tutorials/1D_Poisson.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solving the Poisson's equation using VQE\n",
    "\n",
    "Author: Zhixin Song \\\n",
    "Last update: 01/12/2023"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Introduction\n",
    "\n",
    "Poisson's equation is $\\Delta \\phi = f$, where $\\Delta = \\nabla^2$ is the Laplace operator. It is widely used in theoretical physics, including electrostatistics and gravitational field. In Cartesian coordinates, it takes the form\n",
    "\n",
    "$$\n",
    "\\bigg(\\frac{\\partial^2}{\\partial x^2} +\\frac{\\partial^2}{\\partial y^2} + \\frac{\\partial^2}{\\partial z^2}\\bigg)\\phi(x,y,z) = f(x,y,z).\n",
    "$$\n",
    "\n",
    "Here, we only consider the 1D case for simplicity. We will disscuss how to generalize it to higher dimensions in a different tutorial. Also, we consider the Dirichlet boundary conditions (B.C.s) in domain $[a,b]$.\n",
    "\n",
    "$$\n",
    "\\Delta \\phi(x)= \\frac{\\partial^2 \\phi(x)}{\\partial x^2} = f(x), \\text{  with    } \\phi(a) = \\phi_a, \\phi(b) = \\phi_b.\n",
    "$$\n",
    "\n",
    "The classical approach to solve this type of PDE is discretizing the spatial domain into a grid mesh of $m$ points such that we can approximate the differential operator with a difference operator\n",
    "\n",
    "$$\n",
    "\\Delta \\phi(x) \\approx \\frac{\\phi_{i-1} - 2 \\phi_i + \\phi_{i+1}}{h^2},\n",
    "$$\n",
    "\n",
    "where $h = (b-a)/(m-1)$ is the separation between each grid point. Then, we can tramsform the orginal problem of solving PDE into a linear system $Ax=b$ where\n",
    "\n",
    "$$\n",
    "A = \n",
    "\\begin{bmatrix}\n",
    "    -2 &1 &0 &\\cdots &0 &0 &0\\\\\n",
    "    1 &-2 &1 &\\cdots &0 &0 &0\\\\\n",
    "    \\vdots &\\vdots &\\vdots &\\ddots &\\vdots &\\vdots &\\vdots\\\\\n",
    "    0 &0 &0 &\\cdots &1 &-2 &1\\\\\n",
    "    0 &0 &0 &\\cdots &0 &1 &-2\n",
    "\\end{bmatrix},\n",
    "b = \n",
    "\\begin{bmatrix}\n",
    "    h^2f_1 - \\phi_a \\\\\n",
    "    h^2 f_2 \\\\\n",
    "    \\vdots \\\\\n",
    "    h^2 f_{m-1} \\\\\n",
    "    h^2 f_m - \\phi_b\n",
    "\\end{bmatrix}.\n",
    "$$\n",
    "\n",
    "The error from such an discretization procedure decreases exponentially with the size of grid. On a classical computer, one can solve this linear system using Gaussian elimination with Matlab when the problem size is small. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Quantum Solution\n",
    "\n",
    "Notice that a quantum state actually maps to an **exponential** grid size. Therefore the amount of quantum resources is linear in number of qubits $n$ and the discretization error decreases exponentially $\\mathcal{O}(1/2^n)$. There are multiple routes to solve this linear system. For the long-term view, one would use the HHL algorithm for speedup. On NISQ machines, we consider the Variational Quantum Eigensolver (VQE).\n",
    "\n",
    "First of all, we need to transfer the problem into a Hamiltonian $H$ where\n",
    "\n",
    "$$\n",
    "H = A^\\dagger(I - |b\\rangle\\langle b|) A\n",
    "$$\n",
    "\n",
    "In order to see how this works, we can try to plug into the solution $|x\\rangle$\n",
    "\n",
    "$$\n",
    "H |x\\rangle = A^\\dagger(I - |b\\rangle\\langle b|) A |x\\rangle = A^\\dagger(I - |b\\rangle\\langle b|) |b\\rangle\n",
    "= A^\\dagger(|b\\rangle - |b\\rangle) = 0 |x\\rangle. \n",
    "$$\n",
    "\n",
    "One can see that the solution is encoded in the ground state of $H$ with an eigenvalue of $0$. Then, we can use VQE to find the ground state.\n",
    "\n",
    "**Note**: Quantum solvers uaually find a normalized solution $|x\\rangle/||x||$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Numerics\n",
    "\n",
    "Consider the driving function $f(x)=x$, boundary $[a,b]= [0,1]$ and boundary conditions $\\phi_a = \\phi_b=0$. Then, the analytical solution is $\\phi(x)=(x^3-x)/6$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.display import clear_output\n",
    "!pip install qiskit==0.39.0\n",
    "clear_output()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.0 Setup backend"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qiskit import QuantumCircuit, transpile\n",
    "from qiskit.circuit import Parameter\n",
    "\n",
    "from qiskit import IBMQ\n",
    "# IBMQ.save_account(TOKEN)\n",
    "IBMQ.load_account() # Load account from disk\n",
    "IBMQ.providers()    # List all available providers\n",
    "provider = IBMQ.get_provider(hub='ibm-q-community', group='ibmquantumawards', project='open-science-22')\n",
    "guadalupe = provider.get_backend(\"ibmq_guadalupe\")\n",
    "\n",
    "\n",
    "from qiskit.providers.fake_provider import FakeGuadalupe\n",
    "\n",
    "# Noisy simulator backend\n",
    "noisy_sim = FakeGuadalupe()\n",
    "\n",
    "from qiskit import Aer\n",
    "backend = Aer.get_backend('statevector_simulator')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.1 Encode the problem Hamiltonian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import sparse\n",
    "from scipy.sparse import diags\n",
    "from scipy.sparse.linalg import eigs\n",
    "import matplotlib.pyplot as plt\n",
    "plt.rcParams.update({\"font.size\": 16})  # enlarge matplotlib fonts\n",
    "\n",
    "from qiskit.opflow import MatrixOp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "tG1j2d1-PP3O",
    "outputId": "c3af297d-5d30-4f13-a56f-ba76ebf8d131",
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Classical solution:\n",
      " [-0.04853869 -0.09707737 -0.14440259 -0.18930087 -0.23055876 -0.26696277\n",
      " -0.29729945 -0.32035532 -0.33491693 -0.3397708  -0.33370346 -0.31550146\n",
      " -0.28395131 -0.23783956 -0.17595273 -0.09707737]\n",
      "Eigenvector:\n",
      " [-0.04853869 -0.09707737 -0.14440259 -0.18930087 -0.23055876 -0.26696277\n",
      " -0.29729945 -0.32035532 -0.33491693 -0.3397708  -0.33370346 -0.31550146\n",
      " -0.28395131 -0.23783956 -0.17595273 -0.09707737]\n",
      "Lenth of Pauli String: 38\n"
     ]
    }
   ],
   "source": [
    "total_res = {}\n",
    "\n",
    "N = 4   # number of qubits\n",
    "dim = 2**N # dimension of the operator A\n",
    "\n",
    "# Setup a tridiagonal matrix\n",
    "k = [np.ones(dim-1), -2*np.ones(dim), np.ones(dim-1)]\n",
    "offset = [-1, 0, 1]\n",
    "A = diags(k,offset).toarray()\n",
    "\n",
    "# Setup the driving term f(x) = x\n",
    "b = np.linspace(0, 1, dim)\n",
    "h = 1/(dim-1)\n",
    "sampled_b = b*(h**2)\n",
    "bt = np.linspace(0, 1, dim)\n",
    "\n",
    "# Setup the Dirichlet B.C.s\n",
    "phi_a, phi_b = 0, 0\n",
    "sampled_b[0] -= phi_a\n",
    "sampled_b[dim-1] -= phi_b\n",
    "norm = np.linalg.norm(sampled_b)\n",
    "sampled_b = sampled_b/norm\n",
    "\n",
    "# Solve the linear system of equations\n",
    "x = np.linalg.solve(A, sampled_b)\n",
    "f = np.linalg.norm(x)\n",
    "x = x/f\n",
    "\n",
    "# Build Hamiltonian\n",
    "sampled_b = sampled_b.reshape([dim, 1])\n",
    "Hamiltonian = A@(np.eye(dim)- sampled_b@sampled_b.T)@A\n",
    "# print(Hamiltonian)\n",
    "\n",
    "print(\"Classical solution:\\n\", x)\n",
    "eig_val, eig_state = np.linalg.eig(Hamiltonian)\n",
    "# print(\"Eigenvalues:\\n\", eig_val)\n",
    "# print(min(eig_val))\n",
    "vec = eig_state[:,-1]\n",
    "# print(eig_state)\n",
    "print(\"Eigenvector:\\n\", -vec)\n",
    "\n",
    "# Transform into Pauli operators\n",
    "H_op = MatrixOp(Hamiltonian).to_pauli_op()\n",
    "print(\"Lenth of Pauli String:\",len(H_op))\n",
    "# print(H_op)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.2 Ansatz design"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<pre style=\"word-wrap: normal;white-space: pre;background: #fff0;line-height: 1.1;font-family: &quot;Courier New&quot;,Courier,monospace\">     ┌──────────┐┌──────────┐     ┌──────────┐┌───────────┐                          ┌───────────┐┌───────────┐                          ┌───────────┐┌───────────┐                          ┌───────────┐┌───────────┐                                    \n",
       "q_0: ┤ Ry(θ[0]) ├┤ Rz(θ[4]) ├──■──┤ Ry(θ[8]) ├┤ Rz(θ[12]) ├───────────────────■──────┤ Ry(θ[16]) ├┤ Rz(θ[20]) ├───────────────────■──────┤ Ry(θ[24]) ├┤ Rz(θ[28]) ├───────────────────■──────┤ Ry(θ[32]) ├┤ Rz(θ[36]) ├───────────────────■────────────────\n",
       "     ├──────────┤├──────────┤┌─┴─┐└──────────┘└┬──────────┤┌───────────┐    ┌─┴─┐    └───────────┘├───────────┤┌───────────┐    ┌─┴─┐    └───────────┘├───────────┤┌───────────┐    ┌─┴─┐    └───────────┘├───────────┤┌───────────┐    ┌─┴─┐              \n",
       "q_1: ┤ Ry(θ[1]) ├┤ Rz(θ[5]) ├┤ X ├─────■───────┤ Ry(θ[9]) ├┤ Rz(θ[13]) ├────┤ X ├──────────■──────┤ Ry(θ[17]) ├┤ Rz(θ[21]) ├────┤ X ├──────────■──────┤ Ry(θ[25]) ├┤ Rz(θ[29]) ├────┤ X ├──────────■──────┤ Ry(θ[33]) ├┤ Rz(θ[37]) ├────┤ X ├──────■───────\n",
       "     ├──────────┤├──────────┤└───┘   ┌─┴─┐     └──────────┘├───────────┤┌───┴───┴───┐    ┌─┴─┐    └───────────┘├───────────┤┌───┴───┴───┐    ┌─┴─┐    └───────────┘├───────────┤┌───┴───┴───┐    ┌─┴─┐    └───────────┘├───────────┤┌───┴───┴───┐┌─┴─┐     \n",
       "q_2: ┤ Ry(θ[2]) ├┤ Rz(θ[6]) ├────────┤ X ├──────────■──────┤ Ry(θ[10]) ├┤ Rz(θ[14]) ├────┤ X ├──────────■──────┤ Ry(θ[18]) ├┤ Rz(θ[22]) ├────┤ X ├──────────■──────┤ Ry(θ[26]) ├┤ Rz(θ[30]) ├────┤ X ├──────────■──────┤ Ry(θ[34]) ├┤ Rz(θ[38]) ├┤ X ├──■──\n",
       "     ├──────────┤├──────────┤        └───┘        ┌─┴─┐    ├───────────┤├───────────┤    └───┘        ┌─┴─┐    ├───────────┤├───────────┤    └───┘        ┌─┴─┐    ├───────────┤├───────────┤    └───┘        ┌─┴─┐    ├───────────┤├───────────┤└───┘┌─┴─┐\n",
       "q_3: ┤ Ry(θ[3]) ├┤ Rz(θ[7]) ├─────────────────────┤ X ├────┤ Ry(θ[11]) ├┤ Rz(θ[15]) ├─────────────────┤ X ├────┤ Ry(θ[19]) ├┤ Rz(θ[23]) ├─────────────────┤ X ├────┤ Ry(θ[27]) ├┤ Rz(θ[31]) ├─────────────────┤ X ├────┤ Ry(θ[35]) ├┤ Rz(θ[39]) ├─────┤ X ├\n",
       "     └──────────┘└──────────┘                     └───┘    └───────────┘└───────────┘                 └───┘    └───────────┘└───────────┘                 └───┘    └───────────┘└───────────┘                 └───┘    └───────────┘└───────────┘     └───┘</pre>"
      ],
      "text/plain": [
       "     ┌──────────┐┌──────────┐     ┌──────────┐┌───────────┐                          ┌───────────┐┌───────────┐                          ┌───────────┐┌───────────┐                          ┌───────────┐┌───────────┐                                    \n",
       "q_0: ┤ Ry(θ[0]) ├┤ Rz(θ[4]) ├──■──┤ Ry(θ[8]) ├┤ Rz(θ[12]) ├───────────────────■──────┤ Ry(θ[16]) ├┤ Rz(θ[20]) ├───────────────────■──────┤ Ry(θ[24]) ├┤ Rz(θ[28]) ├───────────────────■──────┤ Ry(θ[32]) ├┤ Rz(θ[36]) ├───────────────────■────────────────\n",
       "     ├──────────┤├──────────┤┌─┴─┐└──────────┘└┬──────────┤┌───────────┐    ┌─┴─┐    └───────────┘├───────────┤┌───────────┐    ┌─┴─┐    └───────────┘├───────────┤┌───────────┐    ┌─┴─┐    └───────────┘├───────────┤┌───────────┐    ┌─┴─┐              \n",
       "q_1: ┤ Ry(θ[1]) ├┤ Rz(θ[5]) ├┤ X ├─────■───────┤ Ry(θ[9]) ├┤ Rz(θ[13]) ├────┤ X ├──────────■──────┤ Ry(θ[17]) ├┤ Rz(θ[21]) ├────┤ X ├──────────■──────┤ Ry(θ[25]) ├┤ Rz(θ[29]) ├────┤ X ├──────────■──────┤ Ry(θ[33]) ├┤ Rz(θ[37]) ├────┤ X ├──────■───────\n",
       "     ├──────────┤├──────────┤└───┘   ┌─┴─┐     └──────────┘├───────────┤┌───┴───┴───┐    ┌─┴─┐    └───────────┘├───────────┤┌───┴───┴───┐    ┌─┴─┐    └───────────┘├───────────┤┌───┴───┴───┐    ┌─┴─┐    └───────────┘├───────────┤┌───┴───┴───┐┌─┴─┐     \n",
       "q_2: ┤ Ry(θ[2]) ├┤ Rz(θ[6]) ├────────┤ X ├──────────■──────┤ Ry(θ[10]) ├┤ Rz(θ[14]) ├────┤ X ├──────────■──────┤ Ry(θ[18]) ├┤ Rz(θ[22]) ├────┤ X ├──────────■──────┤ Ry(θ[26]) ├┤ Rz(θ[30]) ├────┤ X ├──────────■──────┤ Ry(θ[34]) ├┤ Rz(θ[38]) ├┤ X ├──■──\n",
       "     ├──────────┤├──────────┤        └───┘        ┌─┴─┐    ├───────────┤├───────────┤    └───┘        ┌─┴─┐    ├───────────┤├───────────┤    └───┘        ┌─┴─┐    ├───────────┤├───────────┤    └───┘        ┌─┴─┐    ├───────────┤├───────────┤└───┘┌─┴─┐\n",
       "q_3: ┤ Ry(θ[3]) ├┤ Rz(θ[7]) ├─────────────────────┤ X ├────┤ Ry(θ[11]) ├┤ Rz(θ[15]) ├─────────────────┤ X ├────┤ Ry(θ[19]) ├┤ Rz(θ[23]) ├─────────────────┤ X ├────┤ Ry(θ[27]) ├┤ Rz(θ[31]) ├─────────────────┤ X ├────┤ Ry(θ[35]) ├┤ Rz(θ[39]) ├─────┤ X ├\n",
       "     └──────────┘└──────────┘                     └───┘    └───────────┘└───────────┘                 └───┘    └───────────┘└───────────┘                 └───┘    └───────────┘└───────────┘                 └───┘    └───────────┘└───────────┘     └───┘"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from qiskit.circuit.library import EfficientSU2\n",
    "\n",
    "depth = 5 # depth of ansatz\n",
    "ansatz = EfficientSU2(N, entanglement='linear', reps=depth, skip_final_rotation_layer=True).decompose()\n",
    "ansatz.draw(fold=300)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.3 Transpile"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "number and type of gates in the cirucit: OrderedDict([('ry', 20), ('rz', 20), ('cx', 15)])\n",
      "number of parameters in the circuit: 40\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<pre style=\"word-wrap: normal;white-space: pre;background: #fff0;line-height: 1.1;font-family: &quot;Courier New&quot;,Courier,monospace\">     ┌──────────┐┌──────────┐     ┌──────────┐┌───────────┐                          ┌───────────┐┌───────────┐                          ┌───────────┐┌───────────┐                          ┌───────────┐┌───────────┐                                    \n",
       "q_0: ┤ Ry(θ[0]) ├┤ Rz(θ[4]) ├──■──┤ Ry(θ[8]) ├┤ Rz(θ[12]) ├───────────────────■──────┤ Ry(θ[16]) ├┤ Rz(θ[20]) ├───────────────────■──────┤ Ry(θ[24]) ├┤ Rz(θ[28]) ├───────────────────■──────┤ Ry(θ[32]) ├┤ Rz(θ[36]) ├───────────────────■────────────────\n",
       "     ├──────────┤├──────────┤┌─┴─┐└──────────┘└┬──────────┤┌───────────┐    ┌─┴─┐    └───────────┘├───────────┤┌───────────┐    ┌─┴─┐    └───────────┘├───────────┤┌───────────┐    ┌─┴─┐    └───────────┘├───────────┤┌───────────┐    ┌─┴─┐              \n",
       "q_1: ┤ Ry(θ[1]) ├┤ Rz(θ[5]) ├┤ X ├─────■───────┤ Ry(θ[9]) ├┤ Rz(θ[13]) ├────┤ X ├──────────■──────┤ Ry(θ[17]) ├┤ Rz(θ[21]) ├────┤ X ├──────────■──────┤ Ry(θ[25]) ├┤ Rz(θ[29]) ├────┤ X ├──────────■──────┤ Ry(θ[33]) ├┤ Rz(θ[37]) ├────┤ X ├──────■───────\n",
       "     ├──────────┤├──────────┤└───┘   ┌─┴─┐     └──────────┘├───────────┤┌───┴───┴───┐    ┌─┴─┐    └───────────┘├───────────┤┌───┴───┴───┐    ┌─┴─┐    └───────────┘├───────────┤┌───┴───┴───┐    ┌─┴─┐    └───────────┘├───────────┤┌───┴───┴───┐┌─┴─┐     \n",
       "q_2: ┤ Ry(θ[2]) ├┤ Rz(θ[6]) ├────────┤ X ├──────────■──────┤ Ry(θ[10]) ├┤ Rz(θ[14]) ├────┤ X ├──────────■──────┤ Ry(θ[18]) ├┤ Rz(θ[22]) ├────┤ X ├──────────■──────┤ Ry(θ[26]) ├┤ Rz(θ[30]) ├────┤ X ├──────────■──────┤ Ry(θ[34]) ├┤ Rz(θ[38]) ├┤ X ├──■──\n",
       "     ├──────────┤├──────────┤        └───┘        ┌─┴─┐    ├───────────┤├───────────┤    └───┘        ┌─┴─┐    ├───────────┤├───────────┤    └───┘        ┌─┴─┐    ├───────────┤├───────────┤    └───┘        ┌─┴─┐    ├───────────┤├───────────┤└───┘┌─┴─┐\n",
       "q_3: ┤ Ry(θ[3]) ├┤ Rz(θ[7]) ├─────────────────────┤ X ├────┤ Ry(θ[11]) ├┤ Rz(θ[15]) ├─────────────────┤ X ├────┤ Ry(θ[19]) ├┤ Rz(θ[23]) ├─────────────────┤ X ├────┤ Ry(θ[27]) ├┤ Rz(θ[31]) ├─────────────────┤ X ├────┤ Ry(θ[35]) ├┤ Rz(θ[39]) ├─────┤ X ├\n",
       "     └──────────┘└──────────┘                     └───┘    └───────────┘└───────────┘                 └───┘    └───────────┘└───────────┘                 └───┘    └───────────┘└───────────┘                 └───┘    └───────────┘└───────────┘     └───┘</pre>"
      ],
      "text/plain": [
       "     ┌──────────┐┌──────────┐     ┌──────────┐┌───────────┐                          ┌───────────┐┌───────────┐                          ┌───────────┐┌───────────┐                          ┌───────────┐┌───────────┐                                    \n",
       "q_0: ┤ Ry(θ[0]) ├┤ Rz(θ[4]) ├──■──┤ Ry(θ[8]) ├┤ Rz(θ[12]) ├───────────────────■──────┤ Ry(θ[16]) ├┤ Rz(θ[20]) ├───────────────────■──────┤ Ry(θ[24]) ├┤ Rz(θ[28]) ├───────────────────■──────┤ Ry(θ[32]) ├┤ Rz(θ[36]) ├───────────────────■────────────────\n",
       "     ├──────────┤├──────────┤┌─┴─┐└──────────┘└┬──────────┤┌───────────┐    ┌─┴─┐    └───────────┘├───────────┤┌───────────┐    ┌─┴─┐    └───────────┘├───────────┤┌───────────┐    ┌─┴─┐    └───────────┘├───────────┤┌───────────┐    ┌─┴─┐              \n",
       "q_1: ┤ Ry(θ[1]) ├┤ Rz(θ[5]) ├┤ X ├─────■───────┤ Ry(θ[9]) ├┤ Rz(θ[13]) ├────┤ X ├──────────■──────┤ Ry(θ[17]) ├┤ Rz(θ[21]) ├────┤ X ├──────────■──────┤ Ry(θ[25]) ├┤ Rz(θ[29]) ├────┤ X ├──────────■──────┤ Ry(θ[33]) ├┤ Rz(θ[37]) ├────┤ X ├──────■───────\n",
       "     ├──────────┤├──────────┤└───┘   ┌─┴─┐     └──────────┘├───────────┤┌───┴───┴───┐    ┌─┴─┐    └───────────┘├───────────┤┌───┴───┴───┐    ┌─┴─┐    └───────────┘├───────────┤┌───┴───┴───┐    ┌─┴─┐    └───────────┘├───────────┤┌───┴───┴───┐┌─┴─┐     \n",
       "q_2: ┤ Ry(θ[2]) ├┤ Rz(θ[6]) ├────────┤ X ├──────────■──────┤ Ry(θ[10]) ├┤ Rz(θ[14]) ├────┤ X ├──────────■──────┤ Ry(θ[18]) ├┤ Rz(θ[22]) ├────┤ X ├──────────■──────┤ Ry(θ[26]) ├┤ Rz(θ[30]) ├────┤ X ├──────────■──────┤ Ry(θ[34]) ├┤ Rz(θ[38]) ├┤ X ├──■──\n",
       "     ├──────────┤├──────────┤        └───┘        ┌─┴─┐    ├───────────┤├───────────┤    └───┘        ┌─┴─┐    ├───────────┤├───────────┤    └───┘        ┌─┴─┐    ├───────────┤├───────────┤    └───┘        ┌─┴─┐    ├───────────┤├───────────┤└───┘┌─┴─┐\n",
       "q_3: ┤ Ry(θ[3]) ├┤ Rz(θ[7]) ├─────────────────────┤ X ├────┤ Ry(θ[11]) ├┤ Rz(θ[15]) ├─────────────────┤ X ├────┤ Ry(θ[19]) ├┤ Rz(θ[23]) ├─────────────────┤ X ├────┤ Ry(θ[27]) ├┤ Rz(θ[31]) ├─────────────────┤ X ├────┤ Ry(θ[35]) ├┤ Rz(θ[39]) ├─────┤ X ├\n",
       "     └──────────┘└──────────┘                     └───┘    └───────────┘└───────────┘                 └───┘    └───────────┘└───────────┘                 └───┘    └───────────┘└───────────┘                 └───┘    └───────────┘└───────────┘     └───┘"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ansatz_opt = transpile(ansatz, backend=backend, optimization_level=3)\n",
    "\n",
    "print('number and type of gates in the cirucit:', ansatz_opt.count_ops())\n",
    "print('number of parameters in the circuit:', ansatz_opt.num_parameters)\n",
    "ansatz_opt.draw(fold=300)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.4 Optimizer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qiskit.algorithms.optimizers import SPSA, COBYLA, L_BFGS_B, NELDER_MEAD, SLSQP, ADAM, AQGD, CG, POWELL, QNSPSA\n",
    "\n",
    "# optimizer = SPSA(maxiter=500)\n",
    "optimizer  = L_BFGS_B(maxiter=5000)\n",
    "# optimizer  = ADAM(maxiter=200, lr=0.2)\n",
    "# optimizer  = AQGD(maxiter=1000, eta=1.0, tol=1e-06, momentum=0.25, param_tol=1e-06, averaging=10)\n",
    "# optimizer  = POWELL()\n",
    "# optimizer  = COBYLA(maxiter=10000)\n",
    "# optimizer  = SLSQP(maxiter=10000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.5 Define VQE program"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qiskit.utils import QuantumInstance\n",
    "from qiskit.algorithms import VQE\n",
    "\n",
    "quantum_instance = QuantumInstance(backend=backend, seed_simulator=28, seed_transpiler=28,\n",
    "                                        basis_gates=None,\n",
    "                                        optimization_level=3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.04970883 -0.09922035 -0.14737069 -0.19288916 -0.23456106 -0.27113955\n",
      " -0.30067831 -0.32170144 -0.33380658 -0.33663297 -0.32987386 -0.31216645\n",
      " -0.28149713 -0.2364521  -0.17541169 -0.09691468]\n",
      "Current round using ansatz TwoLocal with depth 5, found eigenvalue 1.172034253425e-05. Best so far 1.172034253425e-05\n"
     ]
    }
   ],
   "source": [
    "best_result = 99999\n",
    "\n",
    "vqe = VQE(ansatz_opt, optimizer,quantum_instance=quantum_instance,initial_point=2*np.pi*np.random.rand(ansatz_opt.num_parameters))\n",
    "result = vqe.compute_minimum_eigenvalue(H_op)\n",
    "quantum_solution = -1*np.abs(result.eigenstate).real\n",
    "print(quantum_solution)\n",
    "\n",
    "if result.eigenvalue.real < best_result:\n",
    "    best_result = result.eigenvalue.real\n",
    "    kept_result = result\n",
    "print(\"Current round using ansatz TwoLocal with depth {}, found eigenvalue {}. Best so far {}\".format(depth, result.eigenvalue.real,best_result))\n",
    "total_res.update({(N, depth):kept_result})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "WM5IzuTgLOTz"
   },
   "source": [
    "### 3.6 Convergence Analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "QYVEEusrcJny"
   },
   "source": [
    "### 3.7 Benchmark with analytical solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 295
    },
    "id": "qkxcXsCsN-oH",
    "outputId": "dd1ef506-0ab1-428b-bd66-ac76b0954c70"
   },
   "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": [
    "t = np.arange(0., 1., 0.02)\n",
    "res = (t**3-t)/6\n",
    "norm_res = np.linalg.norm(res)\n",
    "res_norm = res/norm_res\n",
    "\n",
    "xt = np.arange(0,1,1/dim)\n",
    "exact = [1/6*(x**3-x) for x in np.arange(0,1,1/dim)]\n",
    "norm = np.linalg.norm(exact)\n",
    "exact = exact/norm\n",
    "\n",
    "# red dashes, blue squares and green triangles\n",
    "plt.plot(t, res_norm, 'r-', label='analytical')\n",
    "plt.plot(bt, x, 'o-.', label='classical')\n",
    "plt.plot(bt, quantum_solution, 'gx--', label='quantum')\n",
    "# plt.legend()\n",
    "plt.legend(loc=\"lower left\")\n",
    "plt.xlabel('Boundary [0,1]')\n",
    "plt.ylabel('Solution Profile')\n",
    "plt.title(\"4-qubit VQE for Poisson Eqn, TwoLocal, BFGS\")\n",
    "plt.grid(linestyle = '--', linewidth = 0.5)\n",
    "# plt.show()\n",
    "# plt.savefig(\"Poisson.png\", bbox_inches='tight', dpi=300)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "id": "cGJ8L6j6YScY"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<h3>Version Information</h3><table><tr><th>Qiskit Software</th><th>Version</th></tr><tr><td><code>qiskit-terra</code></td><td>0.22.0</td></tr><tr><td><code>qiskit-aer</code></td><td>0.11.0</td></tr><tr><td><code>qiskit-ignis</code></td><td>0.4.0</td></tr><tr><td><code>qiskit-ibmq-provider</code></td><td>0.19.2</td></tr><tr><td><code>qiskit</code></td><td>0.39.0</td></tr><tr><th>System information</th></tr><tr><td>Python version</td><td>3.8.13</td></tr><tr><td>Python compiler</td><td>Clang 12.0.0 </td></tr><tr><td>Python build</td><td>default, Mar 28 2022 06:16:26</td></tr><tr><td>OS</td><td>Darwin</td></tr><tr><td>CPUs</td><td>2</td></tr><tr><td>Memory (Gb)</td><td>8.0</td></tr><tr><td colspan='2'>Fri Jan 13 04:21:09 2023 EST</td></tr></table>"
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   ],
   "source": [
    "import qiskit.tools.jupyter\n",
    "%qiskit_version_table"
   ]
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   "metadata": {},
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