Docs preview for PR #2520.

pr-2520/_sources/applications/python/deutsch_jozsa.ipynb.txt

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "7aa9cc8f-4e42-401f-a1fd-665e5cda19c7",
+   "metadata": {},
+   "source": [
+    "## _*The Deutsch-Jozsa Algorithm*_\n",
+    "\n",
+    "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",
+    "\n",
+    "- **Balanced**, meaning it outputs 0 for exactly half of its possible inputs and 1 for the other half.\n",
+    "- **Constant**, meaning it outputs the same value (either 0 or 1) for all inputs.\n",
+    "\n",
+    "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",
+    "\n",
+    "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 $f(x) = 0$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6edbe9a5-2a81-42e4-ac0c-50a0ef4a0dda",
+   "metadata": {},
+   "source": [
+    "## The Theory\n",
+    "\n",
+    "Here are the steps to implement the algorithm:\n",
+    "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",
+    "$$\n",
+    "|0\\ldots 0\\rangle |0\\rangle\n",
+    "$$\n",
+    "2. Create the superposition of all input qubits by applying the Hadamard gate to each qubit.\n",
+    "$$\n",
+    "H^{\\otimes^n} |0\\ldots 0\\rangle |0\\rangle = \\frac{1}{\\sqrt{2^n}}\\sum_{i=0}^{2^n-1}|i\\rangle |0\\rangle \n",
+    "$$\n",
+    "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",
+    "$$\n",
+    "\\frac{1}{\\sqrt{2^n}}\\sum_{i=0}^{2^n-1}|i\\rangle |0\\rangle \\rightarrow \\frac{1}{\\sqrt{2^{n+1}}}\\sum_{i=0}^{2^n-1}|i\\rangle ( |0\\rangle - |1\\rangle )\n",
+    "$$\n",
+    "4. Query the oracle.\n",
+    "$$\n",
+    "\\frac{1}{\\sqrt{2^{n+1}}}\\sum_{i=0}^{2^n-1}|i\\rangle ( |0\\rangle - |1\\rangle ) \\rightarrow \\frac{1}{\\sqrt{2^{n+1}}}\\sum_{i=0}^{2^n-1}(-1)^{f(i)}|i\\rangle ( |0\\rangle - |1\\rangle ) \n",
+    "$$\n",
+    "5. Apply the Hadamard gate to all input gates.\n",
+    "6. Measure input gates. If measured values are non-zero, then the function is balanced. If not, then it is constant."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5449f9a7-7b1a-4212-8b26-3edaae1fcabd",
+   "metadata": {},
+   "source": [
+    "## The Algorithm Implementation\n",
+    "\n",
+    "Here is the CUDA-Q code following the steps outlined in the above theory section."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "id": "08e04d22-535c-4368-a495-dfe7ed5ff567",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Import the CUDA-Q package and set the target to run on NVIDIA GPUs.\n",
+    "\n",
+    "import cudaq\n",
+    "import random\n",
+    "from typing import List\n",
+    "\n",
+    "cudaq.set_target(\"nvidia\")\n",
+    "\n",
+    "# Number of qubits for the Deutsch-Jozsa algorithm, the last qubit is an auxiliary qubit\n",
+    "qubit_count = 3"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "id": "4d16e25e-d3df-4d07-9e75-a6a046680caa",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Generated fx for function type = constant: [1, 1]\n",
+      "oracleType =  0\n",
+      "oracleValue =  1\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Set the function to be \"constant\" or \"balanced\"\n",
+    "function_type = 'constant'\n",
+    "\n",
+    "# Initialize fx depending on whether the function is constant or balanced\n",
+    "if function_type == 'constant':\n",
+    "    # For a constant function, fx is either all 0's or all 1's\n",
+    "    oracleType = 0  # 0 for constant\n",
+    "    fx_value = random.choice([0, 1])  # Randomly pick 0 or 1\n",
+    "    oracleValue = fx_value  # In constant case, fx_value is passed, for balanced it's not used\n",
+    "    fx = [fx_value] * (qubit_count - 1)\n",
+    "else:\n",
+    "    # For a balanced function, half of fx will be 0's and half will be 1's\n",
+    "    oracleType = 1\n",
+    "    fx = [0] * ((qubit_count - 1) // 2) + [1] * ((qubit_count - 1) - (qubit_count - 1) // 2)\n",
+    "    random.shuffle(fx)  # Shuffle to randomize the positions of 0's and 1's\n",
+    "\n",
+    "# If needed initialize fx, oracleType, and oracleValue manually\n",
+    "#oracleType = 0\n",
+    "#oracleValue = 0\n",
+    "#fx = [0,0]\n",
+    "\n",
+    "print(f\"Generated fx for function type = {function_type}: {fx}\")\n",
+    "print (\"oracleType = \", oracleType)\n",
+    "print (\"oracleValue = \", oracleValue)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 33,
+   "id": "caa90b54-16d3-419d-910f-7a36e4e14829",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "     ╭───╮╭───╮     \n",
+      "q0 : ┤ h ├┤ h ├─────\n",
+      "     ├───┤├───┤     \n",
+      "q1 : ┤ h ├┤ h ├─────\n",
+      "     ├───┤├───┤╭───╮\n",
+      "q2 : ┤ x ├┤ h ├┤ x ├\n",
+      "     ╰───╯╰───╯╰───╯\n",
+      "\n",
+      "Input qubits measurement outcome and frequency = { 00:1 }\n",
+      "\n",
+      "The oracle function is constant.\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Define kernel\n",
+    "@cudaq.kernel\n",
+    "def kernel(fx: List[int], qubit_count: int, oracleType: int, oracleValue: int):\n",
+    "    # Allocate two input qubits\n",
+    "    input_qubits = cudaq.qvector(qubit_count-1)\n",
+    "    # Allocate an auxiliary qubit (initially |0⟩)\n",
+    "    auxiliary_qubit = cudaq.qubit()\n",
+    "\n",
+    "    # Prepare the auxiliary qubit\n",
+    "    x(auxiliary_qubit)\n",
+    "    h(auxiliary_qubit)\n",
+    "\n",
+    "    # Place the rest of the register in a superposition state\n",
+    "    h(input_qubits)\n",
+    "\n",
+    "    # Logic for oracleType == 0 (constant oracle)\n",
+    "    if oracleType == 0:\n",
+    "        if oracleValue == 1:\n",
+    "            # Apply X gate to the auxiliary qubit\n",
+    "            x(auxiliary_qubit)\n",
+    "        elif oracleValue == 0:\n",
+    "            # Apply identity gate (do nothing)\n",
+    "            pass\n",
+    "\n",
+    "    # Logic for oracleType == 1 (balanced oracle)\n",
+    "    elif oracleType == 1:\n",
+    "        for i in range(len(fx)):\n",
+    "            if fx[i] == 1:\n",
+    "                x.ctrl(input_qubits[i], auxiliary_qubit)\n",
+    "    \n",
+    "    # Apply Hadamard to the input qubit again after querying the oracle\n",
+    "    h(input_qubits)\n",
+    "\n",
+    "    # Measure the input qubit to yield if the function is constant or balanced.\n",
+    "    mz(input_qubits)\n",
+    "\n",
+    "print(cudaq.draw(kernel, fx, qubit_count, oracleType, oracleValue))\n",
+    "\n",
+    "result = cudaq.sample(kernel, fx, qubit_count, oracleType, oracleValue, shots_count=1)\n",
+    "\n",
+    "# Debugging: Print the raw result dictionary\n",
+    "print(f\"Input qubits measurement outcome and frequency = {result}\")\n",
+    "\n",
+    "# Define the expected constant results for '00' and '11' for the number of input qubits\n",
+    "expected_constant_results = ['0' * (qubit_count - 1), '1' * (qubit_count - 1)]\n",
+    "\n",
+    "# Check if either '00' or '11' (or their equivalent for more qubits) appears in the result\n",
+    "is_constant = any(result_key in result for result_key in expected_constant_results)\n",
+    "\n",
+    "if is_constant:\n",
+    "    print(\"The oracle function is constant.\")\n",
+    "else:\n",
+    "    print(\"The oracle function is balanced.\")\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "278c6b13-e3a0-4f61-a25b-eed75494b376",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "print(cudaq.__version__)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
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+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
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+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
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