hidden Boolean function, which is provided as an oracle. The function is guaranteed to be either:\n", |
| 11 | + "\n", |
| 12 | + "- **Balanced**, meaning it outputs 0 for exactly half of its possible inputs and 1 for the other half.\n", |
| 13 | + "- **Constant**, meaning it outputs the same value (either 0 or 1) for all inputs.\n", |
| 14 | + "\n", |
| 15 | + "Classically, determining whether the function is balanced or constant requires evaluating the oracle multiple times. In the worst-case scenario, one would need to query at least half of the inputs to distinguish a constant function. However, the Deutsch-Jozsa algorithm demonstrates quantum superiority by solving this problem with a single query to the oracle, regardless of the input size.\n", |
| 16 | + "\n", |
| 17 | + "This notebook implements the Deutsch-Jozsa algorithm as described in [Cleve et al. 1997](https://arxiv.org/pdf/quant-ph/9708016.pdf). The input for the oracle function $f$ is a $n$-bit string. It means that for $x\\ in \\{0,1\\}^n$, the value of $f(x)$ is either constant, i.e., the same for all $x$, or balanced, i.e., exactly half of the $n$-bit string whose $f(x) = 0$." |
| 18 | + ] |
| 19 | + }, |
| 20 | + { |
| 21 | + "cell_type": "markdown", |
| 22 | + "id": "6edbe9a5-2a81-42e4-ac0c-50a0ef4a0dda", |
| 23 | + "metadata": {}, |
| 24 | + "source": [ |
| 25 | + "## The Theory\n", |
| 26 | + "\n", |
| 27 | + "Here are the steps to implement the algorithm:\n", |
| 28 | + "1. Start with initializing all input qubits and single auxiliary qubits to zero. The first $n-1$ input qubits are used for querying the oracle, and the last auxiliary qubit is used for storing the answer of the oracle\n", |
| 29 | + "$$\n", |
| 30 | + "|0\\ldots 0\\rangle |0\\rangle\n", |
| 31 | + "$$\n", |
| 32 | + "2. Create the superposition of all input qubits by applying the Hadamard gate to each qubit.\n", |
| 33 | + "$$\n", |
| 34 | + "H^{\\otimes^n} |0\\ldots 0\\rangle |0\\rangle = \\frac{1}{\\sqrt{2^n}}\\sum_{i=0}^{2^n-1}|i\\rangle |0\\rangle \n", |
| 35 | + "$$\n", |
| 36 | + "3. Apply the Pauli-X gate and apply the Hadamard gate to the auxiliary qubit. This is to store the answer of the oracle in the phase.\n", |
| 37 | + "$$\n", |
| 38 | + "\\frac{1}{\\sqrt{2^n}}\\sum_{i=0}^{2^n-1}|i\\rangle |0\\rangle \\rightarrow \\frac{1}{\\sqrt{2^{n+1}}}\\sum_{i=0}^{2^n-1}|i\\rangle ( |0\\rangle - |1\\rangle )\n", |
| 39 | + "$$\n", |
| 40 | + "4. Query the oracle.\n", |
| 41 | + "$$\n", |
| 42 | + "\\frac{1}{\\sqrt{2^{n+1}}}\\sum_{i=0}^{2^n-1}|i\\rangle ( |0\\rangle - |1\\rangle ) \\rightarrow \\frac{1}{\\sqrt{2^{n+1}}}\\sum_{i=0}^{2^n-1}(-1)^{f(i)}|i\\rangle ( |0\\rangle - |1\\rangle ) \n", |
| 43 | + "$$\n", |
| 44 | + "5. Apply the Hadamard gate to all input gates.\n", |
| 45 | + "6. Measure input gates. If measured values are non-zero, then the function is balanced. If not, then it is constant." |
| 46 | + ] |
| 47 | + }, |
| 48 | + { |
| 49 | + "cell_type": "markdown", |
| 50 | + "id": "5449f9a7-7b1a-4212-8b26-3edaae1fcabd", |
| 51 | + "metadata": {}, |
| 52 | + "source": [ |
| 53 | + "## The Algorithm Implementation\n", |
| 54 | + "\n", |
| 55 | + "Here is the CUDA-Q code following the steps outlined in the above theory section." |
| 56 | + ] |
| 57 | + }, |
| 58 | + { |
| 59 | + "cell_type": "code", |
| 60 | + "execution_count": 31, |
| 61 | + "id": "08e04d22-535c-4368-a495-dfe7ed5ff567", |
| 62 | + "metadata": {}, |
| 63 | + "outputs": [], |
| 64 | + "source": [ |
| 65 | + "# Import the CUDA-Q package and set the target to run on NVIDIA GPUs.\n", |
| 66 | + "\n", |
| 67 | + "import cudaq\n", |
| 68 | + "import random\n", |
| 69 | + "from typing import List\n", |
| 70 | + "\n", |
| 71 | + "cudaq.set_target(\"nvidia\")\n", |
| 72 | + "\n", |
| 73 | + "# Number of qubits for the Deutsch-Jozsa algorithm, the last qubit is an auxiliary qubit\n", |
| 74 | + "qubit_count = 3" |
| 75 | + ] |
| 76 | + }, |
| 77 | + { |
| 78 | + "cell_type": "code", |
| 79 | + "execution_count": 32, |
| 80 | + "id": "4d16e25e-d3df-4d07-9e75-a6a046680caa", |
| 81 | + "metadata": {}, |
| 82 | + "outputs": [ |
| 83 | + { |
| 84 | + "name": "stdout", |
| 85 | + "output_type": "stream", |
| 86 | + "text": [ |
| 87 | + "Generated fx for function type = constant: [1, 1]\n", |
| 88 | + "oracleType = 0\n", |
| 89 | + "oracleValue = 1\n" |
| 90 | + ] |
| 91 | + } |
| 92 | + ], |
| 93 | + "source": [ |
| 94 | + "# Set the function to be \"constant\" or \"balanced\"\n", |
| 95 | + "function_type = 'constant'\n", |
| 96 | + "\n", |
| 97 | + "# Initialize fx depending on whether the function is constant or balanced\n", |
| 98 | + "if function_type == 'constant':\n", |
| 99 | + " # For a constant function, fx is either all 0's or all 1's\n", |
| 100 | + " oracleType = 0 # 0 for constant\n", |
| 101 | + " fx_value = random.choice([0, 1]) # Randomly pick 0 or 1\n", |
| 102 | + " oracleValue = fx_value # In constant case, fx_value is passed, for balanced it's not used\n", |
| 103 | + " fx = [fx_value] * (qubit_count - 1)\n", |
| 104 | + "else:\n", |
| 105 | + " # For a balanced function, half of fx will be 0's and half will be 1's\n", |
| 106 | + " oracleType = 1\n", |
| 107 | + " fx = [0] * ((qubit_count - 1) // 2) + [1] * ((qubit_count - 1) - (qubit_count - 1) // 2)\n", |
| 108 | + " random.shuffle(fx) # Shuffle to randomize the positions of 0's and 1's\n", |
| 109 | + "\n", |
| 110 | + "# If needed initialize fx, oracleType, and oracleValue manually\n", |
| 111 | + "#oracleType = 0\n", |
| 112 | + "#oracleValue = 0\n", |
| 113 | + "#fx = [0,0]\n", |
| 114 | + "\n", |
| 115 | + "print(f\"Generated fx for function type = {function_type}: {fx}\")\n", |
| 116 | + "print (\"oracleType = \", oracleType)\n", |
| 117 | + "print (\"oracleValue = \", oracleValue)" |
| 118 | + ] |
| 119 | + }, |
| 120 | + { |
| 121 | + "cell_type": "code", |
| 122 | + "execution_count": 33, |
| 123 | + "id": "caa90b54-16d3-419d-910f-7a36e4e14829", |
| 124 | + "metadata": {}, |
| 125 | + "outputs": [ |
| 126 | + { |
| 127 | + "name": "stdout", |
| 128 | + "output_type": "stream", |
| 129 | + "text": [ |
| 130 | + " ╭───╮╭───╮ \n", |
| 131 | + "q0 : ┤ h ├┤ h ├─────\n", |
| 132 | + " ├───┤├───┤ \n", |
| 133 | + "q1 : ┤ h ├┤ h ├─────\n", |
| 134 | + " ├───┤├───┤╭───╮\n", |
| 135 | + "q2 : ┤ x ├┤ h ├┤ x ├\n", |
| 136 | + " ╰───╯╰───╯╰───╯\n", |
| 137 | + "\n", |
| 138 | + "Input qubits measurement outcome and frequency = { 00:1 }\n", |
| 139 | + "\n", |
| 140 | + "The oracle function is constant.\n" |
| 141 | + ] |
| 142 | + } |
| 143 | + ], |
| 144 | + "source": [ |
| 145 | + "# Define kernel\n", |
| 146 | + "@cudaq.kernel\n", |
| 147 | + "def kernel(fx: List[int], qubit_count: int, oracleType: int, oracleValue: int):\n", |
| 148 | + " # Allocate two input qubits\n", |
| 149 | + " input_qubits = cudaq.qvector(qubit_count-1)\n", |
| 150 | + " # Allocate an auxiliary qubit (initially |0⟩)\n", |
| 151 | + " auxiliary_qubit = cudaq.qubit()\n", |
| 152 | + "\n", |
| 153 | + " # Prepare the auxiliary qubit\n", |
| 154 | + " x(auxiliary_qubit)\n", |
| 155 | + " h(auxiliary_qubit)\n", |
| 156 | + "\n", |
| 157 | + " # Place the rest of the register in a superposition state\n", |
| 158 | + " h(input_qubits)\n", |
| 159 | + "\n", |
| 160 | + " # Logic for oracleType == 0 (constant oracle)\n", |
| 161 | + " if oracleType == 0:\n", |
| 162 | + " if oracleValue == 1:\n", |
| 163 | + " # Apply X gate to the auxiliary qubit\n", |
| 164 | + " x(auxiliary_qubit)\n", |
| 165 | + " elif oracleValue == 0:\n", |
| 166 | + " # Apply identity gate (do nothing)\n", |
| 167 | + " pass\n", |
| 168 | + "\n", |
| 169 | + " # Logic for oracleType == 1 (balanced oracle)\n", |
| 170 | + " elif oracleType == 1:\n", |
| 171 | + " for i in range(len(fx)):\n", |
| 172 | + " if fx[i] == 1:\n", |
| 173 | + " x.ctrl(input_qubits[i], auxiliary_qubit)\n", |
| 174 | + " \n", |
| 175 | + " # Apply Hadamard to the input qubit again after querying the oracle\n", |
| 176 | + " h(input_qubits)\n", |
| 177 | + "\n", |
| 178 | + " # Measure the input qubit to yield if the function is constant or balanced.\n", |
| 179 | + " mz(input_qubits)\n", |
| 180 | + "\n", |
| 181 | + "print(cudaq.draw(kernel, fx, qubit_count, oracleType, oracleValue))\n", |
| 182 | + "\n", |
| 183 | + "result = cudaq.sample(kernel, fx, qubit_count, oracleType, oracleValue, shots_count=1)\n", |
| 184 | + "\n", |
| 185 | + "# Debugging: Print the raw result dictionary\n", |
| 186 | + "print(f\"Input qubits measurement outcome and frequency = {result}\")\n", |
| 187 | + "\n", |
| 188 | + "# Define the expected constant results for '00' and '11' for the number of input qubits\n", |
| 189 | + "expected_constant_results = ['0' * (qubit_count - 1), '1' * (qubit_count - 1)]\n", |
| 190 | + "\n", |
| 191 | + "# Check if either '00' or '11' (or their equivalent for more qubits) appears in the result\n", |
| 192 | + "is_constant = any(result_key in result for result_key in expected_constant_results)\n", |
| 193 | + "\n", |
| 194 | + "if is_constant:\n", |
| 195 | + " print(\"The oracle function is constant.\")\n", |
| 196 | + "else:\n", |
| 197 | + " print(\"The oracle function is balanced.\")\n" |
| 198 | + ] |
| 199 | + }, |
| 200 | + { |
| 201 | + "cell_type": "code", |
| 202 | + "execution_count": null, |
| 203 | + "id": "278c6b13-e3a0-4f61-a25b-eed75494b376", |
| 204 | + "metadata": {}, |
| 205 | + "outputs": [], |
| 206 | + "source": [ |
| 207 | + "print(cudaq.__version__)" |
| 208 | + ] |
| 209 | + } |
| 210 | + ], |
| 211 | + "metadata": { |
| 212 | + "kernelspec": { |
| 213 | + "display_name": "Python 3", |
| 214 | + "language": "python", |
| 215 | + "name": "python3" |
| 216 | + }, |
| 217 | + "language_info": { |
| 218 | + "codemirror_mode": { |
| 219 | + "name": "ipython", |
| 220 | + "version": 3 |
| 221 | + }, |
| 222 | + "file_extension": ".py", |
| 223 | + "mimetype": "text/x-python", |
| 224 | + "name": "python", |
| 225 | + "nbconvert_exporter": "python", |
| 226 | + "pygments_lexer": "ipython3", |
| 227 | + "version": "3.10.12" |
| 228 | + } |
| 229 | + }, |
| 230 | + "nbformat": 4, |
| 231 | + "nbformat_minor": 5 |
| 232 | +} |
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