Task4/VQE.Noiseless.Simulator.Code1.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from scipy.optimize import minimize\n",
    "from qiskit import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h1 align=\"center\">\n",
    "Quantum simulation of a many-body system\n",
    "</h1>\n",
    "\n",
    "**The problem:** Simulating quantum systems with several interacting bodies on a classical computer is very difficult.\n",
    "\n",
    "<img src=\"manybody.png\" alt=\"drawing\" width=\"500\"/>\n",
    "\n",
    "**The solution:** One of the important things to know about this system is its ground energy. For such very large systems, this task becomes more difficult for a conventional computer, and here where the quantum computer is advantageous.\n",
    "\n",
    "Quantum computer uses the variational quantum eigensolver (VQE); a hybrid classical-quantum algorithm that variationally determines the ground state energy of a Hamiltonian using the variational principle:\n",
    "$\\langle \\psi | H | \\psi \\rangle \\geq E_i \\rightarrow$ the expectation value of a Hamiltonian will always be greater than or equal to the energy of the ground state of this system.\n",
    "\n",
    "&#10170; **For the sake of simplicity, let us focus on a system of two spins, and study the interaction between them:**\n",
    "\n",
    "<img src=\"twobody.png\" alt=\"drawing\" width=\"300\"/>\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us find out the matrix representation of the two $spin\\;1/2$ particles Hamiltonian that describe their interaction, an example of this system is the hydrogen atom, which has a $spin\\;1/2 $proton in the nucleus and a $spin\\;1/2$ electron located around.\n",
    "\n",
    "Along the z-axis this system forms basis states which contains 4 states:\n",
    "\n",
    "$$\n",
    "|\\uparrow \\uparrow\\rangle=|\\uparrow\\rangle_{1}|\\uparrow\\rangle_{2},\\quad |\\uparrow \\downarrow\\rangle=|\\uparrow\\rangle_{1}|\\downarrow\\rangle_{2},\\quad |\\downarrow \\uparrow\\rangle=|\\downarrow\\rangle_{1}|\\uparrow\\rangle_{2},\\quad |\\downarrow \\downarrow\\rangle=|\\downarrow\\rangle_{1}|\\downarrow\\rangle_{2}\n",
    "$$\n",
    "\n",
    "Where: $\\quad |\\quad\\rangle_1$ : state for the particle 1 and $\\quad |\\quad\\rangle_2$: state for the particle 2.\n",
    "\n",
    "Some requirements:\n",
    "\n",
    "$$\n",
    "\\begin{array}{l}\n",
    "\\checkmark \\quad \\hat{\\vec{S}}=\\hat{\\vec{S}}_{x}+\\hat{\\vec{S}}_{y}+\\hat{\\vec{S}}_{z} \\\\\n",
    "\\checkmark \\quad S_{+}=S_{x}+i S_{y}, \\quad S_{-}=S_{x}-i S_{y} \\\\\n",
    "\\checkmark \\quad S_{z}|\\uparrow\\rangle=+ \\frac{\\hbar}{2}|\\uparrow\\rangle,\\;\\; S_{z}|\\downarrow\\rangle=- \\frac{\\hbar}{2}|\\downarrow\\rangle ,\\;\\; S_{+}|\\downarrow\\rangle=\\hbar|\\uparrow\\rangle, \\;\\; S_{-}|\\uparrow\\rangle=\\hbar|\\downarrow\\rangle, \\;\\; S_{+}|\\uparrow\\rangle=0, \\;\\; S_{-}|\\downarrow\\rangle=0\n",
    "\\end{array}\n",
    "$$\n",
    "\n",
    "We define the Hamiltonian that describes this interaction as:\n",
    "\n",
    "$$\n",
    "\\widehat{H}=\\frac{2 A}{\\hbar^{2}} \\hat{\\vec{S}}_{1} \\cdot \\hat{\\vec{S}}_{2}\n",
    "$$\n",
    "\n",
    "Where he is proportional to the spin operator for the particles 1 and 2 $(\\hat{\\vec{S}}_{1} \\text { and } \\hat{\\vec{S}}_{2} \\; \\text{respectively})$.\n",
    "The $\\frac{2 A}{\\hbar^{2}}$ is a normalization factor, where $A$ is the measure of the strength of this interaction and has unit\n",
    "energy.\n",
    "\n",
    "Now let us go ahead and find the matrix representation relying on the requirements above:\n",
    "\n",
    "$$\n",
    "\\widehat{H}=\\frac{2 A}{\\hbar^{2}} \\hat{\\vec{S}}_{1} \\hat{\\vec{S}}_{2}=\\frac{2 A}{\\hbar^{2}}\\left(\\hat{\\vec{S}}_{1 x} \\hat{\\vec{S}}_{2 x}+\\hat{\\vec{S}}_{1 y} \\hat{\\vec{S}}_{2 y}+\\hat{\\vec{S}}_{12} \\hat{\\vec{S}}_{2 z}\\right)=\\frac{A}{\\hbar^{2}}\\left(\\hat{\\vec{S}}_{1+} \\hat{\\vec{S}}_{2-}+\\hat{\\vec{S}}_{1-} \\hat{\\vec{S}}_{2+}+2 \\hat{\\vec{S}}_{12} \\hat{\\vec{S}}_{2 z}\\right)\n",
    "$$\n",
    "\n",
    "What are the components of this Hamiltonian matrix? To know that, we need to compute all the combinations of the system basis states $\\{|\\uparrow \\uparrow\\rangle, |\\uparrow \\downarrow\\rangle, |\\downarrow \\uparrow\\rangle, |\\downarrow \\downarrow\\rangle\\}$ of the following expression:\n",
    "\n",
    "$$\\langle\\quad| \\widehat{H} |\\quad\\rangle$$\n",
    "\n",
    "By Using the requirements we can conclude the elements of this matrix; we end up with:\n",
    "\n",
    "$$\\widehat{H}=A\\left(\\begin{array}{cccc}\n",
    "\\frac{1}{2} & 0 & 0 & 0 \\\\\n",
    "0 & -\\frac{1}{2} & 1 & 0 \\\\\n",
    "0 & 1 & -\\frac{1}{2} & 0 \\\\\n",
    "0 & 0 & 0 & \\frac{1}{2}\n",
    "\\end{array}\\right)$$\n",
    "\n",
    "Looking at the matrix we can deduce easily two eigenvalues: $\\frac{A}{2}$ and $\\frac{A}{2}$.And the two other come from the middle $2\\times2$ submatrix: $\\frac{A}{2}$ and $-\\frac{3 A}{2}$\n",
    "\n",
    "The states of the eigenvalue $\\frac{A}{2}$ form a spin 1 system, the state of the ground state energy $-\\frac{3 A}{2},$ form a spin 0 system. The following figure makes it easy to understand:\n",
    "\n",
    "<img src=\"spin_alignement.png\" alt=\"drawing\" width=\"500\"/>\n",
    "\n",
    "[Image source](https://chemistry.stackexchange.com/a/54259/85866)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* And know let's use VQE to find the groud state of the Hamiltonian:$H=\\begin{pmatrix}\n",
    "\\frac{1}{2} & 0 &0  & 0 \\\\\n",
    "0 & -\\frac{1}{2} & 1 & 0 \\\\\n",
    "0 & 1 & -\\frac{1}{2} & 0 \\\\\n",
    "0 & 0 & 0 & \\frac{1}{2}\n",
    "\\end{pmatrix}.$\n",
    "\n",
    "I'm ignoring the constante $A$ from the matrix to simplify the work, I will multiply the eigenvalue by it later.\n",
    "\n",
    "## #VQE ON A **IDEAL** SIMULATOR\n",
    "[See it also in the noisy case](https://github.com/walid-mk/VQE/blob/master/Task4/VQE.Noisy.Simulator.Code2.ipynb).\n",
    "\n",
    "* Let first have the eigenvalues of $H$ and find the lowest one which corresponds to the lowest energy. We will use for this some ``` Numpy ``` tools:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The eigenvalues of H : [ 0.5 -1.5  0.5  0.5]\n",
      "The exact ground state energy is:  -1.5\n"
     ]
    }
   ],
   "source": [
    "H = np.mat(\"0.5 0 0 0; 0 -0.5 1 0; 0 1 -0.5 0; 0 0 0 0.5\") #The Hamiltonian matrix\n",
    "print(\"The eigenvalues of H :\", np.linalg.eigvals(H))\n",
    "print('The exact ground state energy is: ', np.linalg.eigvals(H)[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* As we can see the lowest eigenvalue is $-\\frac{3}{2}$.\n",
    "\n",
    "## Variational Quantum Eingensolver (VQE):"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### - Now let us find this value using VQE and see the accuracy of its result\n",
    "\n",
    "* We need to be able to express the Hamiltonian matrix in a quantum circuit, so that the expectations values can be measured using the quantum computer.\n",
    "\n",
    "* To do that we need to decompose $H$ into the sum of tensor products of Pauli operators, multiplied by some constant coefficients: $H = a \\cdot II + b \\cdot XX + c \\cdot ZZ + d \\cdot YY.$\n",
    "\n",
    "    This equation leads to $4$ equations with $4$ variables, which can be written as $Ax=C$ :\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    "1 & 0 & 1  & 0 \\\\\n",
    "0 & 1 & 0 & -1 \\\\\n",
    "1 & 0 & -1 & 0 \\\\\n",
    "0 & 1 & 0 & 1\n",
    "\\end{pmatrix}.\n",
    "\\begin{pmatrix}\n",
    "a \\\\\n",
    "b \\\\\n",
    "c \\\\\n",
    "d\n",
    "\\end{pmatrix} = \n",
    "\\begin{pmatrix}\n",
    "1/2 \\\\\n",
    "0 \\\\\n",
    "-1/2 \\\\\n",
    "1\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "    We can use ```Numpy``` for this task to compute the coefficients $a$, $b$, $c$, and $d$ :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a: 0.0 \n",
      "b: 0.5 \n",
      "c: 0.5 \n",
      "d: 0.5\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[1,0,1,0],[0,1,0,-1],[1,0,-1,0],[0,1,0,1]])\n",
    "C = np.array([1/2,0,-1/2,1])\n",
    "S = np.linalg.solve(A,C) # x = A^-1 * C\n",
    "a, b, c, d = S[0], S[1], S[2], S[3]\n",
    "print(\"a:\", a,\"\\nb:\", b, \"\\nc:\", c, \"\\nd:\", d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Therefore  $H = 0 \\cdot II + 0.5 \\cdot XX + 0.5 \\cdot ZZ + 0.5 \\cdot YY.$\n",
    "\n",
    "And this is another way to compute those coefficients:\n",
    "* Since the Pauli matrices form an orthogonal basis in the Hilbert space, and our $4\\times4$ hermitian matrix is described in the same space, then: \n",
    "$$H = \\sum_{i,j=1,x,y,z} a_{i,j} \\left( \\sigma_i \\otimes \\sigma_j \\right),\n",
    "$$\n",
    "Therefore the coefficients are\n",
    "$$\n",
    "a_{i,j} = \\frac{1}{4} Tr\\left[\\left( \\sigma_i \\otimes \\sigma_j \\right) H \\right]\n",
    "$$\n",
    "the factor $\\frac{1}{4}$ is due to normalization, [see the code for this method](https://michaelgoerz.net/notes/decomposing-two-qubit-hamiltonians-into-pauli-matrices.html). And for more detail [see this](https://quantumcomputing.stackexchange.com/questions/8725/can-arbitrary-matrices-be-decomposed-using-the-pauli-basis/8726#8726).\n",
    "\n",
    "I'm going to convert $H$ to a dictionary for a purposeful reason:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'II': 0.0, 'XX': 0.5, 'ZZ': 0.5, 'YY': 0.5}\n"
     ]
    }
   ],
   "source": [
    "H ={'II':a, 'XX':b, 'ZZ':c,'YY':d}\n",
    "print(H)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### - The Ansatz:\n",
    "\n",
    "* The variational method consists of calculating the expectation value of $H$, over multiple trial wavefunctions to get the right one which corresponds to the minimum energy $min_\\theta\\left\\langle \\psi \\right| H \\left| \\psi \\right\\rangle = lowest\\; energy$.\n",
    "\n",
    "* So we initialize our circuit with a best guess, or ansatz: $$(I \\otimes X).(C_{NOT}).(R_z(\\theta) \\otimes I).(H \\otimes I) \\left| 0 \\right\\rangle \\otimes \\left| 0 \\right\\rangle$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# The ansatz initialization\n",
    "def ansatz_init(circuit, parameter):\n",
    "    q = circuit.qregs[0]\n",
    "    circuit.h(q[0])\n",
    "    circuit.rz(parameter, q[0])\n",
    "    circuit.cx(q[0], q[1])\n",
    "    circuit.x(q[1])\n",
    "    return circuit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### - Transfer to the $Z\\; basis$:\n",
    "* Now to calculate the expectation value, we need to be very careful about the basis.\n",
    "* The only physical measurement that a quantum computer can do, is in the Z basis {${\\left| 0 \\right\\rangle; \\left| 1 \\right\\rangle}$}. And as we can see, the Hamiltonian is decomposed into a linear combination of Pauli's matrices, as these form a basis for hermitian matrices. Hence, we have three kinds of measurement basis:\n",
    "$$Z basis: {\\left| 0 \\right\\rangle; \\left| 1 \\right\\rangle},\\qquad X basis: {\\left| + \\right\\rangle; \\left| - \\right\\rangle},\\qquad Y basis: {\\left| +i \\right\\rangle; \\left| -i \\right\\rangle}.$$\n",
    "Thus, we need to express the last two basis in the $Z\\;basis$ :\n",
    "\n",
    " * $X\\;basis$ in term of $Z\\;basis$; we get the matrix: $H^{'} = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n",
    "1 & 1\\\\\n",
    "1 & -1\n",
    "\\end{pmatrix}$.\n",
    "\n",
    " * $Y\\;basis$ in term of $Z\\;basis$; we get the matrix: $Y^{'} = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n",
    "1 & 1\\\\\n",
    "i & -i\n",
    "\\end{pmatrix}$.\n",
    "\n",
    "<b> Those two matrices transfer from {${\\left| 0 \\right\\rangle; \\left| 1 \\right\\rangle}$} to {${\\left| + \\right\\rangle; \\left| - \\right\\rangle}$} and {${\\left| +i \\right\\rangle; \\left| -i \\right\\rangle}$} respectevly.\n",
    "    \n",
    "* Every quantum state $\\left| \\psi \\right\\rangle$ can be expressed in different sets of basis, so if we want it to be written in the {$\\left| 0 \\right\\rangle; \\left| 1 \\right\\rangle$} basis, to get the count of the measurement to calculate the expectations values, we process as follow:\n",
    "\n",
    " * We apply the inverse of $H^{'}$ to go from the {${\\left| + \\right\\rangle; \\left| - \\right\\rangle}$} basis to the {${\\left| 0 \\right\\rangle; \\left| 1 \\right\\rangle}$} basis; consequently, we get the matrix: $H_{gate} = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n",
    "1 & 1\\\\\n",
    "1 & -1\n",
    "\\end{pmatrix}$. (It is the Hadamard gate)\n",
    " * We apply the inverse of $Y^{'}$ to go from the {${\\left| +i \\right\\rangle; \\left| -i \\right\\rangle}$} basis to the {${\\left| 0 \\right\\rangle; \\left| 1 \\right\\rangle}$} basis; then, we get the matrix: $Y_{gate} = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n",
    "1 & -i\\\\\n",
    "1 & i\n",
    "\\end{pmatrix}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# transfer to the Z basis measurement circuit\n",
    "def z_measure_circ(parameter, measure):\n",
    "    q = QuantumRegister(2)\n",
    "    c = ClassicalRegister(2)\n",
    "    circuit = QuantumCircuit(q, c)\n",
    "\n",
    "    # implement the ansatz in the circuit\n",
    "    circuit = ansatz_init(circuit, parameter)\n",
    "\n",
    "    # measurement\n",
    "    if measure == 'XX':\n",
    "        circuit.barrier(q[0],q[1])\n",
    "        circuit.u(np.pi/2, 0, np.pi, q[0])\n",
    "        circuit.u(np.pi/2, 0, np.pi, q[1])\n",
    "        circuit.measure(q[0], c[0])\n",
    "        circuit.measure(q[1], c[1])\n",
    "    elif measure == 'ZZ':\n",
    "        circuit.measure(q[0], c[0])\n",
    "        circuit.measure(q[1], c[1])\n",
    "    elif measure == 'YY':\n",
    "        circuit.barrier(q[0],q[1])\n",
    "        circuit.u(np.pi/2, 0, np.pi/2, q[0])\n",
    "        circuit.u(np.pi/2,0, np.pi/2, q[1])\n",
    "        circuit.measure(q[0], c[0])\n",
    "        circuit.measure(q[1], c[1])\n",
    "    else:\n",
    "         raise ValueError('Input should be \"XX\" or \"YY\" or \"ZZ\"')\n",
    "\n",
    "    return circuit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* The counts result is returned in qiskit, in a form of a dictionary, so I will define a function ```get_from``` that returns the values in the dictionary. Also, it will be useful for the Hamiltonian dictionary created above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# If key is missing than return 0 otherwise the corresponding value.\n",
    "def get_from(d: dict, key: str):\n",
    "\n",
    "    value = 0\n",
    "    if key in d:\n",
    "        value = d[key]\n",
    "    return value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### - The expectation value:\n",
    "\n",
    "* The expectation value in the case of two qubits, is calculated considering the computational basis (Z-Axis) as follow: eigenvalues of the Pauli's $Z$ matrix times the corresponding probabilities\n",
    "$$\\frac{1}{N}.(N_{00}\\left\\langle 00 \\right| Z \\otimes Z \\left| 00 \\right\\rangle + N_{11}\\left\\langle 11 \\right| Z \\otimes Z \\left| 11 \\right\\rangle + N_{10}\\left\\langle 10 \\right| Z \\otimes Z \\left| 10 \\right\\rangle + N_{01}\\left\\langle 01 \\right| Z \\otimes Z \\left| 01 \\right\\rangle)$$\n",
    "In which: \n",
    "$$\\left\\langle 00 \\right| Z \\otimes Z \\left| 00 \\right\\rangle = \\left\\langle 11 \\right| Z \\otimes Z \\left| 11 \\right\\rangle = 1$$\n",
    "and \n",
    "$$\\left\\langle 10 \\right| Z \\otimes Z \\left| 10 \\right\\rangle = \\left\\langle 01 \\right| Z \\otimes Z \\left| 01 \\right\\rangle = -1$$\n",
    "therefore the expectation value become:\n",
    "$$\\frac{1}{N}.((N_{00} + N_{11}) - (N_{10} + N_{01}))$$\n",
    "where $N$ is the number of shots, and $N_{00}$, $N_{11}$, $N_{10}$, $N_{01}$ are the number of time the state $\\left| 00 \\right\\rangle$, $\\left| 11 \\right\\rangle$, $\\left| 10 \\right\\rangle$, $\\left| 01 \\right\\rangle$ are measured respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "#calculate the expectation value for each Pauli's gate\n",
    "def expec_value(parameter, measure):\n",
    "    # measure in the right basis, then use the counts to compute the expectation value.\n",
    "    if measure == 'II':\n",
    "        return 1\n",
    "    if measure == 'XX':\n",
    "        circuit = z_measure_circ(parameter, 'XX')\n",
    "    elif measure == 'ZZ':\n",
    "        circuit = z_measure_circ(parameter, 'ZZ')\n",
    "    elif measure == 'YY':\n",
    "        circuit = z_measure_circ(parameter, 'YY')\n",
    "    else:\n",
    "        raise ValueError('Input should be \"II\" or \"XX\" or \"ZZ\" or \"YY\"')\n",
    "    \n",
    "    shots = 1000\n",
    "    backend = BasicAer.get_backend('qasm_simulator')\n",
    "    job = execute(circuit, backend, shots=shots)\n",
    "    result = job.result()\n",
    "    counts = result.get_counts()\n",
    "    \n",
    "    expectation_value = ((get_from(counts, '00')+get_from(counts, '11')) - \n",
    "                         (get_from(counts,'10')+get_from(counts, '01'))) / shots\n",
    "\n",
    "    return expectation_value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* create a function ```sum_expec``` that sums up all the expectation value inside the one of the Hamiltonian, multiply them by their coefficients, and returns the final result.\n",
    "$$\\left\\langle H \\right\\rangle = a \\cdot \\left\\langle \\psi \\right| II \\left| \\psi \\right\\rangle + b \\cdot \\left\\langle \\psi \\right| XX \\left| \\psi \\right\\rangle + c \\cdot \\left\\langle \\psi \\right| ZZ \\left| \\psi \\right\\rangle + d \\cdot \\left\\langle \\psi \\right| YY \\left| \\psi \\right\\rangle.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sum_expec(parameter):\n",
    "    if isinstance(parameter, np.ndarray): \n",
    "        parameter = parameter[0] \n",
    "    expec_value_II = get_from(H, 'II') * expec_value(parameter, 'II') #a*<II>\n",
    "    expec_value_XX = get_from(H, 'XX') * expec_value(parameter, 'XX') #b*<XX>\n",
    "    expec_value_ZZ = get_from(H, 'ZZ') * expec_value(parameter, 'ZZ') #c*<ZZ>\n",
    "    expec_value_YY = get_from(H, 'YY') * expec_value(parameter, 'YY') #d*<YY>\n",
    "    # summing the expectations results\n",
    "    sum_result = expec_value_II + expec_value_XX + expec_value_ZZ + expec_value_YY\n",
    "    \n",
    "    return sum_result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Using the optimizer:\n",
    "* the ansatz we had initialize have one degree of freedom i.e. it depends in one parameter $\\theta$ (rotation) on which she can vary, so the trials wavefunctions depend on $\\theta$; $\\left| \\psi(\\theta) \\right\\rangle$.\n",
    "\n",
    "<b> So at the first time, we are going to count on an optimizer called ```minimize``` from ```scipy.optimize```, to search the best angle $\\theta$, therefore the best wavefunction $\\left| \\psi(\\theta) \\right\\rangle$, which minimize the expectation value: $min_\\theta\\left\\langle \\psi(\\theta) \\right| H \\left| \\psi(\\theta) \\right\\rangle = lowest\\; energy\\;$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The exact ground state energy is: -1.5\n",
      "The estimated ground state energy using VQE algorithm is: -1.5\n",
      "\n",
      "The optimal parameter theta is : 3.1454328877068733 \n"
     ]
    }
   ],
   "source": [
    "from scipy.optimize import minimize_scalar\n",
    "parameter = 1 #initialize an arbitrary angle\n",
    "\n",
    "tol = 1e-3 #tolerance for optimization precision.\n",
    "#Inject the sum_expec result inside the minimizer function\n",
    "sum_expec_result = minimize(sum_expec, parameter, method=\"Powell\", tol=tol)\n",
    "\n",
    "print('The exact ground state energy is: {}'.format(-1.5))\n",
    "print('The estimated ground state energy using VQE algorithm is: {}'.format(sum_expec_result.fun))\n",
    "print(\"\\nThe optimal parameter theta is : {} \".format(sum_expec_result.x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* As can be seen, the optimizer can ﬁnd the best angle $\\theta = 3.1454328877068733 \\approx \\pi \\;$ for the global minimum energy $ E = \\left\\langle \\psi \\right| H \\left| \\psi \\right\\rangle = -1.5$ (in arbitrary units) as expected."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Using **```for```**  loop:\n",
    "<b> Here, classical optimization is not necessary, since we have only one parameter to sweep over. Therefore it is obvious that we are going to loop over a set of values in the range $[0, 2\\pi)$ :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "angles = np.linspace(0.0, 2.0 * np.pi, 25) # an array of angles in [0,2pi)\n",
    "\n",
    "lowest_array=[] #  empty regular list\n",
    "for parameter in angles:\n",
    "    lowest = sum_expec(parameter)\n",
    "    lowest_array.append(lowest)\n",
    "    low = np.array(lowest_array) # transformed to a numpy array"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some ```matplotlib``` functions to plot a nice graph :)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The intersection between the angle theta = pi and the constructed graph is the lowest energy:  -1.5\n"
     ]
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from matplotlib.ticker import (FormatStrFormatter, AutoMinorLocator, FuncFormatter)\n",
    "\n",
    "fig = plt.figure(figsize=(15, 10))\n",
    "ax = fig.add_subplot()\n",
    "ax.plot(angles, low, linewidth=3, color='mediumseagreen')\n",
    "\n",
    "ax.annotate('The lowest energy', xy=(3.141, -1.5), xytext=(3.141, -1.25),\n",
    "             arrowprops=dict(facecolor='black', shrink=0.05),)\n",
    "#make the minor graduation\n",
    "def minor_tick(x, pos):\n",
    "    if not x % 1.0:\n",
    "        return \"\"\n",
    "    return \"%.2f\" % x\n",
    "ax.xaxis.set_minor_locator(AutoMinorLocator()) #plot the minor graduation\n",
    "ax.xaxis.set_minor_formatter(FuncFormatter(minor_tick)) #plot the minor graduation numbers\n",
    "\n",
    "ax.tick_params(which='both', width=2) #controle both graduation for x and y axis\n",
    "ax.tick_params(which='major', length=7) #controle the major graduation; the number of grad\n",
    "ax.tick_params(which='minor', length=4, color='black') #control the minor graduation the number of grad\n",
    "ax.tick_params(labelcolor='black', labelsize=20, width=3) #controle the major graduation\n",
    "\n",
    "plt.xlabel(r'$Angles \\; \\theta ∈ [0,2\\pi)$', {'color': 'C0', 'fontsize': 18})\n",
    "plt.ylabel(r'$Expectation \\; value \\; \\left\\langle \\psi(\\theta) \\right| H \\left| \\psi(\\theta) \\right\\rangle$',\n",
    "           {'color': 'C0', 'fontsize': 18})\n",
    "plt.title(r'$min_\\theta \\left\\langle \\psi(\\theta) \\right| H \\left| \\psi(\\theta) \\right\\rangle$',\n",
    "           {'color': 'C0', 'fontsize': 18})\n",
    "plt.grid(linestyle='-.')\n",
    "plt.show()\n",
    "# Use the interpolation Numpy function, where it utilises the best angle given by the optimizer ≈𝜋, to see if\n",
    "# it give us the right ground state in Y-axis in the graph below: (f(angle) = ?)\n",
    "inter = np.interp( np.pi ,angles,low)\n",
    "print('The intersection between the angle theta = pi and the constructed graph is the lowest energy: ',inter)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## &#10173; Final results: \n",
    "\n",
    "**Enumerating the expectation values for all angles, lead to the optimal angle $\\theta=\\pi$, therefore gives the lowest eigenstate $|\\psi(\\theta)\\rangle$ (the right ansatz), found as follow:**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 327.252x144.48 with 1 Axes>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "######## Ansatz Circuit ########\n",
    "qc = QuantumCircuit(2)\n",
    "qc.h(0)\n",
    "qc.rz(np.pi,0)\n",
    "qc.cx(0,1)\n",
    "qc.x(1)\n",
    "qc.draw('mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \n",
       "\\\\Statevector: \\; |\\psi(\\theta)\\rangle = \\begin{bmatrix}\n",
       "0 \\\\\n",
       "\\tfrac{1}{\\sqrt{2}}i \\\\\n",
       "-\\tfrac{1}{\\sqrt{2}}i \\\\\n",
       "0\n",
       "\\end{bmatrix}\n",
       "$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from qiskit_textbook.tools import array_to_latex\n",
    "######## Ansatz State-vector ########\n",
    "backend2 = Aer.get_backend('statevector_simulator')\n",
    "final_state = execute(qc,backend2).result().get_statevector()\n",
    "array_to_latex(final_state, pretext=r\"\\\\Statevector: \\; |\\psi(\\theta)\\rangle = \")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$Statevector: \\; |\\psi(\\theta)\\rangle = \\begin{bmatrix}\n",
    "0 \\\\\n",
    "\\tfrac{1}{\\sqrt{2}}i \\\\\n",
    "-\\tfrac{1}{\\sqrt{2}}i \\\\\n",
    "0\n",
    "\\end{bmatrix}$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**The corresponding lowest energy is $\\left\\langle \\psi(\\pi)| H |\\psi(\\pi)\\right\\rangle = -1.5 \\times A$, This energy match with an antisymmetric spin configuration, with a total angular momentum $𝑚_𝑠 = 0$, hence an antiparallel alignment between our two studied $spins\\;1/2$, see below figure:**\n",
    "\n",
    "<img src=\"groundstate.png\" alt=\"drawing\" width=\"300\"/>\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I searched the lowest energy for the same Hamiltonian, using the VQE in a noisy environment. Moreover, I used another code implementation  and take the same step I toke here to get the values, [see it in my github](https://github.com/walid-mk/VQE/blob/master/Task4/VQE.Noisy.Simulator.Code2.ipynb)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h1 align=\"center\">\n",
    "References\n",
    "</h1>\n",
    "\n",
    "https://arxiv.org/abs/1304.3061\n",
    "\n",
    "https://github.com/DavitKhach/quantum-algorithms-tutorials/blob/master/variational_quantum_eigensolver.ipynb"
   ]
  }
 ],
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