examples/variational_quantum_eigensolver.ipynb

{
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   "cell_type": "markdown",
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   "source": [
    "# Implementation of a Variational Quantum Eigensolver (VQE).\n",
    "\n",
    "The example shown here is from the paper \"Scalable Quantum Simulation of\n",
    "Molecular Energies\" by P.J.J. O'Malley et al. [arXiv:1512.06860v2](https://arxiv.org/abs/1512.06860v2)\n",
    "(Note that only the latest arXiv version contains the correct coefficients of\n",
    " the Hamiltonian)\n",
    "\n",
    "Eq. 2 of the paper shows the functional which one needs to minimize and Eq. 3\n",
    "shows the coupled cluster ansatz for the trial wavefunction (using the unitary\n",
    "coupled cluster approach). The Hamiltonian is given in Eq. 1. The coefficients\n",
    "can be found in Table 1. Note that both the ansatz and the Hamiltonian can be\n",
    "calculated using FermiLib which is a library for simulating quantum systems\n",
    "on top of ProjectQ."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import projectq\n",
    "from projectq.ops import All, Measure, QubitOperator, TimeEvolution, X\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.optimize import minimize_scalar\n",
    "\n",
    "# Data from paper (arXiv:1512.06860v2) table 1: R, I, Z0, Z1, Z0Z1, X0X1, Y0Y1\n",
    "raw_data_table_1 = [\n",
    "    [0.20,  2.8489, 0.5678, -1.4508, 0.6799, 0.0791, 0.0791],\n",
    "    [0.25,  2.1868, 0.5449, -1.2870, 0.6719, 0.0798, 0.0798],\n",
    "    [0.30,  1.7252, 0.5215, -1.1458, 0.6631, 0.0806, 0.0806],\n",
    "    [0.35,  1.3827, 0.4982, -1.0226, 0.6537, 0.0815, 0.0815],\n",
    "    [0.40,  1.1182, 0.4754, -0.9145, 0.6438, 0.0825, 0.0825],\n",
    "    [0.45,  0.9083, 0.4534, -0.8194, 0.6336, 0.0835, 0.0835],\n",
    "    [0.50,  0.7381, 0.4325, -0.7355, 0.6233, 0.0846, 0.0846],\n",
    "    [0.55,  0.5979, 0.4125, -0.6612, 0.6129, 0.0858, 0.0858],\n",
    "    [0.60,  0.4808, 0.3937, -0.5950, 0.6025, 0.0870, 0.0870],\n",
    "    [0.65,  0.3819, 0.3760, -0.5358, 0.5921, 0.0883, 0.0883],\n",
    "    [0.70,  0.2976, 0.3593, -0.4826, 0.5818, 0.0896, 0.0896],\n",
    "    [0.75,  0.2252, 0.3435, -0.4347, 0.5716, 0.0910, 0.0910],\n",
    "    [0.80,  0.1626, 0.3288, -0.3915, 0.5616, 0.0925, 0.0925],\n",
    "    [0.85,  0.1083, 0.3149, -0.3523, 0.5518, 0.0939, 0.0939],\n",
    "    [0.90,  0.0609, 0.3018, -0.3168, 0.5421, 0.0954, 0.0954],\n",
    "    [0.95,  0.0193, 0.2895, -0.2845, 0.5327, 0.0970, 0.0970],\n",
    "    [1.00, -0.0172, 0.2779, -0.2550, 0.5235, 0.0986, 0.0986],\n",
    "    [1.05, -0.0493, 0.2669, -0.2282, 0.5146, 0.1002, 0.1002],\n",
    "    [1.10, -0.0778, 0.2565, -0.2036, 0.5059, 0.1018, 0.1018],\n",
    "    [1.15, -0.1029, 0.2467, -0.1810, 0.4974, 0.1034, 0.1034],\n",
    "    [1.20, -0.1253, 0.2374, -0.1603, 0.4892, 0.1050, 0.1050],\n",
    "    [1.25, -0.1452, 0.2286, -0.1413, 0.4812, 0.1067, 0.1067],\n",
    "    [1.30, -0.1629, 0.2203, -0.1238, 0.4735, 0.1083, 0.1083],\n",
    "    [1.35, -0.1786, 0.2123, -0.1077, 0.4660, 0.1100, 0.1100],\n",
    "    [1.40, -0.1927, 0.2048, -0.0929, 0.4588, 0.1116, 0.1116],\n",
    "    [1.45, -0.2053, 0.1976, -0.0792, 0.4518, 0.1133, 0.1133],\n",
    "    [1.50, -0.2165, 0.1908, -0.0666, 0.4451, 0.1149, 0.1149],\n",
    "    [1.55, -0.2265, 0.1843, -0.0549, 0.4386, 0.1165, 0.1165],\n",
    "    [1.60, -0.2355, 0.1782, -0.0442, 0.4323, 0.1181, 0.1181],\n",
    "    [1.65, -0.2436, 0.1723, -0.0342, 0.4262, 0.1196, 0.1196],\n",
    "    [1.70, -0.2508, 0.1667, -0.0251, 0.4204, 0.1211, 0.1211],\n",
    "    [1.75, -0.2573, 0.1615, -0.0166, 0.4148, 0.1226, 0.1226],\n",
    "    [1.80, -0.2632, 0.1565, -0.0088, 0.4094, 0.1241, 0.1241],\n",
    "    [1.85, -0.2684, 0.1517, -0.0015, 0.4042, 0.1256, 0.1256],\n",
    "    [1.90, -0.2731, 0.1472,  0.0052, 0.3992, 0.1270, 0.1270],\n",
    "    [1.95, -0.2774, 0.1430,  0.0114, 0.3944, 0.1284, 0.1284],\n",
    "    [2.00, -0.2812, 0.1390,  0.0171, 0.3898, 0.1297, 0.1297],\n",
    "    [2.05, -0.2847, 0.1352,  0.0223, 0.3853, 0.1310, 0.1310],\n",
    "    [2.10, -0.2879, 0.1316,  0.0272, 0.3811, 0.1323, 0.1323],\n",
    "    [2.15, -0.2908, 0.1282,  0.0317, 0.3769, 0.1335, 0.1335],\n",
    "    [2.20, -0.2934, 0.1251,  0.0359, 0.3730, 0.1347, 0.1347],\n",
    "    [2.25, -0.2958, 0.1221,  0.0397, 0.3692, 0.1359, 0.1359],\n",
    "    [2.30, -0.2980, 0.1193,  0.0432, 0.3655, 0.1370, 0.1370],\n",
    "    [2.35, -0.3000, 0.1167,  0.0465, 0.3620, 0.1381, 0.1381],\n",
    "    [2.40, -0.3018, 0.1142,  0.0495, 0.3586, 0.1392, 0.1392],\n",
    "    [2.45, -0.3035, 0.1119,  0.0523, 0.3553, 0.1402, 0.1402],\n",
    "    [2.50, -0.3051, 0.1098,  0.0549, 0.3521, 0.1412, 0.1412],\n",
    "    [2.55, -0.3066, 0.1078,  0.0572, 0.3491, 0.1422, 0.1422],\n",
    "    [2.60, -0.3079, 0.1059,  0.0594, 0.3461, 0.1432, 0.1432],\n",
    "    [2.65, -0.3092, 0.1042,  0.0614, 0.3433, 0.1441, 0.1441],\n",
    "    [2.70, -0.3104, 0.1026,  0.0632, 0.3406, 0.1450, 0.1450],\n",
    "    [2.75, -0.3115, 0.1011,  0.0649, 0.3379, 0.1458, 0.1458],\n",
    "    [2.80, -0.3125, 0.0997,  0.0665, 0.3354, 0.1467, 0.1467],\n",
    "    [2.85, -0.3135, 0.0984,  0.0679, 0.3329, 0.1475, 0.1475]]\n",
    "\n",
    "\n",
    "def variational_quantum_eigensolver(theta, hamiltonian):\n",
    "    \"\"\"\n",
    "    Args:\n",
    "        theta (float): variational parameter for ansatz wavefunction\n",
    "        hamiltonian (QubitOperator): Hamiltonian of the system\n",
    "    Returns:\n",
    "        energy of the wavefunction for parameter theta\n",
    "    \"\"\"\n",
    "    # Create a ProjectQ compiler with a simulator as a backend\n",
    "    eng = projectq.MainEngine()\n",
    "    # Allocate 2 qubits in state |00>\n",
    "    wavefunction = eng.allocate_qureg(2)\n",
    "    # Initialize the Hartree Fock state |01>\n",
    "    X | wavefunction[0]\n",
    "    # build the operator for ansatz wavefunction\n",
    "    ansatz_op = QubitOperator('X0 Y1')\n",
    "    # Apply the unitary e^{-i * ansatz_op * t}\n",
    "    TimeEvolution(theta, ansatz_op) | wavefunction\n",
    "    # flush all gates\n",
    "    eng.flush()\n",
    "    # Calculate the energy.\n",
    "    # The simulator can directly return expectation values, while on a\n",
    "    # real quantum devices one would have to measure each term of the\n",
    "    # Hamiltonian.\n",
    "    energy = eng.backend.get_expectation_value(hamiltonian, wavefunction)\n",
    "    # Measure in order to return to return to a classical state\n",
    "    # (as otherwise the simulator will give an error)\n",
    "    All(Measure) | wavefunction\n",
    "    return energy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106362ed0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lowest_energies = []\n",
    "bond_distances = []\n",
    "for i in range(len(raw_data_table_1)):\n",
    "    # Use data of paper to construct the Hamiltonian\n",
    "    bond_distances.append(raw_data_table_1[i][0])\n",
    "    hamiltonian = raw_data_table_1[i][1] * QubitOperator(())  # == identity\n",
    "    hamiltonian += raw_data_table_1[i][2] * QubitOperator(\"Z0\")\n",
    "    hamiltonian += raw_data_table_1[i][3] * QubitOperator(\"Z1\")\n",
    "    hamiltonian += raw_data_table_1[i][4] * QubitOperator(\"Z0 Z1\")\n",
    "    hamiltonian += raw_data_table_1[i][5] * QubitOperator(\"X0 X1\")\n",
    "    hamiltonian += raw_data_table_1[i][6] * QubitOperator(\"Y0 Y1\")\n",
    "\n",
    "    # Use Scipy to perform the classical outerloop of the variational\n",
    "    # eigensolver, i.e., the minimization of the parameter theta.\n",
    "    # See documentation of Scipy for different optimizers.\n",
    "    minimum = minimize_scalar(lambda theta: variational_quantum_eigensolver(theta, hamiltonian))\n",
    "    lowest_energies.append(minimum.fun)\n",
    "\n",
    "# print result\n",
    "plt.xlabel(\"Bond length [Angstrom]\")\n",
    "plt.ylabel(\"Total Energy [Hartree]\")\n",
    "plt.title(\"Variational Quantum Eigensolver\")\n",
    "plt.plot(bond_distances, lowest_energies, \"b.-\",\n",
    "         label=\"Ground-state energy of molecular hydrogen\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.15"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}