qrng.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "#initialization\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import math\n",
    "from qiskit import QuantumCircuit\n",
    "from qiskit.circuit.library import QFT\n",
    "from qiskit.quantum_info import Statevector\n",
    "from qiskit.primitives import Sampler\n",
    "from qiskit.visualization import plot_histogram, plot_bloch_vector, plot_state_qsphere\n",
    "from qiskit.quantum_info import Operator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Function to apply QPE to generate a random number\n",
    "def qpe_random_number(n_qubits):\n",
    "    # Create a quantum circuit with n qubits for the phase register and 1 qubit for the target\n",
    "    qc = QuantumCircuit(n_qubits + 1, n_qubits)\n",
    "    \n",
    "    # Apply Hadamard gate on all phase qubits (create superposition)\n",
    "    for qubit in range(n_qubits):\n",
    "        qc.h(qubit)\n",
    "    \n",
    "    # Apply controlled Rz gates with random phases\n",
    "    for qubit in range(n_qubits):\n",
    "        random_angle = np.random.uniform(0, 2 * np.pi)\n",
    "        qc.crz(random_angle, qubit, n_qubits)  # Apply controlled-Rz with random phase to each qubit\n",
    "    \n",
    "    # Apply inverse Quantum Fourier Transform (QFT) to the phase register\n",
    "    iqft = QFT(num_qubits=n_qubits, inverse=True, insert_barriers=True)\n",
    "    qc.append(iqft, range(n_qubits))\n",
    "    \n",
    "    # Measure the phase register qubits\n",
    "    qc.measure(range(n_qubits), range(n_qubits))\n",
    "    sampler = Sampler()\n",
    "    job = sampler.run(circuits=qc, shots=1024)\n",
    "    result = job.result()\n",
    "\n",
    "# Get the result counts and convert quasi-probabilities to probabilities\n",
    "    counts = result.quasi_dists[0].binary_probabilities()\n",
    "    random_binary = max(counts, key=counts.get)  # The most frequent outcome\n",
    "    random_number = int(random_binary, 2)\n",
    "    \n",
    "    return random_number"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/var/folders/sn/4ym_qb8x0cs_f961d78kvvkc0000gn/T/ipykernel_61361/2706292456.py:21: DeprecationWarning: The class ``qiskit.primitives.sampler.Sampler`` is deprecated as of qiskit 1.2. It will be removed no earlier than 3 months after the release date. All implementations of the `BaseSamplerV1` interface have been deprecated in favor of their V2 counterparts. The V2 alternative for the `Sampler` class is `StatevectorSampler`.\n",
      "  sampler = Sampler()\n"
     ]
    }
   ],
   "source": [
    "samples = 1000\n",
    "sample_random = []\n",
    "for i in range (samples):\n",
    "    n = 4\n",
    "    sample_random.append(qpe_random_number(n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 53.,   0.,   2.,   0., 186.,  67.,  14.,  94., 149., 435.]),\n",
       " array([ 0. ,  1.5,  3. ,  4.5,  6. ,  7.5,  9. , 10.5, 12. , 13.5, 15. ]),\n",
       " <BarContainer object of 10 artists>)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(sample_random)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def rng_qpe(nqubits):\n",
    "    # create quantum circuit\n",
    "    qc = QuantumCircuit(nqubits+1, nqubits)\n",
    "    # prepare eigenstate\n",
    "    qc.x(nqubits)\n",
    "    qc.h(nqubits)\n",
    "    # step 1: add hadamard\n",
    "    for qubit in range(nqubits):\n",
    "        qc.h(qubit)\n",
    "        qc.barrier()\n",
    "    # step 2: controlled unitary operations\n",
    "    for reg1_index in range(nqubits):\n",
    "        repetitions = 2**reg1_index\n",
    "        random_angle = np.random.uniform(0, 4*np.pi)\n",
    "        qc.crx(repetitions* random_angle, reg1_index, nqubits)\n",
    "    qc.barrier()\n",
    "    # step 3: IQFT on the first register\n",
    "    iqft = QFT(num_qubits=nqubits, inverse=True, insert_barriers=True)\n",
    "    qc.append(iqft, range(nqubits))\n",
    "    qc.barrier()\n",
    "    # step 4: measurement\n",
    "    for reg1_index in range(nqubits):\n",
    "        qc.measure(reg1_index, reg1_index)\n",
    "    # qc.draw(output='mpl').savefig('qpe_4.pdf')\n",
    "    sampler = Sampler()\n",
    "    job = sampler.run(circuits=qc, shots=5000)\n",
    "    result = job.result()\n",
    "    counts = result.quasi_dists[0].binary_probabilities()\n",
    "    random_binary = max(counts, key=counts.get)  # The most frequent outcome\n",
    "    random_number = int(random_binary, 2)\n",
    "    \n",
    "    return random_number, random_angle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/var/folders/sn/4ym_qb8x0cs_f961d78kvvkc0000gn/T/ipykernel_61361/410450505.py:25: DeprecationWarning: The class ``qiskit.primitives.sampler.Sampler`` is deprecated as of qiskit 1.2. It will be removed no earlier than 3 months after the release date. All implementations of the `BaseSamplerV1` interface have been deprecated in favor of their V2 counterparts. The V2 alternative for the `Sampler` class is `StatevectorSampler`.\n",
      "  sampler = Sampler()\n"
     ]
    }
   ],
   "source": [
    "samples = 2000\n",
    "sample_random = []\n",
    "angles = []\n",
    "for i in range (samples):\n",
    "    n = 4\n",
    "    rn,ra = rng_qpe(n)\n",
    "    sample_random.append(rn)\n",
    "    angles.append(ra)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([255., 129., 277., 131., 239., 123., 248., 119., 264., 215.]),\n",
       " array([ 0. ,  1.5,  3. ,  4.5,  6. ,  7.5,  9. , 10.5, 12. , 13.5, 15. ]),\n",
       " <BarContainer object of 10 artists>)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(sample_random)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[1., 0., 0., ..., 0., 0., 1.],\n",
       "        [0., 0., 0., ..., 0., 0., 2.],\n",
       "        [0., 0., 1., ..., 1., 0., 0.],\n",
       "        ...,\n",
       "        [2., 0., 0., ..., 0., 0., 0.],\n",
       "        [1., 0., 0., ..., 0., 0., 1.],\n",
       "        [1., 0., 0., ..., 0., 0., 0.]]),\n",
       " array([2.67870401e-05, 6.28299607e-01, 1.25657243e+00, 1.88484525e+00,\n",
       "        2.51311807e+00, 3.14139089e+00, 3.76966370e+00, 4.39793652e+00,\n",
       "        5.02620934e+00, 5.65448216e+00, 6.28275498e+00]),\n",
       " <a list of 1000 BarContainer objects>)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(random_angles)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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