vqe_example.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Esimerkki VQE algoritmista\n",
    "\n",
    "https://qiskit.org/textbook/ch-applications/vqe-molecules.html#VQE-Implementation-in-Qiskit\n",
    "\n",
    "Hyvä kuvaus miten VQE algoritmi pääpiirteissään toimii:\n",
    "\n",
    "https://qiskit.org/textbook/ch-algorithms/defining-quantum-circuits.html#4.-Example:-Variational-Quantum-Eigensolvers-\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Huomioita (Laurilta)\n",
    "\n",
    "- Käyttää suoraan qiskit kirjastoa, ei hyödynnä tequilaa\n",
    "- laskee molekyylin perustason energian LiH molekyylin kahden atomin eri etäisyyksillä\n",
    "- backend: lokaalisti ajettava qiskitin oma kvanttikonesimulaattori \"statevector_simulator\"\n",
    "\n",
    "### Miten about toimii??\n",
    "\n",
    "Tämä määrittelee kvanttikoneessa pyöräytettävän algoritmin\n",
    "\n",
    "```vqe = VQE(qubitOp, var_form, optimizer)```\n",
    "\n",
    "- qubitOp = Qubit operator of the Observable (tämä on molekyylin perusteella rakennettu Hamiltonian?)\n",
    "- optimizer = klassisella koneella ajettava optimointialgoritmi, jota ajetaan jokaisen iteraation lopuksi jotta saadaan seuraavan iteraation konfiguraatio?\n",
    "- var_form = A parameterized circuit used as Ansatz for the wave function (Ikäänkuin aloituskonfiguraatio?)\n",
    "\n",
    "### Algoritmin tiedot\n",
    "\n",
    "- Nimi: Ciskit VQE example\n",
    "- Algoritmin tyyppi: VQE(SLSQP optimizer, UCCSD ansatz)\n",
    "- Molekyyli: LiH (Lithium hydride)\n",
    "\n",
    "### Metriikat\n",
    "\n",
    "- iterations: max 5?, koska ```optimizer = SLSQP(maxiter=5)```. Mutta todellisen määrän saa tietää vain simuloimalla? ```vqe.run(backend)``` palauttaa dictin jossa dataa, voiko siitä päätellä?\n",
    "- measurements: ?\n",
    "- circuit_depth: 100? Tuon mukaan Hamiltonianissa on 100 pauli-porttia? Mutta onko circuit paljon isompi kun päälle tulee ansatz\n",
    "- accuracy: algoritmin tulos on -7.881014174031252, mikä on oikea arvo?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Riippuvuudet (ainakin): qiskit, PySCF"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qiskit.aqua.algorithms import VQE, NumPyEigensolver\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from qiskit.chemistry.components.variational_forms import UCCSD\n",
    "from qiskit.chemistry.components.initial_states import HartreeFock\n",
    "from qiskit.circuit.library import EfficientSU2\n",
    "from qiskit.aqua.components.optimizers import COBYLA, SPSA, SLSQP\n",
    "from qiskit.aqua.operators import Z2Symmetries\n",
    "from qiskit.aqua import QuantumInstance\n",
    "from qiskit import IBMQ, BasicAer, Aer, execute\n",
    "from qiskit.chemistry.drivers import PySCFDriver, UnitsType\n",
    "from qiskit.chemistry import FermionicOperator\n",
    "from qiskit.ignis.mitigation.measurement import CompleteMeasFitter\n",
    "from qiskit.providers.aer.noise import NoiseModel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_qubit_op(dist):\n",
    "    driver = PySCFDriver(atom=\"Li .0 .0 .0; H .0 .0 \" + str(dist), unit=UnitsType.ANGSTROM, \n",
    "                         charge=0, spin=0, basis='sto3g')\n",
    "    molecule = driver.run()\n",
    "    freeze_list = [0]\n",
    "    remove_list = [-3, -2]\n",
    "    repulsion_energy = molecule.nuclear_repulsion_energy\n",
    "    num_particles = molecule.num_alpha + molecule.num_beta\n",
    "    num_spin_orbitals = molecule.num_orbitals * 2\n",
    "    remove_list = [x % molecule.num_orbitals for x in remove_list]\n",
    "    freeze_list = [x % molecule.num_orbitals for x in freeze_list]\n",
    "    remove_list = [x - len(freeze_list) for x in remove_list]\n",
    "    remove_list += [x + molecule.num_orbitals - len(freeze_list)  for x in remove_list]\n",
    "    freeze_list += [x + molecule.num_orbitals for x in freeze_list]\n",
    "    ferOp = FermionicOperator(h1=molecule.one_body_integrals, h2=molecule.two_body_integrals)\n",
    "    ferOp, energy_shift = ferOp.fermion_mode_freezing(freeze_list)\n",
    "    num_spin_orbitals -= len(freeze_list)\n",
    "    num_particles -= len(freeze_list)\n",
    "    ferOp = ferOp.fermion_mode_elimination(remove_list)\n",
    "    num_spin_orbitals -= len(remove_list)\n",
    "    qubitOp = ferOp.mapping(map_type='parity', threshold=0.00000001)\n",
    "    qubitOp = Z2Symmetries.two_qubit_reduction(qubitOp, num_particles)\n",
    "    shift = energy_shift + repulsion_energy\n",
    "    return qubitOp, num_particles, num_spin_orbitals, shift"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Atomien etäisyydeksi on ihahasta repäisty 1.5. Alkuperäisessä koodissa algoritmi ajettiin usealla eri etäisyydellä"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hamiltonian: Representation: paulis, qubits: 4, size: 100\n",
      "IIII\t(-0.19975412837028714+0j)\n",
      "IIIZ\t(-0.09179660692475007+0j)\n",
      "IIZX\t(-0.002734107511296195+0j)\n",
      "IIIX\t(0.002734107511296195+0j)\n",
      "IIXX\t(-0.0003098950346342288+0j)\n",
      "IIYY\t(0.0003098950346342288+0j)\n",
      "IIZZ\t(-0.2119593401711971+0j)\n",
      "IIXZ\t(0.019577653842563426+0j)\n",
      "IIXI\t(0.019577653842563426+0j)\n",
      "IIZI\t(0.371356403772398+0j)\n",
      "IZII\t(0.0917966069247502+0j)\n",
      "ZXII\t(0.00273410751129621+0j)\n",
      "IXII\t(0.00273410751129621+0j)\n",
      "XXII\t(-0.00030989503463423225+0j)\n",
      "YYII\t(0.00030989503463423225+0j)\n",
      "ZZII\t(-0.2119593401711971+0j)\n",
      "XZII\t(-0.01957765384256343+0j)\n",
      "XIII\t(0.01957765384256343+0j)\n",
      "ZIII\t(-0.371356403772398+0j)\n",
      "IZIZ\t(-0.12357087224898831+0j)\n",
      "IZZX\t(0.011733623912066815+0j)\n",
      "IZIX\t(-0.011733623912066815+0j)\n",
      "IZXX\t(0.033058728587758685+0j)\n",
      "IZYY\t(-0.033058728587758685+0j)\n",
      "IXIZ\t(0.011733623912066815+0j)\n",
      "ZXIZ\t(0.011733623912066815+0j)\n",
      "IXZX\t(-0.0030346568302017447+0j)\n",
      "ZXZX\t(-0.0030346568302017447+0j)\n",
      "IXIX\t(0.0030346568302017447+0j)\n",
      "ZXIX\t(0.0030346568302017447+0j)\n",
      "IXXX\t(-0.008373361424260768+0j)\n",
      "ZXXX\t(-0.008373361424260768+0j)\n",
      "IXYY\t(0.008373361424260768+0j)\n",
      "ZXYY\t(0.008373361424260768+0j)\n",
      "YYIZ\t(0.033058728587758685+0j)\n",
      "XXIZ\t(-0.033058728587758685+0j)\n",
      "YYZX\t(-0.008373361424260768+0j)\n",
      "XXZX\t(0.008373361424260768+0j)\n",
      "YYIX\t(0.008373361424260768+0j)\n",
      "XXIX\t(-0.008373361424260768+0j)\n",
      "YYXX\t(-0.030738327177315564+0j)\n",
      "XXXX\t(0.030738327177315564+0j)\n",
      "YYYY\t(0.030738327177315564+0j)\n",
      "XXYY\t(-0.030738327177315564+0j)\n",
      "ZZIZ\t(0.05665606755281661+0j)\n",
      "ZZZX\t(0.0015406700897050557+0j)\n",
      "ZZIX\t(-0.0015406700897050557+0j)\n",
      "ZZXX\t(0.0023679368995874277+0j)\n",
      "ZZYY\t(-0.0023679368995874277+0j)\n",
      "XIIZ\t(0.012733913979291384+0j)\n",
      "XZIZ\t(-0.012733913979291384+0j)\n",
      "XIZX\t(-0.0021111376685944965+0j)\n",
      "XZZX\t(0.0021111376685944965+0j)\n",
      "XIIX\t(0.0021111376685944965+0j)\n",
      "XZIX\t(-0.0021111376685944965+0j)\n",
      "XIXX\t(-0.007764441182118521+0j)\n",
      "XZXX\t(0.007764441182118521+0j)\n",
      "XIYY\t(0.007764441182118521+0j)\n",
      "XZYY\t(-0.007764441182118521+0j)\n",
      "ZIIZ\t(0.11433954684977805+0j)\n",
      "ZIZX\t(-0.010540187409020258+0j)\n",
      "ZIIX\t(0.010540187409020258+0j)\n",
      "ZIXX\t(-0.03511677040241387+0j)\n",
      "ZIYY\t(0.03511677040241387+0j)\n",
      "IZZZ\t(-0.05665606755281661+0j)\n",
      "IZXZ\t(-0.012733913979291382+0j)\n",
      "IZXI\t(-0.012733913979291382+0j)\n",
      "IXZZ\t(-0.0015406700897050557+0j)\n",
      "ZXZZ\t(-0.0015406700897050557+0j)\n",
      "IXXZ\t(0.002111137668594497+0j)\n",
      "ZXXZ\t(0.002111137668594497+0j)\n",
      "IXXI\t(0.002111137668594497+0j)\n",
      "ZXXI\t(0.002111137668594497+0j)\n",
      "YYZZ\t(-0.0023679368995874277+0j)\n",
      "XXZZ\t(0.0023679368995874277+0j)\n",
      "YYXZ\t(0.007764441182118521+0j)\n",
      "XXXZ\t(-0.007764441182118521+0j)\n",
      "YYXI\t(0.007764441182118521+0j)\n",
      "XXXI\t(-0.007764441182118521+0j)\n",
      "ZZZZ\t(0.08470391802239646+0j)\n",
      "ZZXZ\t(-0.009012042792638016+0j)\n",
      "ZZXI\t(-0.009012042792638016+0j)\n",
      "XIZZ\t(-0.009012042792638016+0j)\n",
      "XZZZ\t(0.009012042792638016+0j)\n",
      "XIXZ\t(0.006575744899180413+0j)\n",
      "XZXZ\t(-0.006575744899180413+0j)\n",
      "XIXI\t(0.006575744899180413+0j)\n",
      "XZXI\t(-0.006575744899180413+0j)\n",
      "ZIZZ\t(0.06050560567276786+0j)\n",
      "ZIXZ\t(0.010889407716090486+0j)\n",
      "ZIXI\t(0.010889407716090486+0j)\n",
      "IZZI\t(0.11433954684977805+0j)\n",
      "IXZI\t(-0.010540187409020258+0j)\n",
      "ZXZI\t(-0.010540187409020258+0j)\n",
      "YYZI\t(-0.03511677040241387+0j)\n",
      "XXZI\t(0.03511677040241387+0j)\n",
      "ZZZI\t(-0.06050560567276786+0j)\n",
      "XIZI\t(-0.010889407716090484+0j)\n",
      "XZZI\t(0.010889407716090484+0j)\n",
      "ZIZI\t(-0.11409163501020889+0j)\n",
      "\n"
     ]
    }
   ],
   "source": [
    "\n",
    "optimizer = SLSQP(maxiter=5)\n",
    "qubitOp, num_particles, num_spin_orbitals, shift = get_qubit_op(1.5) # atomien väli 1.5 ångströmiä \n",
    "initial_state = HartreeFock(\n",
    "    num_spin_orbitals,\n",
    "    num_particles,\n",
    "    qubit_mapping='parity'\n",
    ") \n",
    "var_form = UCCSD(\n",
    "    num_orbitals=num_spin_orbitals,\n",
    "    num_particles=num_particles,\n",
    "    initial_state=initial_state,\n",
    "    qubit_mapping='parity'\n",
    ")\n",
    "print('Hamiltonian:',qubitOp)\n",
    "print(qubitOp.print_details())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Todellista kvanttikonetta simuloiva simulaattori: 'qasm_simulator'\n",
    "\n",
    "Ideaali kvanttikone: 'statevector_simulator'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "result: {'optimal_parameters': {Parameter(θ[0]): 0.03057109084458127, Parameter(θ[1]): 0.004609630654785288, Parameter(θ[2]): 0.03056473142345731, Parameter(θ[3]): 0.004610918583535204, Parameter(θ[4]): -0.034432624465397556, Parameter(θ[5]): 0.054337483939551325, Parameter(θ[6]): 0.05442190584670642, Parameter(θ[7]): -0.10964156455313295}, 'optimal_point': array([ 0.03057109,  0.00460963,  0.03056473,  0.00461092, -0.03443262,\n",
      "        0.05433748,  0.05442191, -0.10964156]), 'optimal_value': -1.0990625467099198, 'optimizer_evals': 61, 'optimizer_time': 1.5300019880005493, 'eigenvalue': (-1.0990625467099198+0j), 'eigenstate': array([-3.26731735e-03+3.24546645e-03j, -5.82867088e-16-1.31838984e-15j,\n",
      "       -2.16756929e-02+2.15307319e-02j, -7.02057047e-01+6.97361886e-01j,\n",
      "       -3.85760361e-02+3.83180504e-02j,  0.00000000e+00+0.00000000e+00j,\n",
      "        2.37385999e-02-2.35798428e-02j, -2.16711812e-02+2.15262504e-02j,\n",
      "        0.00000000e+00+0.00000000e+00j,  0.00000000e+00+0.00000000e+00j,\n",
      "        0.00000000e+00+0.00000000e+00j, -1.22124533e-15-2.22044605e-16j,\n",
      "        7.72694114e-02-7.67526552e-02j,  0.00000000e+00+0.00000000e+00j,\n",
      "       -3.85735366e-02+3.83155676e-02j, -3.26823025e-03+3.24637324e-03j]), 'cost_function_evals': 61}\n",
      "\n",
      "eigenvalue: -7.881014173659004\n"
     ]
    }
   ],
   "source": [
    "# hamiltonian, optimizer ja ansatz yhteen\n",
    "backend = BasicAer.get_backend('statevector_simulator')\n",
    "vqe = VQE(qubitOp, var_form, optimizer)\n",
    "result = vqe.run(backend)\n",
    "print('result:',result)\n",
    "print()\n",
    "print('eigenvalue:', np.real(result['eigenvalue'] + shift))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Alkuperäinen koodi jossa algoritmi ajetaan useammalla etäisyydellä"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "backend = BasicAer.get_backend(\"statevector_simulator\")\n",
    "distances = np.arange(0.5, 4.0, 0.1)\n",
    "exact_energies = []\n",
    "vqe_energies = []\n",
    "optimizer = SLSQP(maxiter=5)\n",
    "for dist in distances:\n",
    "    qubitOp, num_particles, num_spin_orbitals, shift = get_qubit_op(dist)\n",
    "    result = NumPyEigensolver(qubitOp).run()\n",
    "    exact_energies.append(np.real(result.eigenvalues) + shift)\n",
    "    initial_state = HartreeFock(\n",
    "        num_spin_orbitals,\n",
    "        num_particles,\n",
    "        qubit_mapping='parity'\n",
    "    ) \n",
    "    var_form = UCCSD(\n",
    "        num_orbitals=num_spin_orbitals,\n",
    "        num_particles=num_particles,\n",
    "        initial_state=initial_state,\n",
    "        qubit_mapping='parity'\n",
    "    )\n",
    "    vqe = VQE(qubitOp, var_form, optimizer)\n",
    "    vqe_result = np.real(vqe.run(backend)['eigenvalue'] + shift)\n",
    "    vqe_energies.append(vqe_result)\n",
    "    print(\"Interatomic Distance:\", np.round(dist, 2), \"VQE Result:\", vqe_result, \"Exact Energy:\", exact_energies[-1])\n",
    "    \n",
    "print(\"All energies have been calculated\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(distances, exact_energies, label=\"Exact Energy\")\n",
    "plt.plot(distances, vqe_energies, label=\"VQE Energy\")\n",
    "plt.xlabel('Atomic distance (Angstrom)')\n",
    "plt.ylabel('Energy')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}