VQE_EN.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Variational Quantum Eigensolver\n",
    "\n",
    "<em> Copyright (c) 2021 Institute for Quantum Computing, Baidu Inc. All Rights Reserved. </em>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Overview\n",
    "\n",
    "It is widely believed that one of the most promising applications of quantum computing in the near future is solving quantum chemistry problems [1-2]. **Variational Quantum Eigensolver** (VQE) is a strong proof to this possibility of studying quantum chemistry with **Noisy Intermediate-Scale Quantum** (NISQ) devices [1-4]. The core task is to solve the energy ground state of any molecular Hamiltonian $\\hat{H}$ by preparing a parametrized wave function ansatz $|\\Psi(\\boldsymbol\\theta)\\rangle$ on a quantum computer and adopt classical optimization methods (e.g. gradient descent) to adjust the parameters $\\boldsymbol\\theta$ to minimize the expectation value $\\langle \\Psi(\\boldsymbol\\theta)|\\hat{H}|\\Psi(\\boldsymbol\\theta)\\rangle$. This approach is based on the **Rayleigh-Ritz variational principle**. \n",
    "\n",
    "$$\n",
    "E_0 = \\min_{\\boldsymbol\\theta} \\langle \\Psi(\\boldsymbol\\theta)|\\hat{H}|\\Psi(\\boldsymbol\\theta)\\rangle.\n",
    "\\tag{1}\n",
    "$$\n",
    "\n",
    "where $E_0$ denotes the ground state energy. Numerically, it can be understood as finding the smallest eigenvalue $\\lambda_{\\min}$ of a **discretized** Hamiltonian $H$ (hermitian matrix) and its corresponding eigenvector $|\\Psi_0\\rangle$. How such a discretization can be done on a classical computer belongs to the art of quantum chemistry and is far beyond the scope of this tutorial. We will discuss this part with a few words in the background section. In general, such a Hamiltonian $H$ is expressed as a weighted sum of Pauli spin operators $\\{X,Y,Z\\}$ (native to quantum devices) such that this information can be processed on a quantum computer.\n",
    "\n",
    "$$\n",
    "H = \\sum_k c_k ~ \\bigg( \\bigotimes_{j=0}^{M-1} \\sigma_j^{(k)} \\bigg),\n",
    "\\tag{2}\n",
    "$$\n",
    "\n",
    "where $\\sigma_j^{(k)} \\in \\{I,X,Y,Z\\}$ and $M$ stands for qubit number. We refer this form of Hamiltonian as **Pauli strings**. For example, \n",
    "\n",
    "$$\n",
    "H= 0.12~Y_0 \\otimes I_1-0.04~X_0\\otimes Z_1.\n",
    "\\tag{3}\n",
    "$$\n",
    "\n",
    "In the next section, we will provide a brief review on the electronic structure problem which essentially tells us where does the Hamiltonian $H$ come from. For those who are already familiar with this topic or only interested in how to implement VQE on Paddle Quantum, please skip this part and jump into the illustrative example of hydrogen molecule $H_2$.\n",
    "\n",
    "## Background: the electronic structure problem\n",
    "\n",
    "In this section, we focus on one of the fundamental problems in quantum chemistry --  **the electronic structure problem**. To be more specific, we are interested in the low lying energy eigenstates of any given molecule. These knowledge could help predict reaction rates and location of stable structures [5]. Suppose a molecule consists of $N_n$ nuclei and $N_e$ electrons, the first quantized (canonical quantization) Hamiltonian operator $\\hat{H}_{mol}$ describing the total energy of this molecular system can be written as\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\hat{H}_{\\text{mol}} & = -\\sum_{i}\\frac{\\nabla_{R_i}^2}{2M_i} - \\sum_{i} \\frac{\\nabla_{r_i}^2}{2} -\\sum_{i,j}\\frac{Z_i}{\\lvert R_i - r_j\\lvert} + \\sum_{i,j>i}\\frac{Z_iZ_j}{\\lvert R_i - R_j\\lvert} + \\sum_{i, j>i}\\frac{1}{\\lvert r_i - r_j\\lvert}, \n",
    "\\tag{4}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "where $R_i, M_i,$ and $Z_i$ denote the position, mass and atomic number (the number of protons) of the $i^{th}$ nucleus, and the positions of electrons are $r_i$. The first two sums describe the kinetic energy of nuclei and electrons, respectively. The third sum describes the attractive Coulomb interaction between the positively charged nuclei and the negatively charged electrons. The last two terms represent the repulsive nuclei-nuclei and electron-electron interactions.  Here, the molecular Hamiltonian $\\hat{H}_\\text{mol}$ is already in atomic units of energy, **Hartree**. 1 Hartree is $[\\hbar^2/(m_ee^2a_0^2)] = 27.2$ eV or 630 kcal/mol, where $m_e, e,$ and $a_0$ stand for the mass of electron, charge of electron, and Bohr radius. \n",
    "\n",
    "\n",
    "**Note:** The spin-orbit interaction and hyperfine interaction are not considered in this picture. They can be treated as perturbations if necessary. \n",
    "\n",
    "### Born-Oppenheimer approximation\n",
    "\n",
    "Since the nuclei are much heavier than electrons, the electrons will move much faster than the nuclei. It is reasonable to treat the positions of nuclei as fixed, $R_i =$ constants. This is known as the Born-Oppenheimer approximation by decoupling the behavior of nuclei and electrons in time scale. Consequently, the kinetic energy term of nuclei will disappear and the nuclei-nuclei repulsive interaction term can be viewed as an energy shift (independent of electron positions $r_i$). We could derive the electronic Hamiltonian $\\hat{H}_{\\text{electron}}$ as\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\hat{H}_{\\text{electron}} & =  - \\sum_{i} \\frac{\\nabla_{r_i}^2}{2} -\\sum_{i,j}\\frac{Z_i}{\\lvert R_i - r_j\\lvert} + \\sum_{i, j>i}\\frac{1}{\\lvert r_i - r_j\\lvert} \n",
    "\\tag{5},\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "The energy levels of the electrons in the molecule can be found by solving the time independent Schrödinger equation\n",
    "$$\n",
    "\\hat{H}_{\\text{electron}} |\\Psi_n \\rangle = E_n |\\Psi_n \\rangle,\n",
    "\\tag{6}\n",
    "$$\n",
    "\n",
    "where $n$ stands for the energy level. Notice the electron repulsion terms scale as $N_e(N_e-1)/2$ which means for the Oxygen molecule $O_2$ carrying 16 electrons there will be 120 electron repulsion terms in total! In general, this problem cannot be solved analytically. As Dirac concluded in [Quantum mechanics of many-electron systems](https://royalsocietypublishing.org/doi/10.1098/rspa.1929.0094) [6],\n",
    "\n",
    "> *The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.*                                                                                                                                                                  \n",
    ">\n",
    "> ​\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t-- Paul Dirac (1929)\n",
    "\n",
    "A straightforward numerical approach is discretizing the infinite-dimensional Hilbert space into equidistant grid points where linear algebra guides the whole calculation. Suppose each axis of space is discretized into $k$ points, the $N$-electron (drop the subscript e for simplicity) wave function can be written as [2]\n",
    "\n",
    "$$\n",
    "|\\Psi \\rangle = \\sum_{\\mathbf{x_1}, \\ldots, \\mathbf{x_N}} \\psi(\\mathbf{x_1}, \\ldots, \\mathbf{x_N}) \\mathcal{A}(|\\mathbf{x_1}, \\ldots, \\mathbf{x_N}\\rangle).\n",
    "\\tag{7}\n",
    "$$\n",
    "\n",
    "where coordinate $|\\mathbf{x_j}\\rangle = |r_j\\rangle |\\sigma_j\\rangle$ records the spatial location and spin of the $j^{th}$ electron, $|r_j\\rangle  = |x_j,y_j,z_j\\rangle$ for $j\\in \\{1,2,\\cdots,N\\}$, $x_j,y_j,z_j \\in \\{0,1,\\cdots,k-1\\}$ and  $\\sigma_j \\in \\{\\downarrow,\\uparrow\\}$ for spin down or up.  There will be $k^{3N}\\times 2^{N}$ complex amplitudes in total. Here, $\\mathcal{A}$ denotes anti-symmetrization and a consequence of the Pauli exclusion principle (electrons are fermion), and $\\psi(\\mathbf{x_1}, \\mathbf{x_2}, \\ldots, \\mathbf{x_N})=\\langle\\mathbf{x_1}, \\mathbf{x_2}, \\ldots, \\mathbf{x_N}|\\Psi\\rangle$. One can see that storing such a wave function already requires **exponentially growing memory** with respect to the number of electrons $N$. This would make classical simulation methods based on this naive numerical approach intractable for systems size larger than few tens of electrons. Now, the question becomes can we prepare such a wave function $|\\Psi\\rangle$ directly on a quantum computer and measure the expectation value $E_0$? In the next section, let's take the simplest molecular system -- hydrogen molecule $H_2$ as a concrete example.\n",
    "\n",
    "\n",
    "\n",
    "**Note:** A detailed review on quantum chemistry and existing classical computational methods are far beyond the scope of this tutorial, we refer the enthusiastic readers to the standard textbooks *'Molecular Electronic-Structure Theory'* [5] by Helgaker and *'Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory'* [7] by Szabo & Ostlund. To bridge to knowledge gap between quantum chemistry and quantum computing, please check the following review papers [Quantum computational chemistry](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.92.015003) [2] and [Quantum chemistry in the age of quantum computing](https://pubs.acs.org/doi/10.1021/acs.chemrev.8b00803) [1].\n",
    "\n",
    "**Note:** For energy calculation, it is desired to reach the **chemical accuracy** of $1.6\\times10^{-3}$ Hartree or 1 kcal/mol . "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Energy ground state of the hydrogen molecule $H_2$\n",
    "\n",
    "### Building electronic Hamiltonian\n",
    "\n",
    "First of all, let us import the necessary libraries and packages.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-01-09T07:26:04.537224Z",
     "start_time": "2021-01-09T07:26:00.836470Z"
    }
   },
   "outputs": [],
   "source": [
    "import os\n",
    "import platform\n",
    "import matplotlib.pyplot as plt\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "from IPython.display import clear_output\n",
    "\n",
    "import numpy\n",
    "from numpy import concatenate\n",
    "from numpy import pi as PI\n",
    "from numpy import savez, zeros\n",
    "\n",
    "from paddle import fluid\n",
    "from paddle.complex import matmul, transpose\n",
    "from paddle_quantum.circuit import UAnsatz\n",
    "from paddle_quantum.utils import pauli_str_to_matrix\n",
    "from paddle_quantum.VQE.chemistrysub import H2_generator"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To analyze specific molecules, we need several key information such as **geometry**, **basis set** (such as STO-3G), **multiplicity**, and **charge** to obtain the discretized Hamiltonian $H$. Specifically, through our built-in quantum chemistry toolkit, fermion-to-qubit mapping technology can be used to output the qubit Hamiltonian of hydrogen molecule $H_2$,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-01-09T07:26:04.570840Z",
     "start_time": "2021-01-09T07:26:04.544640Z"
    }
   },
   "outputs": [],
   "source": [
    "Hamiltonian, N = H2_generator()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For more advanced users, we provide a simple tutorial on how to generate such a Hamiltonian. Install the following two packages first (**only available for Mac/Linux users, not available to Windows users temporarily**):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-01-09T07:26:07.190974Z",
     "start_time": "2021-01-09T07:26:05.431945Z"
    }
   },
   "outputs": [],
   "source": [
    "!pip install openfermion==0.11.0\n",
    "clear_output()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-01-09T07:26:09.219268Z",
     "start_time": "2021-01-09T07:26:07.195371Z"
    }
   },
   "outputs": [],
   "source": [
    "!pip install openfermionpyscf==0.4\n",
    "clear_output()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-01-09T07:26:10.031901Z",
     "start_time": "2021-01-09T07:26:09.224389Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The generated h2 Hamiltonian is \n",
      " (-0.04207897647782281+0j) [] +\n",
      "(-0.04475014401535161+0j) [X0 X1 Y2 Y3] +\n",
      "(0.04475014401535161+0j) [X0 Y1 Y2 X3] +\n",
      "(0.04475014401535161+0j) [Y0 X1 X2 Y3] +\n",
      "(-0.04475014401535161+0j) [Y0 Y1 X2 X3] +\n",
      "(0.17771287465139946+0j) [Z0] +\n",
      "(0.17059738328801055+0j) [Z0 Z1] +\n",
      "(0.12293305056183798+0j) [Z0 Z2] +\n",
      "(0.16768319457718958+0j) [Z0 Z3] +\n",
      "(0.1777128746513994+0j) [Z1] +\n",
      "(0.16768319457718958+0j) [Z1 Z2] +\n",
      "(0.12293305056183798+0j) [Z1 Z3] +\n",
      "(-0.24274280513140456+0j) [Z2] +\n",
      "(0.17627640804319586+0j) [Z2 Z3] +\n",
      "(-0.24274280513140456+0j) [Z3]\n"
     ]
    }
   ],
   "source": [
    "# Operating system information\n",
    "sysStr = platform.system()\n",
    "\n",
    "# Decide which operating system the user is using\n",
    "if sysStr in ('Linux', 'Darwin'):\n",
    "\n",
    "    import openfermion\n",
    "    import openfermionpyscf\n",
    "\n",
    "    # Please check whether the geometric configuration file of h2 is downloaded correctly\n",
    "    geo = 'h2.xyz'\n",
    "    charge = 0\n",
    "    multiplicity = 1\n",
    "\n",
    "    # Generate Hamiltonian\n",
    "    mol = openfermion.hamiltonians.MolecularData(geo, 'sto-3g', multiplicity, charge)\n",
    "    openfermionpyscf.run_pyscf(mol)\n",
    "    terms_molecular_hamiltonian = mol.get_molecular_hamiltonian()\n",
    "    fermionic_hamiltonian = openfermion.transforms.get_fermion_operator(terms_molecular_hamiltonian)\n",
    "    qubit_op = openfermion.transforms.jordan_wigner(fermionic_hamiltonian)\n",
    "\n",
    "    # Print result\n",
    "    print(\"The generated h2 Hamiltonian is \\n\", qubit_op)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Note:** This Hamiltonian is generated with an interatomic distance of $d = 74$ pm. \n",
    "\n",
    "In addition to hydrogen molecule $H_2$, we also provide the geometric configuration file of hydrogen fluoride (HF) molecule `hf.xyz`. If you need to test the geometric configuration of more molecules, please check out this [database](http://smart.sns.it/molecules/index.html). In addition, we also need to convert the Hamiltonian into the Pauli string format supported by Paddle Quantum. Here we provide this interface.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-01-09T07:26:11.884870Z",
     "start_time": "2021-01-09T07:26:11.875694Z"
    }
   },
   "outputs": [],
   "source": [
    "def Hamiltonian_str_convert(qubit_op):\n",
    "    '''\n",
    "    Convert the Hamiltonian information provided above into Pauli strings supported by Paddle Quantum\n",
    "\n",
    "    H = [[1.0, \"z0,x1\"], [-1.0, \"y0,z1\"], ...]\n",
    "    '''\n",
    "    info_dic = qubit_op.terms\n",
    "    \n",
    "    def process_tuple(tup):\n",
    "        if len(tup) == 0:\n",
    "            return 'i0'\n",
    "        else:\n",
    "            res = ''\n",
    "            for ele in tup:\n",
    "                res += ele[1].lower()\n",
    "                res += str(ele[0])\n",
    "                res += ','\n",
    "            return res[:-1]\n",
    "    H_info = []\n",
    "    \n",
    "    for key, value in qubit_op.terms.items():\n",
    "        H_info.append([value.real, process_tuple(key)])\n",
    "    \n",
    "    return H_info\n",
    "\n",
    "if sysStr in ('Linux', 'Darwin'):\n",
    "    Hamiltonian = Hamiltonian_str_convert(qubit_op)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Building QNN and trial wave function\n",
    "\n",
    "To implement VQE, we firstly need to design a quantum neural network QNN to prepare the wave function ansatz $|\\Psi(\\boldsymbol\\theta)\\rangle$. Here, we provide a 4-qubit quantum circuit template with a depth of $D$ blocks. The dotted frame in the figure below denotes a single block:\n",
    "\n",
    "\n",
    "![Utheta.jpg](https://release-data.cdn.bcebos.com/PIC%2FUtheta.jpg)\n",
    "\n",
    "Next, we use the `UAnsatz` class and the built-in `real_entangled_layer(theta, D)` circuit template in Paddle Quantum to realize this QNN.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-01-09T07:26:13.717206Z",
     "start_time": "2021-01-09T07:26:13.709627Z"
    }
   },
   "outputs": [],
   "source": [
    "def U_theta(theta, Hamiltonian, N, D):\n",
    "    \"\"\"\n",
    "    Quantum Neural Network\n",
    "    \"\"\"\n",
    "    \n",
    "    # Initialize the quantum neural network according to the number of qubits N\n",
    "    cir = UAnsatz(N)\n",
    "    \n",
    "    # Built-in {R_y + CNOT} circuit template\n",
    "    cir.real_entangled_layer(theta[:D], D)\n",
    "    \n",
    "    # Lay R_y gates in the last row\n",
    "    for i in range(N):\n",
    "        cir.ry(theta=theta[D][i][0], which_qubit=i)\n",
    "        \n",
    "    # The quantum neural network acts on the default initial state |0000>\n",
    "    cir.run_state_vector()\n",
    "    \n",
    "    # Calculate the expected value of a given Hamiltonian\n",
    "    expectation_val = cir.expecval(Hamiltonian)\n",
    "\n",
    "    return expectation_val"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Setting up the loss function and model\n",
    "\n",
    "Now that we have the target Hamiltonian and QNN, we will further define the training model and loss function. By applying the QNN $U(\\theta)$ on the initial state $|0..0\\rangle$, we get the output state $|\\psi(\\boldsymbol\\theta)\\rangle $. Then, the loss function to be minimized is the expectation value, \n",
    "\n",
    "\n",
    "$$\n",
    "\\min_{\\boldsymbol\\theta}  \\mathcal{L}(\\boldsymbol \\theta) = \\min_{\\boldsymbol\\theta} \\langle \\Psi(\\boldsymbol\\theta)|H |\\Psi(\\boldsymbol\\theta)\\rangle\n",
    "= \\min_{\\boldsymbol\\theta} \\sum_k c_k~\\langle \\Psi(\\boldsymbol\\theta)| \\bigotimes_j \\sigma_j^{(k)}|\\Psi(\\boldsymbol\\theta)\\rangle.\n",
    "\\tag{8}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-01-09T07:26:15.661433Z",
     "start_time": "2021-01-09T07:26:15.654903Z"
    }
   },
   "outputs": [],
   "source": [
    "class StateNet(fluid.dygraph.Layer):\n",
    "\n",
    "    def __init__(self, shape, param_attr=fluid.initializer.Uniform(\n",
    "        low=0.0, high=2 * PI), dtype=\"float64\"):\n",
    "        super(StateNet, self).__init__()\n",
    "        \n",
    "        # Initialize the theta parameter list and fill the initial value with a uniform distribution of [0, 2*pi]\n",
    "        self.theta = self.create_parameter(shape=shape, attr=param_attr, dtype=dtype, is_bias=False)\n",
    "        \n",
    "    # Define loss function and forward propagation mechanism\n",
    "    def forward(self, N, D):\n",
    "        \n",
    "        # Calculate the loss function/expected value\n",
    "        loss = U_theta(self.theta, Hamiltonian, N, D)\n",
    "\n",
    "        return loss"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Hyper-parameters\n",
    "\n",
    "Before training the QNN, we also need to set some training hyper-parameters, mainly the learning rate (LR), the number of iterations (ITR), and the depth (D) of repeated blocks. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-01-09T07:26:17.770511Z",
     "start_time": "2021-01-09T07:26:17.767150Z"
    }
   },
   "outputs": [],
   "source": [
    "ITR = 100  # Set the number of optimization iterations\n",
    "LR = 0.2   # Set the learning rate\n",
    "D = 2      # Set the depth of the repetitive calculation module in QNN"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Training\n",
    "\n",
    "After all the training model parameters are set, we convert the data into variables in the Paddle dynamic computational graph, and then train the quantum neural network. The results of the training process is stored in the summary_data file.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-01-09T07:26:33.571665Z",
     "start_time": "2021-01-09T07:26:19.157308Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter: 20 loss: -0.8523\n",
      "iter: 20 Ground state energy: -0.8523 Ha\n",
      "iter: 40 loss: -1.1264\n",
      "iter: 40 Ground state energy: -1.1264 Ha\n",
      "iter: 60 loss: -1.1323\n",
      "iter: 60 Ground state energy: -1.1323 Ha\n",
      "iter: 80 loss: -1.1360\n",
      "iter: 80 Ground state energy: -1.1360 Ha\n",
      "iter: 100 loss: -1.1361\n",
      "iter: 100 Ground state energy: -1.1361 Ha\n"
     ]
    }
   ],
   "source": [
    "# Initialize Paddle dynamic computational graph\n",
    "with fluid.dygraph.guard():\n",
    "\n",
    "\n",
    "    # Determine the parameter dimension of the network\n",
    "    net = StateNet(shape=[D + 1, N, 1])\n",
    "\n",
    "    \n",
    "    # Generally speaking, we use Adam optimizer to obtain relatively good convergence,\n",
    "    # You can change it to SGD or RMS prop.\n",
    "    opt = fluid.optimizer.AdamOptimizer(\n",
    "          learning_rate=LR, parameter_list=net.parameters())\n",
    "\n",
    "    # Record optimization results\n",
    "    summary_iter, summary_loss = [], []\n",
    "    \n",
    "    # Optimization loop\n",
    "    for itr in range(1, ITR + 1):\n",
    "        \n",
    "        # Forward propagation to calculate loss function\n",
    "        loss = net(N, D)\n",
    "\n",
    "        # Use back propagation to minimize the loss function\n",
    "        loss.backward()\n",
    "        opt.minimize(loss)\n",
    "        net.clear_gradients()\n",
    "        \n",
    "        # Record optimization results\n",
    "        summary_loss.append(loss.numpy())\n",
    "        summary_iter.append(itr)\n",
    "        \n",
    "        # Print result\n",
    "        if itr % 20 == 0:\n",
    "            print(\"iter:\", itr, \"loss:\", \"%.4f\" % loss.numpy())\n",
    "            print(\"iter:\", itr, \"Ground state energy:\", \"%.4f Ha\" \n",
    "                                                % loss.numpy())\n",
    "\n",
    "    # Save the training results to the output folder\n",
    "    os.makedirs(\"output\", exist_ok=True)\n",
    "    savez(\"./output/summary_data\", iter = summary_iter, \n",
    "                                   energy=summary_loss)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Benchmarking\n",
    "We have now completed the training of the quantum neural network, and the estimated value of the ground state energy obtained is $E_0 \\approx -1.1361$ Hartree, we compare it with the theoretical value $E_0 = -1.13618$ to benchmark our model. The estimation obtained with VQE agree with a full configuration-interaction (FCI) calculation within chemical accuracy $\\varepsilon = 1.6 \\times 10^{-3}$ Hartree.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-01-09T07:26:42.456121Z",
     "start_time": "2021-01-09T07:26:41.495984Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Created with matplotlib (https://matplotlib.org/) -->\n",
       "<svg height=\"262.19625pt\" version=\"1.1\" viewBox=\"0 0 394.160937 262.19625\" width=\"394.160937pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       " <metadata>\n",
       "  <rdf:RDF xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
       "   <cc:Work>\n",
       "    <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
       "    <dc:date>2021-01-09T15:26:42.336203</dc:date>\n",
       "    <dc:format>image/svg+xml</dc:format>\n",
       "    <dc:creator>\n",
       "     <cc:Agent>\n",
       "      <dc:title>Matplotlib v3.3.1, https://matplotlib.org/</dc:title>\n",
       "     </cc:Agent>\n",
       "    </dc:creator>\n",
       "   </cc:Work>\n",
       "  </rdf:RDF>\n",
       " </metadata>\n",
       " <defs>\n",
       "  <style type=\"text/css\">*{stroke-linecap:butt;stroke-linejoin:round;}</style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"M 0 262.19625 \n",
       "L 394.160937 262.19625 \n",
       "L 394.160937 0 \n",
       "L 0 0 \n",
       "z\n",
       "\" style=\"fill:none;\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\"M 52.160938 224.64 \n",
       "L 386.960938 224.64 \n",
       "L 386.960938 7.2 \n",
       "L 52.160938 7.2 \n",
       "z\n",
       "\" style=\"fill:#ffffff;\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_1\">\n",
       "      <defs>\n",
       "       <path d=\"M 0 0 \n",
       "L 0 3.5 \n",
       "\" id=\"mb15aed2d89\" style=\"stroke:#000000;stroke-width:0.8;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"64.304739\" xlink:href=\"#mb15aed2d89\" y=\"224.64\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- 0 -->\n",
       "      <g transform=\"translate(61.123489 239.238437)scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path d=\"M 31.78125 66.40625 \n",
       "Q 24.171875 66.40625 20.328125 58.90625 \n",
       "Q 16.5 51.421875 16.5 36.375 \n",
       "Q 16.5 21.390625 20.328125 13.890625 \n",
       "Q 24.171875 6.390625 31.78125 6.390625 \n",
       "Q 39.453125 6.390625 43.28125 13.890625 \n",
       "Q 47.125 21.390625 47.125 36.375 \n",
       "Q 47.125 51.421875 43.28125 58.90625 \n",
       "Q 39.453125 66.40625 31.78125 66.40625 \n",
       "z\n",
       "M 31.78125 74.21875 \n",
       "Q 44.046875 74.21875 50.515625 64.515625 \n",
       "Q 56.984375 54.828125 56.984375 36.375 \n",
       "Q 56.984375 17.96875 50.515625 8.265625 \n",
       "Q 44.046875 -1.421875 31.78125 -1.421875 \n",
       "Q 19.53125 -1.421875 13.0625 8.265625 \n",
       "Q 6.59375 17.96875 6.59375 36.375 \n",
       "Q 6.59375 54.828125 13.0625 64.515625 \n",
       "Q 19.53125 74.21875 31.78125 74.21875 \n",
       "z\n",
       "\" id=\"DejaVuSans-48\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_2\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"125.792342\" xlink:href=\"#mb15aed2d89\" y=\"224.64\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 20 -->\n",
       "      <g transform=\"translate(119.429842 239.238437)scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path d=\"M 19.1875 8.296875 \n",
       "L 53.609375 8.296875 \n",
       "L 53.609375 0 \n",
       "L 7.328125 0 \n",
       "L 7.328125 8.296875 \n",
       "Q 12.9375 14.109375 22.625 23.890625 \n",
       "Q 32.328125 33.6875 34.8125 36.53125 \n",
       "Q 39.546875 41.84375 41.421875 45.53125 \n",
       "Q 43.3125 49.21875 43.3125 52.78125 \n",
       "Q 43.3125 58.59375 39.234375 62.25 \n",
       "Q 35.15625 65.921875 28.609375 65.921875 \n",
       "Q 23.96875 65.921875 18.8125 64.3125 \n",
       "Q 13.671875 62.703125 7.8125 59.421875 \n",
       "L 7.8125 69.390625 \n",
       "Q 13.765625 71.78125 18.9375 73 \n",
       "Q 24.125 74.21875 28.421875 74.21875 \n",
       "Q 39.75 74.21875 46.484375 68.546875 \n",
       "Q 53.21875 62.890625 53.21875 53.421875 \n",
       "Q 53.21875 48.921875 51.53125 44.890625 \n",
       "Q 49.859375 40.875 45.40625 35.40625 \n",
       "Q 44.1875 33.984375 37.640625 27.21875 \n",
       "Q 31.109375 20.453125 19.1875 8.296875 \n",
       "z\n",
       "\" id=\"DejaVuSans-50\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-50\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"187.279946\" xlink:href=\"#mb15aed2d89\" y=\"224.64\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 40 -->\n",
       "      <g transform=\"translate(180.917446 239.238437)scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path d=\"M 37.796875 64.3125 \n",
       "L 12.890625 25.390625 \n",
       "L 37.796875 25.390625 \n",
       "z\n",
       "M 35.203125 72.90625 \n",
       "L 47.609375 72.90625 \n",
       "L 47.609375 25.390625 \n",
       "L 58.015625 25.390625 \n",
       "L 58.015625 17.1875 \n",
       "L 47.609375 17.1875 \n",
       "L 47.609375 0 \n",
       "L 37.796875 0 \n",
       "L 37.796875 17.1875 \n",
       "L 4.890625 17.1875 \n",
       "L 4.890625 26.703125 \n",
       "z\n",
       "\" id=\"DejaVuSans-52\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-52\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_4\">\n",
       "     <g id=\"line2d_4\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"248.767549\" xlink:href=\"#mb15aed2d89\" y=\"224.64\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_4\">\n",
       "      <!-- 60 -->\n",
       "      <g transform=\"translate(242.405049 239.238437)scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path d=\"M 33.015625 40.375 \n",
       "Q 26.375 40.375 22.484375 35.828125 \n",
       "Q 18.609375 31.296875 18.609375 23.390625 \n",
       "Q 18.609375 15.53125 22.484375 10.953125 \n",
       "Q 26.375 6.390625 33.015625 6.390625 \n",
       "Q 39.65625 6.390625 43.53125 10.953125 \n",
       "Q 47.40625 15.53125 47.40625 23.390625 \n",
       "Q 47.40625 31.296875 43.53125 35.828125 \n",
       "Q 39.65625 40.375 33.015625 40.375 \n",
       "z\n",
       "M 52.59375 71.296875 \n",
       "L 52.59375 62.3125 \n",
       "Q 48.875 64.0625 45.09375 64.984375 \n",
       "Q 41.3125 65.921875 37.59375 65.921875 \n",
       "Q 27.828125 65.921875 22.671875 59.328125 \n",
       "Q 17.53125 52.734375 16.796875 39.40625 \n",
       "Q 19.671875 43.65625 24.015625 45.921875 \n",
       "Q 28.375 48.1875 33.59375 48.1875 \n",
       "Q 44.578125 48.1875 50.953125 41.515625 \n",
       "Q 57.328125 34.859375 57.328125 23.390625 \n",
       "Q 57.328125 12.15625 50.6875 5.359375 \n",
       "Q 44.046875 -1.421875 33.015625 -1.421875 \n",
       "Q 20.359375 -1.421875 13.671875 8.265625 \n",
       "Q 6.984375 17.96875 6.984375 36.375 \n",
       "Q 6.984375 53.65625 15.1875 63.9375 \n",
       "Q 23.390625 74.21875 37.203125 74.21875 \n",
       "Q 40.921875 74.21875 44.703125 73.484375 \n",
       "Q 48.484375 72.75 52.59375 71.296875 \n",
       "z\n",
       "\" id=\"DejaVuSans-54\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-54\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_5\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"310.255152\" xlink:href=\"#mb15aed2d89\" y=\"224.64\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- 80 -->\n",
       "      <g transform=\"translate(303.892652 239.238437)scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path d=\"M 31.78125 34.625 \n",
       "Q 24.75 34.625 20.71875 30.859375 \n",
       "Q 16.703125 27.09375 16.703125 20.515625 \n",
       "Q 16.703125 13.921875 20.71875 10.15625 \n",
       "Q 24.75 6.390625 31.78125 6.390625 \n",
       "Q 38.8125 6.390625 42.859375 10.171875 \n",
       "Q 46.921875 13.96875 46.921875 20.515625 \n",
       "Q 46.921875 27.09375 42.890625 30.859375 \n",
       "Q 38.875 34.625 31.78125 34.625 \n",
       "z\n",
       "M 21.921875 38.8125 \n",
       "Q 15.578125 40.375 12.03125 44.71875 \n",
       "Q 8.5 49.078125 8.5 55.328125 \n",
       "Q 8.5 64.0625 14.71875 69.140625 \n",
       "Q 20.953125 74.21875 31.78125 74.21875 \n",
       "Q 42.671875 74.21875 48.875 69.140625 \n",
       "Q 55.078125 64.0625 55.078125 55.328125 \n",
       "Q 55.078125 49.078125 51.53125 44.71875 \n",
       "Q 48 40.375 41.703125 38.8125 \n",
       "Q 48.828125 37.15625 52.796875 32.3125 \n",
       "Q 56.78125 27.484375 56.78125 20.515625 \n",
       "Q 56.78125 9.90625 50.3125 4.234375 \n",
       "Q 43.84375 -1.421875 31.78125 -1.421875 \n",
       "Q 19.734375 -1.421875 13.25 4.234375 \n",
       "Q 6.78125 9.90625 6.78125 20.515625 \n",
       "Q 6.78125 27.484375 10.78125 32.3125 \n",
       "Q 14.796875 37.15625 21.921875 38.8125 \n",
       "z\n",
       "M 18.3125 54.390625 \n",
       "Q 18.3125 48.734375 21.84375 45.5625 \n",
       "Q 25.390625 42.390625 31.78125 42.390625 \n",
       "Q 38.140625 42.390625 41.71875 45.5625 \n",
       "Q 45.3125 48.734375 45.3125 54.390625 \n",
       "Q 45.3125 60.0625 41.71875 63.234375 \n",
       "Q 38.140625 66.40625 31.78125 66.40625 \n",
       "Q 25.390625 66.40625 21.84375 63.234375 \n",
       "Q 18.3125 60.0625 18.3125 54.390625 \n",
       "z\n",
       "\" id=\"DejaVuSans-56\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-56\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_6\">\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"371.742756\" xlink:href=\"#mb15aed2d89\" y=\"224.64\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- 100 -->\n",
       "      <g transform=\"translate(362.199006 239.238437)scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path d=\"M 12.40625 8.296875 \n",
       "L 28.515625 8.296875 \n",
       "L 28.515625 63.921875 \n",
       "L 10.984375 60.40625 \n",
       "L 10.984375 69.390625 \n",
       "L 28.421875 72.90625 \n",
       "L 38.28125 72.90625 \n",
       "L 38.28125 8.296875 \n",
       "L 54.390625 8.296875 \n",
       "L 54.390625 0 \n",
       "L 12.40625 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-49\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"127.246094\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_7\">\n",
       "     <!-- Number of iteration -->\n",
       "     <g transform=\"translate(170.354688 252.916562)scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path d=\"M 9.8125 72.90625 \n",
       "L 23.09375 72.90625 \n",
       "L 55.421875 11.921875 \n",
       "L 55.421875 72.90625 \n",
       "L 64.984375 72.90625 \n",
       "L 64.984375 0 \n",
       "L 51.703125 0 \n",
       "L 19.390625 60.984375 \n",
       "L 19.390625 0 \n",
       "L 9.8125 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-78\"/>\n",
       "       <path d=\"M 8.5 21.578125 \n",
       "L 8.5 54.6875 \n",
       "L 17.484375 54.6875 \n",
       "L 17.484375 21.921875 \n",
       "Q 17.484375 14.15625 20.5 10.265625 \n",
       "Q 23.53125 6.390625 29.59375 6.390625 \n",
       "Q 36.859375 6.390625 41.078125 11.03125 \n",
       "Q 45.3125 15.671875 45.3125 23.6875 \n",
       "L 45.3125 54.6875 \n",
       "L 54.296875 54.6875 \n",
       "L 54.296875 0 \n",
       "L 45.3125 0 \n",
       "L 45.3125 8.40625 \n",
       "Q 42.046875 3.421875 37.71875 1 \n",
       "Q 33.40625 -1.421875 27.6875 -1.421875 \n",
       "Q 18.265625 -1.421875 13.375 4.4375 \n",
       "Q 8.5 10.296875 8.5 21.578125 \n",
       "z\n",
       "M 31.109375 56 \n",
       "z\n",
       "\" id=\"DejaVuSans-117\"/>\n",
       "       <path d=\"M 52 44.1875 \n",
       "Q 55.375 50.25 60.0625 53.125 \n",
       "Q 64.75 56 71.09375 56 \n",
       "Q 79.640625 56 84.28125 50.015625 \n",
       "Q 88.921875 44.046875 88.921875 33.015625 \n",
       "L 88.921875 0 \n",
       "L 79.890625 0 \n",
       "L 79.890625 32.71875 \n",
       "Q 79.890625 40.578125 77.09375 44.375 \n",
       "Q 74.3125 48.1875 68.609375 48.1875 \n",
       "Q 61.625 48.1875 57.5625 43.546875 \n",
       "Q 53.515625 38.921875 53.515625 30.90625 \n",
       "L 53.515625 0 \n",
       "L 44.484375 0 \n",
       "L 44.484375 32.71875 \n",
       "Q 44.484375 40.625 41.703125 44.40625 \n",
       "Q 38.921875 48.1875 33.109375 48.1875 \n",
       "Q 26.21875 48.1875 22.15625 43.53125 \n",
       "Q 18.109375 38.875 18.109375 30.90625 \n",
       "L 18.109375 0 \n",
       "L 9.078125 0 \n",
       "L 9.078125 54.6875 \n",
       "L 18.109375 54.6875 \n",
       "L 18.109375 46.1875 \n",
       "Q 21.1875 51.21875 25.484375 53.609375 \n",
       "Q 29.78125 56 35.6875 56 \n",
       "Q 41.65625 56 45.828125 52.96875 \n",
       "Q 50 49.953125 52 44.1875 \n",
       "z\n",
       "\" id=\"DejaVuSans-109\"/>\n",
       "       <path d=\"M 48.6875 27.296875 \n",
       "Q 48.6875 37.203125 44.609375 42.84375 \n",
       "Q 40.53125 48.484375 33.40625 48.484375 \n",
       "Q 26.265625 48.484375 22.1875 42.84375 \n",
       "Q 18.109375 37.203125 18.109375 27.296875 \n",
       "Q 18.109375 17.390625 22.1875 11.75 \n",
       "Q 26.265625 6.109375 33.40625 6.109375 \n",
       "Q 40.53125 6.109375 44.609375 11.75 \n",
       "Q 48.6875 17.390625 48.6875 27.296875 \n",
       "z\n",
       "M 18.109375 46.390625 \n",
       "Q 20.953125 51.265625 25.265625 53.625 \n",
       "Q 29.59375 56 35.59375 56 \n",
       "Q 45.5625 56 51.78125 48.09375 \n",
       "Q 58.015625 40.1875 58.015625 27.296875 \n",
       "Q 58.015625 14.40625 51.78125 6.484375 \n",
       "Q 45.5625 -1.421875 35.59375 -1.421875 \n",
       "Q 29.59375 -1.421875 25.265625 0.953125 \n",
       "Q 20.953125 3.328125 18.109375 8.203125 \n",
       "L 18.109375 0 \n",
       "L 9.078125 0 \n",
       "L 9.078125 75.984375 \n",
       "L 18.109375 75.984375 \n",
       "z\n",
       "\" id=\"DejaVuSans-98\"/>\n",
       "       <path d=\"M 56.203125 29.59375 \n",
       "L 56.203125 25.203125 \n",
       "L 14.890625 25.203125 \n",
       "Q 15.484375 15.921875 20.484375 11.0625 \n",
       "Q 25.484375 6.203125 34.421875 6.203125 \n",
       "Q 39.59375 6.203125 44.453125 7.46875 \n",
       "Q 49.3125 8.734375 54.109375 11.28125 \n",
       "L 54.109375 2.78125 \n",
       "Q 49.265625 0.734375 44.1875 -0.34375 \n",
       "Q 39.109375 -1.421875 33.890625 -1.421875 \n",
       "Q 20.796875 -1.421875 13.15625 6.1875 \n",
       "Q 5.515625 13.8125 5.515625 26.8125 \n",
       "Q 5.515625 40.234375 12.765625 48.109375 \n",
       "Q 20.015625 56 32.328125 56 \n",
       "Q 43.359375 56 49.78125 48.890625 \n",
       "Q 56.203125 41.796875 56.203125 29.59375 \n",
       "z\n",
       "M 47.21875 32.234375 \n",
       "Q 47.125 39.59375 43.09375 43.984375 \n",
       "Q 39.0625 48.390625 32.421875 48.390625 \n",
       "Q 24.90625 48.390625 20.390625 44.140625 \n",
       "Q 15.875 39.890625 15.1875 32.171875 \n",
       "z\n",
       "\" id=\"DejaVuSans-101\"/>\n",
       "       <path d=\"M 41.109375 46.296875 \n",
       "Q 39.59375 47.171875 37.8125 47.578125 \n",
       "Q 36.03125 48 33.890625 48 \n",
       "Q 26.265625 48 22.1875 43.046875 \n",
       "Q 18.109375 38.09375 18.109375 28.8125 \n",
       "L 18.109375 0 \n",
       "L 9.078125 0 \n",
       "L 9.078125 54.6875 \n",
       "L 18.109375 54.6875 \n",
       "L 18.109375 46.1875 \n",
       "Q 20.953125 51.171875 25.484375 53.578125 \n",
       "Q 30.03125 56 36.53125 56 \n",
       "Q 37.453125 56 38.578125 55.875 \n",
       "Q 39.703125 55.765625 41.0625 55.515625 \n",
       "z\n",
       "\" id=\"DejaVuSans-114\"/>\n",
       "       <path id=\"DejaVuSans-32\"/>\n",
       "       <path d=\"M 30.609375 48.390625 \n",
       "Q 23.390625 48.390625 19.1875 42.75 \n",
       "Q 14.984375 37.109375 14.984375 27.296875 \n",
       "Q 14.984375 17.484375 19.15625 11.84375 \n",
       "Q 23.34375 6.203125 30.609375 6.203125 \n",
       "Q 37.796875 6.203125 41.984375 11.859375 \n",
       "Q 46.1875 17.53125 46.1875 27.296875 \n",
       "Q 46.1875 37.015625 41.984375 42.703125 \n",
       "Q 37.796875 48.390625 30.609375 48.390625 \n",
       "z\n",
       "M 30.609375 56 \n",
       "Q 42.328125 56 49.015625 48.375 \n",
       "Q 55.71875 40.765625 55.71875 27.296875 \n",
       "Q 55.71875 13.875 49.015625 6.21875 \n",
       "Q 42.328125 -1.421875 30.609375 -1.421875 \n",
       "Q 18.84375 -1.421875 12.171875 6.21875 \n",
       "Q 5.515625 13.875 5.515625 27.296875 \n",
       "Q 5.515625 40.765625 12.171875 48.375 \n",
       "Q 18.84375 56 30.609375 56 \n",
       "z\n",
       "\" id=\"DejaVuSans-111\"/>\n",
       "       <path d=\"M 37.109375 75.984375 \n",
       "L 37.109375 68.5 \n",
       "L 28.515625 68.5 \n",
       "Q 23.6875 68.5 21.796875 66.546875 \n",
       "Q 19.921875 64.59375 19.921875 59.515625 \n",
       "L 19.921875 54.6875 \n",
       "L 34.71875 54.6875 \n",
       "L 34.71875 47.703125 \n",
       "L 19.921875 47.703125 \n",
       "L 19.921875 0 \n",
       "L 10.890625 0 \n",
       "L 10.890625 47.703125 \n",
       "L 2.296875 47.703125 \n",
       "L 2.296875 54.6875 \n",
       "L 10.890625 54.6875 \n",
       "L 10.890625 58.5 \n",
       "Q 10.890625 67.625 15.140625 71.796875 \n",
       "Q 19.390625 75.984375 28.609375 75.984375 \n",
       "z\n",
       "\" id=\"DejaVuSans-102\"/>\n",
       "       <path d=\"M 9.421875 54.6875 \n",
       "L 18.40625 54.6875 \n",
       "L 18.40625 0 \n",
       "L 9.421875 0 \n",
       "z\n",
       "M 9.421875 75.984375 \n",
       "L 18.40625 75.984375 \n",
       "L 18.40625 64.59375 \n",
       "L 9.421875 64.59375 \n",
       "z\n",
       "\" id=\"DejaVuSans-105\"/>\n",
       "       <path d=\"M 18.3125 70.21875 \n",
       "L 18.3125 54.6875 \n",
       "L 36.8125 54.6875 \n",
       "L 36.8125 47.703125 \n",
       "L 18.3125 47.703125 \n",
       "L 18.3125 18.015625 \n",
       "Q 18.3125 11.328125 20.140625 9.421875 \n",
       "Q 21.96875 7.515625 27.59375 7.515625 \n",
       "L 36.8125 7.515625 \n",
       "L 36.8125 0 \n",
       "L 27.59375 0 \n",
       "Q 17.1875 0 13.234375 3.875 \n",
       "Q 9.28125 7.765625 9.28125 18.015625 \n",
       "L 9.28125 47.703125 \n",
       "L 2.6875 47.703125 \n",
       "L 2.6875 54.6875 \n",
       "L 9.28125 54.6875 \n",
       "L 9.28125 70.21875 \n",
       "z\n",
       "\" id=\"DejaVuSans-116\"/>\n",
       "       <path d=\"M 34.28125 27.484375 \n",
       "Q 23.390625 27.484375 19.1875 25 \n",
       "Q 14.984375 22.515625 14.984375 16.5 \n",
       "Q 14.984375 11.71875 18.140625 8.90625 \n",
       "Q 21.296875 6.109375 26.703125 6.109375 \n",
       "Q 34.1875 6.109375 38.703125 11.40625 \n",
       "Q 43.21875 16.703125 43.21875 25.484375 \n",
       "L 43.21875 27.484375 \n",
       "z\n",
       "M 52.203125 31.203125 \n",
       "L 52.203125 0 \n",
       "L 43.21875 0 \n",
       "L 43.21875 8.296875 \n",
       "Q 40.140625 3.328125 35.546875 0.953125 \n",
       "Q 30.953125 -1.421875 24.3125 -1.421875 \n",
       "Q 15.921875 -1.421875 10.953125 3.296875 \n",
       "Q 6 8.015625 6 15.921875 \n",
       "Q 6 25.140625 12.171875 29.828125 \n",
       "Q 18.359375 34.515625 30.609375 34.515625 \n",
       "L 43.21875 34.515625 \n",
       "L 43.21875 35.40625 \n",
       "Q 43.21875 41.609375 39.140625 45 \n",
       "Q 35.0625 48.390625 27.6875 48.390625 \n",
       "Q 23 48.390625 18.546875 47.265625 \n",
       "Q 14.109375 46.140625 10.015625 43.890625 \n",
       "L 10.015625 52.203125 \n",
       "Q 14.9375 54.109375 19.578125 55.046875 \n",
       "Q 24.21875 56 28.609375 56 \n",
       "Q 40.484375 56 46.34375 49.84375 \n",
       "Q 52.203125 43.703125 52.203125 31.203125 \n",
       "z\n",
       "\" id=\"DejaVuSans-97\"/>\n",
       "       <path d=\"M 54.890625 33.015625 \n",
       "L 54.890625 0 \n",
       "L 45.90625 0 \n",
       "L 45.90625 32.71875 \n",
       "Q 45.90625 40.484375 42.875 44.328125 \n",
       "Q 39.84375 48.1875 33.796875 48.1875 \n",
       "Q 26.515625 48.1875 22.3125 43.546875 \n",
       "Q 18.109375 38.921875 18.109375 30.90625 \n",
       "L 18.109375 0 \n",
       "L 9.078125 0 \n",
       "L 9.078125 54.6875 \n",
       "L 18.109375 54.6875 \n",
       "L 18.109375 46.1875 \n",
       "Q 21.34375 51.125 25.703125 53.5625 \n",
       "Q 30.078125 56 35.796875 56 \n",
       "Q 45.21875 56 50.046875 50.171875 \n",
       "Q 54.890625 44.34375 54.890625 33.015625 \n",
       "z\n",
       "\" id=\"DejaVuSans-110\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-78\"/>\n",
       "      <use x=\"74.804688\" xlink:href=\"#DejaVuSans-117\"/>\n",
       "      <use x=\"138.183594\" xlink:href=\"#DejaVuSans-109\"/>\n",
       "      <use x=\"235.595703\" xlink:href=\"#DejaVuSans-98\"/>\n",
       "      <use x=\"299.072266\" xlink:href=\"#DejaVuSans-101\"/>\n",
       "      <use x=\"360.595703\" xlink:href=\"#DejaVuSans-114\"/>\n",
       "      <use x=\"401.708984\" xlink:href=\"#DejaVuSans-32\"/>\n",
       "      <use x=\"433.496094\" xlink:href=\"#DejaVuSans-111\"/>\n",
       "      <use x=\"494.677734\" xlink:href=\"#DejaVuSans-102\"/>\n",
       "      <use x=\"529.882812\" xlink:href=\"#DejaVuSans-32\"/>\n",
       "      <use x=\"561.669922\" xlink:href=\"#DejaVuSans-105\"/>\n",
       "      <use x=\"589.453125\" xlink:href=\"#DejaVuSans-116\"/>\n",
       "      <use x=\"628.662109\" xlink:href=\"#DejaVuSans-101\"/>\n",
       "      <use x=\"690.185547\" xlink:href=\"#DejaVuSans-114\"/>\n",
       "      <use x=\"731.298828\" xlink:href=\"#DejaVuSans-97\"/>\n",
       "      <use x=\"792.578125\" xlink:href=\"#DejaVuSans-116\"/>\n",
       "      <use x=\"831.787109\" xlink:href=\"#DejaVuSans-105\"/>\n",
       "      <use x=\"859.570312\" xlink:href=\"#DejaVuSans-111\"/>\n",
       "      <use x=\"920.751953\" xlink:href=\"#DejaVuSans-110\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <defs>\n",
       "       <path d=\"M 0 0 \n",
       "L -3.5 0 \n",
       "\" id=\"m82ff1f3097\" style=\"stroke:#000000;stroke-width:0.8;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m82ff1f3097\" y=\"183.328577\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- −1.0 -->\n",
       "      <g transform=\"translate(20.878125 187.127796)scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path d=\"M 10.59375 35.5 \n",
       "L 73.1875 35.5 \n",
       "L 73.1875 27.203125 \n",
       "L 10.59375 27.203125 \n",
       "z\n",
       "\" id=\"DejaVuSans-8722\"/>\n",
       "        <path d=\"M 10.6875 12.40625 \n",
       "L 21 12.40625 \n",
       "L 21 0 \n",
       "L 10.6875 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-46\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-8722\"/>\n",
       "       <use x=\"83.789062\" xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"147.412109\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"179.199219\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m82ff1f3097\" y=\"137.175537\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- −0.8 -->\n",
       "      <g transform=\"translate(20.878125 140.974756)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-8722\"/>\n",
       "       <use x=\"83.789062\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"147.412109\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"179.199219\" xlink:href=\"#DejaVuSans-56\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m82ff1f3097\" y=\"91.022497\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_10\">\n",
       "      <!-- −0.6 -->\n",
       "      <g transform=\"translate(20.878125 94.821716)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-8722\"/>\n",
       "       <use x=\"83.789062\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"147.412109\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"179.199219\" xlink:href=\"#DejaVuSans-54\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m82ff1f3097\" y=\"44.869457\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- −0.4 -->\n",
       "      <g transform=\"translate(20.878125 48.668676)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-8722\"/>\n",
       "       <use x=\"83.789062\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"147.412109\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"179.199219\" xlink:href=\"#DejaVuSans-52\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_12\">\n",
       "     <!-- Energy (Ha) -->\n",
       "     <g transform=\"translate(14.798437 145.741094)rotate(-90)scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path d=\"M 9.8125 72.90625 \n",
       "L 55.90625 72.90625 \n",
       "L 55.90625 64.59375 \n",
       "L 19.671875 64.59375 \n",
       "L 19.671875 43.015625 \n",
       "L 54.390625 43.015625 \n",
       "L 54.390625 34.71875 \n",
       "L 19.671875 34.71875 \n",
       "L 19.671875 8.296875 \n",
       "L 56.78125 8.296875 \n",
       "L 56.78125 0 \n",
       "L 9.8125 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-69\"/>\n",
       "       <path d=\"M 45.40625 27.984375 \n",
       "Q 45.40625 37.75 41.375 43.109375 \n",
       "Q 37.359375 48.484375 30.078125 48.484375 \n",
       "Q 22.859375 48.484375 18.828125 43.109375 \n",
       "Q 14.796875 37.75 14.796875 27.984375 \n",
       "Q 14.796875 18.265625 18.828125 12.890625 \n",
       "Q 22.859375 7.515625 30.078125 7.515625 \n",
       "Q 37.359375 7.515625 41.375 12.890625 \n",
       "Q 45.40625 18.265625 45.40625 27.984375 \n",
       "z\n",
       "M 54.390625 6.78125 \n",
       "Q 54.390625 -7.171875 48.1875 -13.984375 \n",
       "Q 42 -20.796875 29.203125 -20.796875 \n",
       "Q 24.46875 -20.796875 20.265625 -20.09375 \n",
       "Q 16.0625 -19.390625 12.109375 -17.921875 \n",
       "L 12.109375 -9.1875 \n",
       "Q 16.0625 -11.328125 19.921875 -12.34375 \n",
       "Q 23.78125 -13.375 27.78125 -13.375 \n",
       "Q 36.625 -13.375 41.015625 -8.765625 \n",
       "Q 45.40625 -4.15625 45.40625 5.171875 \n",
       "L 45.40625 9.625 \n",
       "Q 42.625 4.78125 38.28125 2.390625 \n",
       "Q 33.9375 0 27.875 0 \n",
       "Q 17.828125 0 11.671875 7.65625 \n",
       "Q 5.515625 15.328125 5.515625 27.984375 \n",
       "Q 5.515625 40.671875 11.671875 48.328125 \n",
       "Q 17.828125 56 27.875 56 \n",
       "Q 33.9375 56 38.28125 53.609375 \n",
       "Q 42.625 51.21875 45.40625 46.390625 \n",
       "L 45.40625 54.6875 \n",
       "L 54.390625 54.6875 \n",
       "z\n",
       "\" id=\"DejaVuSans-103\"/>\n",
       "       <path d=\"M 32.171875 -5.078125 \n",
       "Q 28.375 -14.84375 24.75 -17.8125 \n",
       "Q 21.140625 -20.796875 15.09375 -20.796875 \n",
       "L 7.90625 -20.796875 \n",
       "L 7.90625 -13.28125 \n",
       "L 13.1875 -13.28125 \n",
       "Q 16.890625 -13.28125 18.9375 -11.515625 \n",
       "Q 21 -9.765625 23.484375 -3.21875 \n",
       "L 25.09375 0.875 \n",
       "L 2.984375 54.6875 \n",
       "L 12.5 54.6875 \n",
       "L 29.59375 11.921875 \n",
       "L 46.6875 54.6875 \n",
       "L 56.203125 54.6875 \n",
       "z\n",
       "\" id=\"DejaVuSans-121\"/>\n",
       "       <path d=\"M 31 75.875 \n",
       "Q 24.46875 64.65625 21.28125 53.65625 \n",
       "Q 18.109375 42.671875 18.109375 31.390625 \n",
       "Q 18.109375 20.125 21.3125 9.0625 \n",
       "Q 24.515625 -2 31 -13.1875 \n",
       "L 23.1875 -13.1875 \n",
       "Q 15.875 -1.703125 12.234375 9.375 \n",
       "Q 8.59375 20.453125 8.59375 31.390625 \n",
       "Q 8.59375 42.28125 12.203125 53.3125 \n",
       "Q 15.828125 64.359375 23.1875 75.875 \n",
       "z\n",
       "\" id=\"DejaVuSans-40\"/>\n",
       "       <path d=\"M 9.8125 72.90625 \n",
       "L 19.671875 72.90625 \n",
       "L 19.671875 43.015625 \n",
       "L 55.515625 43.015625 \n",
       "L 55.515625 72.90625 \n",
       "L 65.375 72.90625 \n",
       "L 65.375 0 \n",
       "L 55.515625 0 \n",
       "L 55.515625 34.71875 \n",
       "L 19.671875 34.71875 \n",
       "L 19.671875 0 \n",
       "L 9.8125 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-72\"/>\n",
       "       <path d=\"M 8.015625 75.875 \n",
       "L 15.828125 75.875 \n",
       "Q 23.140625 64.359375 26.78125 53.3125 \n",
       "Q 30.421875 42.28125 30.421875 31.390625 \n",
       "Q 30.421875 20.453125 26.78125 9.375 \n",
       "Q 23.140625 -1.703125 15.828125 -13.1875 \n",
       "L 8.015625 -13.1875 \n",
       "Q 14.5 -2 17.703125 9.0625 \n",
       "Q 20.90625 20.125 20.90625 31.390625 \n",
       "Q 20.90625 42.671875 17.703125 53.65625 \n",
       "Q 14.5 64.65625 8.015625 75.875 \n",
       "z\n",
       "\" id=\"DejaVuSans-41\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-69\"/>\n",
       "      <use x=\"63.183594\" xlink:href=\"#DejaVuSans-110\"/>\n",
       "      <use x=\"126.5625\" xlink:href=\"#DejaVuSans-101\"/>\n",
       "      <use x=\"188.085938\" xlink:href=\"#DejaVuSans-114\"/>\n",
       "      <use x=\"227.449219\" xlink:href=\"#DejaVuSans-103\"/>\n",
       "      <use x=\"290.925781\" xlink:href=\"#DejaVuSans-121\"/>\n",
       "      <use x=\"350.105469\" xlink:href=\"#DejaVuSans-32\"/>\n",
       "      <use x=\"381.892578\" xlink:href=\"#DejaVuSans-40\"/>\n",
       "      <use x=\"420.90625\" xlink:href=\"#DejaVuSans-72\"/>\n",
       "      <use x=\"496.101562\" xlink:href=\"#DejaVuSans-97\"/>\n",
       "      <use x=\"557.380859\" xlink:href=\"#DejaVuSans-41\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_11\">\n",
       "    <path clip-path=\"url(#pd4f85b6dad)\" d=\"M 67.379119 17.083636 \n",
       "L 70.453499 52.067317 \n",
       "L 73.52788 73.893781 \n",
       "L 76.60226 89.403365 \n",
       "L 79.67664 101.103015 \n",
       "L 82.75102 106.747158 \n",
       "L 85.8254 106.460241 \n",
       "L 88.89978 105.507185 \n",
       "L 91.974161 108.730875 \n",
       "L 95.048541 114.828986 \n",
       "L 98.122921 121.317281 \n",
       "L 101.197301 127.271118 \n",
       "L 104.271681 132.722947 \n",
       "L 107.346061 136.290317 \n",
       "L 110.420442 136.491529 \n",
       "L 113.494822 135.229009 \n",
       "L 116.569202 136.040872 \n",
       "L 119.643582 139.270694 \n",
       "L 122.717962 143.632285 \n",
       "L 125.792342 149.250196 \n",
       "L 128.866723 156.475322 \n",
       "L 131.941103 164.321024 \n",
       "L 135.015483 171.975289 \n",
       "L 138.089863 180.051056 \n",
       "L 141.164243 188.816374 \n",
       "L 144.238623 196.68898 \n",
       "L 147.313004 201.880479 \n",
       "L 150.387384 204.341765 \n",
       "L 153.461764 205.809501 \n",
       "L 156.536144 207.470168 \n",
       "L 159.610524 208.717602 \n",
       "L 162.684904 208.33442 \n",
       "L 165.759285 206.106599 \n",
       "L 168.833665 203.368367 \n",
       "L 171.908045 202.193985 \n",
       "L 174.982425 203.821426 \n",
       "L 178.056805 207.6248 \n",
       "L 181.131185 211.442128 \n",
       "L 184.205566 213.178232 \n",
       "L 187.279946 212.501548 \n",
       "L 190.354326 210.881746 \n",
       "L 193.428706 209.83521 \n",
       "L 196.503086 209.713236 \n",
       "L 199.577466 210.116092 \n",
       "L 202.651847 210.667376 \n",
       "L 205.726227 211.253538 \n",
       "L 208.800607 211.874728 \n",
       "L 211.874987 212.461954 \n",
       "L 214.949367 212.89122 \n",
       "L 218.023747 213.121225 \n",
       "L 221.098128 213.196632 \n",
       "L 224.172508 213.159061 \n",
       "L 227.246888 213.040632 \n",
       "L 230.321268 212.924493 \n",
       "L 233.395648 212.956861 \n",
       "L 236.470028 213.213594 \n",
       "L 239.544409 213.568351 \n",
       "L 242.618789 213.799369 \n",
       "L 245.693169 213.844271 \n",
       "L 248.767549 213.858923 \n",
       "L 251.841929 213.990464 \n",
       "L 254.916309 214.191296 \n",
       "L 257.99069 214.30897 \n",
       "L 261.06507 214.272577 \n",
       "L 264.13945 214.151579 \n",
       "L 267.21383 214.087872 \n",
       "L 270.28821 214.171999 \n",
       "L 273.36259 214.356773 \n",
       "L 276.436971 214.517026 \n",
       "L 279.511351 214.588477 \n",
       "L 282.585731 214.597824 \n",
       "L 285.660111 214.587599 \n",
       "L 288.734491 214.580132 \n",
       "L 291.808871 214.588033 \n",
       "L 294.883252 214.607926 \n",
       "L 297.957632 214.624079 \n",
       "L 301.032012 214.635498 \n",
       "L 304.106392 214.655124 \n",
       "L 307.180772 214.683389 \n",
       "L 310.255152 214.704753 \n",
       "L 313.329533 214.707474 \n",
       "L 316.403913 214.697921 \n",
       "L 319.478293 214.695318 \n",
       "L 322.552673 214.706996 \n",
       "L 325.627053 214.716078 \n",
       "L 328.701433 214.708365 \n",
       "L 331.775814 214.695788 \n",
       "L 334.850194 214.696601 \n",
       "L 337.924574 214.71113 \n",
       "L 340.998954 214.727622 \n",
       "L 344.073334 214.736059 \n",
       "L 347.147714 214.734302 \n",
       "L 350.222095 214.729354 \n",
       "L 353.296475 214.728984 \n",
       "L 356.370855 214.732614 \n",
       "L 359.445235 214.735857 \n",
       "L 362.519615 214.737722 \n",
       "L 365.593995 214.739443 \n",
       "L 368.668376 214.742204 \n",
       "L 371.742756 214.746107 \n",
       "\" style=\"fill:none;stroke:#ff0000;stroke-linecap:square;stroke-opacity:0.7;stroke-width:1.5;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_12\">\n",
       "    <path clip-path=\"url(#pd4f85b6dad)\" d=\"M 67.379119 214.756364 \n",
       "L 70.453499 214.756364 \n",
       "L 73.52788 214.756364 \n",
       "L 76.60226 214.756364 \n",
       "L 79.67664 214.756364 \n",
       "L 82.75102 214.756364 \n",
       "L 85.8254 214.756364 \n",
       "L 88.89978 214.756364 \n",
       "L 91.974161 214.756364 \n",
       "L 95.048541 214.756364 \n",
       "L 98.122921 214.756364 \n",
       "L 101.197301 214.756364 \n",
       "L 104.271681 214.756364 \n",
       "L 107.346061 214.756364 \n",
       "L 110.420442 214.756364 \n",
       "L 113.494822 214.756364 \n",
       "L 116.569202 214.756364 \n",
       "L 119.643582 214.756364 \n",
       "L 122.717962 214.756364 \n",
       "L 125.792342 214.756364 \n",
       "L 128.866723 214.756364 \n",
       "L 131.941103 214.756364 \n",
       "L 135.015483 214.756364 \n",
       "L 138.089863 214.756364 \n",
       "L 141.164243 214.756364 \n",
       "L 144.238623 214.756364 \n",
       "L 147.313004 214.756364 \n",
       "L 150.387384 214.756364 \n",
       "L 153.461764 214.756364 \n",
       "L 156.536144 214.756364 \n",
       "L 159.610524 214.756364 \n",
       "L 162.684904 214.756364 \n",
       "L 165.759285 214.756364 \n",
       "L 168.833665 214.756364 \n",
       "L 171.908045 214.756364 \n",
       "L 174.982425 214.756364 \n",
       "L 178.056805 214.756364 \n",
       "L 181.131185 214.756364 \n",
       "L 184.205566 214.756364 \n",
       "L 187.279946 214.756364 \n",
       "L 190.354326 214.756364 \n",
       "L 193.428706 214.756364 \n",
       "L 196.503086 214.756364 \n",
       "L 199.577466 214.756364 \n",
       "L 202.651847 214.756364 \n",
       "L 205.726227 214.756364 \n",
       "L 208.800607 214.756364 \n",
       "L 211.874987 214.756364 \n",
       "L 214.949367 214.756364 \n",
       "L 218.023747 214.756364 \n",
       "L 221.098128 214.756364 \n",
       "L 224.172508 214.756364 \n",
       "L 227.246888 214.756364 \n",
       "L 230.321268 214.756364 \n",
       "L 233.395648 214.756364 \n",
       "L 236.470028 214.756364 \n",
       "L 239.544409 214.756364 \n",
       "L 242.618789 214.756364 \n",
       "L 245.693169 214.756364 \n",
       "L 248.767549 214.756364 \n",
       "L 251.841929 214.756364 \n",
       "L 254.916309 214.756364 \n",
       "L 257.99069 214.756364 \n",
       "L 261.06507 214.756364 \n",
       "L 264.13945 214.756364 \n",
       "L 267.21383 214.756364 \n",
       "L 270.28821 214.756364 \n",
       "L 273.36259 214.756364 \n",
       "L 276.436971 214.756364 \n",
       "L 279.511351 214.756364 \n",
       "L 282.585731 214.756364 \n",
       "L 285.660111 214.756364 \n",
       "L 288.734491 214.756364 \n",
       "L 291.808871 214.756364 \n",
       "L 294.883252 214.756364 \n",
       "L 297.957632 214.756364 \n",
       "L 301.032012 214.756364 \n",
       "L 304.106392 214.756364 \n",
       "L 307.180772 214.756364 \n",
       "L 310.255152 214.756364 \n",
       "L 313.329533 214.756364 \n",
       "L 316.403913 214.756364 \n",
       "L 319.478293 214.756364 \n",
       "L 322.552673 214.756364 \n",
       "L 325.627053 214.756364 \n",
       "L 328.701433 214.756364 \n",
       "L 331.775814 214.756364 \n",
       "L 334.850194 214.756364 \n",
       "L 337.924574 214.756364 \n",
       "L 340.998954 214.756364 \n",
       "L 344.073334 214.756364 \n",
       "L 347.147714 214.756364 \n",
       "L 350.222095 214.756364 \n",
       "L 353.296475 214.756364 \n",
       "L 356.370855 214.756364 \n",
       "L 359.445235 214.756364 \n",
       "L 362.519615 214.756364 \n",
       "L 365.593995 214.756364 \n",
       "L 368.668376 214.756364 \n",
       "L 371.742756 214.756364 \n",
       "\" style=\"fill:none;stroke:#0000ff;stroke-dasharray:1.5,2.475;stroke-dashoffset:0;stroke-opacity:0.7;stroke-width:1.5;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\"M 52.160938 224.64 \n",
       "L 52.160938 7.2 \n",
       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\"M 386.960938 224.64 \n",
       "L 386.960938 7.2 \n",
       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\"M 52.160938 224.64 \n",
       "L 386.960938 224.64 \n",
       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\"M 52.160938 7.2 \n",
       "L 386.960938 7.2 \n",
       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "   <g id=\"legend_1\">\n",
       "    <g id=\"patch_7\">\n",
       "     <path d=\"M 244.220312 44.978125 \n",
       "L 379.960938 44.978125 \n",
       "Q 381.960938 44.978125 381.960938 42.978125 \n",
       "L 381.960938 14.2 \n",
       "Q 381.960938 12.2 379.960938 12.2 \n",
       "L 244.220312 12.2 \n",
       "Q 242.220312 12.2 242.220312 14.2 \n",
       "L 242.220312 42.978125 \n",
       "Q 242.220312 44.978125 244.220312 44.978125 \n",
       "z\n",
       "\" style=\"fill:#ffffff;opacity:0.8;stroke:#cccccc;stroke-linejoin:miter;\"/>\n",
       "    </g>\n",
       "    <g id=\"line2d_13\">\n",
       "     <path d=\"M 246.220312 20.4 \n",
       "L 266.220312 20.4 \n",
       "\" style=\"fill:none;stroke:#ff0000;stroke-linecap:square;stroke-opacity:0.7;stroke-width:1.5;\"/>\n",
       "    </g>\n",
       "    <g id=\"line2d_14\"/>\n",
       "    <g id=\"text_13\">\n",
       "     <!-- $\\left\\langle {\\psi \\left( {\\theta } \\right)} \\right|H\\left| {\\psi \\left( {\\theta } \\right)} \\right\\rangle $ -->\n",
       "     <g transform=\"translate(274.220312 23.9)scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path d=\"M 8.9375 31.34375 \n",
       "L 22.703125 75.875 \n",
       "L 31 75.875 \n",
       "L 17.234375 31.34375 \n",
       "L 31 -13.1875 \n",
       "L 22.703125 -13.1875 \n",
       "z\n",
       "\" id=\"DejaVuSans-10216\"/>\n",
       "       <path d=\"M 24.859375 -1.21875 \n",
       "Q 13.921875 0.59375 9.625 5.328125 \n",
       "Q 4.34375 11.140625 6.640625 23 \n",
       "L 12.796875 54.6875 \n",
       "L 21.875 54.6875 \n",
       "L 15.765625 23.34375 \n",
       "Q 14.0625 14.40625 17.484375 10.6875 \n",
       "Q 20.515625 7.46875 26.421875 6.78125 \n",
       "L 35.6875 54.6875 \n",
       "L 44.625 54.6875 \n",
       "L 35.359375 6.84375 \n",
       "Q 41.890625 7.515625 45.75 10.75 \n",
       "Q 50.734375 14.84375 52.390625 23.390625 \n",
       "L 58.453125 54.6875 \n",
       "L 67.53125 54.6875 \n",
       "L 61.375 23.046875 \n",
       "Q 58.984375 10.75 51.5625 5.375 \n",
       "Q 44.875 0.53125 33.796875 -1.171875 \n",
       "L 29.984375 -20.796875 \n",
       "L 21.046875 -20.796875 \n",
       "z\n",
       "\" id=\"DejaVuSans-Oblique-968\"/>\n",
       "       <path d=\"M 45.515625 34.671875 \n",
       "L 14.453125 34.671875 \n",
       "Q 12.359375 20.0625 14.5 13.875 \n",
       "Q 17.1875 6.25 24.46875 6.25 \n",
       "Q 31.78125 6.25 37.359375 13.921875 \n",
       "Q 42.234375 20.65625 45.515625 34.671875 \n",
       "z\n",
       "M 47.015625 42.96875 \n",
       "Q 48.34375 56.84375 46.625 61.71875 \n",
       "Q 43.953125 69.4375 36.765625 69.4375 \n",
       "Q 29.296875 69.4375 23.828125 61.8125 \n",
       "Q 19.53125 55.671875 16.15625 42.96875 \n",
       "z\n",
       "M 38.1875 76.765625 \n",
       "Q 49.90625 76.765625 54.59375 66.40625 \n",
       "Q 59.28125 56.109375 55.71875 37.84375 \n",
       "Q 52.203125 19.625 43.453125 9.28125 \n",
       "Q 34.765625 -1.125 23.046875 -1.125 \n",
       "Q 11.28125 -1.125 6.640625 9.28125 \n",
       "Q 2 19.625 5.515625 37.84375 \n",
       "Q 9.078125 56.109375 17.71875 66.40625 \n",
       "Q 26.421875 76.765625 38.1875 76.765625 \n",
       "z\n",
       "\" id=\"DejaVuSans-Oblique-952\"/>\n",
       "       <path d=\"M 21 76.421875 \n",
       "L 21 -23.578125 \n",
       "L 12.703125 -23.578125 \n",
       "L 12.703125 76.421875 \n",
       "z\n",
       "\" id=\"DejaVuSans-124\"/>\n",
       "       <path d=\"M 16.890625 72.90625 \n",
       "L 26.8125 72.90625 \n",
       "L 21 43.015625 \n",
       "L 56.78125 43.015625 \n",
       "L 62.59375 72.90625 \n",
       "L 72.515625 72.90625 \n",
       "L 58.296875 0 \n",
       "L 48.390625 0 \n",
       "L 55.171875 34.71875 \n",
       "L 19.390625 34.71875 \n",
       "L 12.59375 0 \n",
       "L 2.6875 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-Oblique-72\"/>\n",
       "       <path d=\"M 30.078125 31.34375 \n",
       "L 16.3125 -13.1875 \n",
       "L 8.015625 -13.1875 \n",
       "L 21.78125 31.34375 \n",
       "L 8.015625 75.875 \n",
       "L 16.3125 75.875 \n",
       "z\n",
       "\" id=\"DejaVuSans-10217\"/>\n",
       "      </defs>\n",
       "      <use transform=\"translate(0 0.234375)\" xlink:href=\"#DejaVuSans-10216\"/>\n",
       "      <use transform=\"translate(39.013672 0.234375)\" xlink:href=\"#DejaVuSans-Oblique-968\"/>\n",
       "      <use transform=\"translate(104.980469 0.234375)\" xlink:href=\"#DejaVuSans-40\"/>\n",
       "      <use transform=\"translate(143.994141 0.234375)\" xlink:href=\"#DejaVuSans-Oblique-952\"/>\n",
       "      <use transform=\"translate(205.175781 0.234375)\" xlink:href=\"#DejaVuSans-41\"/>\n",
       "      <use transform=\"translate(244.189453 0.234375)\" xlink:href=\"#DejaVuSans-124\"/>\n",
       "      <use transform=\"translate(277.880859 0.234375)\" xlink:href=\"#DejaVuSans-Oblique-72\"/>\n",
       "      <use transform=\"translate(353.076172 0.234375)\" xlink:href=\"#DejaVuSans-124\"/>\n",
       "      <use transform=\"translate(386.767578 0.234375)\" xlink:href=\"#DejaVuSans-Oblique-968\"/>\n",
       "      <use transform=\"translate(452.734375 0.234375)\" xlink:href=\"#DejaVuSans-40\"/>\n",
       "      <use transform=\"translate(491.748047 0.234375)\" xlink:href=\"#DejaVuSans-Oblique-952\"/>\n",
       "      <use transform=\"translate(552.929688 0.234375)\" xlink:href=\"#DejaVuSans-41\"/>\n",
       "      <use transform=\"translate(591.943359 0.234375)\" xlink:href=\"#DejaVuSans-10217\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"line2d_15\">\n",
       "     <path d=\"M 246.220312 35.398437 \n",
       "L 266.220312 35.398437 \n",
       "\" style=\"fill:none;stroke:#0000ff;stroke-dasharray:1.5,2.475;stroke-dashoffset:0;stroke-opacity:0.7;stroke-width:1.5;\"/>\n",
       "    </g>\n",
       "    <g id=\"line2d_16\"/>\n",
       "    <g id=\"text_14\">\n",
       "     <!-- Ground-state energy -->\n",
       "     <g transform=\"translate(274.220312 38.898437)scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path d=\"M 59.515625 10.40625 \n",
       "L 59.515625 29.984375 \n",
       "L 43.40625 29.984375 \n",
       "L 43.40625 38.09375 \n",
       "L 69.28125 38.09375 \n",
       "L 69.28125 6.78125 \n",
       "Q 63.578125 2.734375 56.6875 0.65625 \n",
       "Q 49.8125 -1.421875 42 -1.421875 \n",
       "Q 24.90625 -1.421875 15.25 8.5625 \n",
       "Q 5.609375 18.5625 5.609375 36.375 \n",
       "Q 5.609375 54.25 15.25 64.234375 \n",
       "Q 24.90625 74.21875 42 74.21875 \n",
       "Q 49.125 74.21875 55.546875 72.453125 \n",
       "Q 61.96875 70.703125 67.390625 67.28125 \n",
       "L 67.390625 56.78125 \n",
       "Q 61.921875 61.421875 55.765625 63.765625 \n",
       "Q 49.609375 66.109375 42.828125 66.109375 \n",
       "Q 29.4375 66.109375 22.71875 58.640625 \n",
       "Q 16.015625 51.171875 16.015625 36.375 \n",
       "Q 16.015625 21.625 22.71875 14.15625 \n",
       "Q 29.4375 6.6875 42.828125 6.6875 \n",
       "Q 48.046875 6.6875 52.140625 7.59375 \n",
       "Q 56.25 8.5 59.515625 10.40625 \n",
       "z\n",
       "\" id=\"DejaVuSans-71\"/>\n",
       "       <path d=\"M 45.40625 46.390625 \n",
       "L 45.40625 75.984375 \n",
       "L 54.390625 75.984375 \n",
       "L 54.390625 0 \n",
       "L 45.40625 0 \n",
       "L 45.40625 8.203125 \n",
       "Q 42.578125 3.328125 38.25 0.953125 \n",
       "Q 33.9375 -1.421875 27.875 -1.421875 \n",
       "Q 17.96875 -1.421875 11.734375 6.484375 \n",
       "Q 5.515625 14.40625 5.515625 27.296875 \n",
       "Q 5.515625 40.1875 11.734375 48.09375 \n",
       "Q 17.96875 56 27.875 56 \n",
       "Q 33.9375 56 38.25 53.625 \n",
       "Q 42.578125 51.265625 45.40625 46.390625 \n",
       "z\n",
       "M 14.796875 27.296875 \n",
       "Q 14.796875 17.390625 18.875 11.75 \n",
       "Q 22.953125 6.109375 30.078125 6.109375 \n",
       "Q 37.203125 6.109375 41.296875 11.75 \n",
       "Q 45.40625 17.390625 45.40625 27.296875 \n",
       "Q 45.40625 37.203125 41.296875 42.84375 \n",
       "Q 37.203125 48.484375 30.078125 48.484375 \n",
       "Q 22.953125 48.484375 18.875 42.84375 \n",
       "Q 14.796875 37.203125 14.796875 27.296875 \n",
       "z\n",
       "\" id=\"DejaVuSans-100\"/>\n",
       "       <path d=\"M 4.890625 31.390625 \n",
       "L 31.203125 31.390625 \n",
       "L 31.203125 23.390625 \n",
       "L 4.890625 23.390625 \n",
       "z\n",
       "\" id=\"DejaVuSans-45\"/>\n",
       "       <path d=\"M 44.28125 53.078125 \n",
       "L 44.28125 44.578125 \n",
       "Q 40.484375 46.53125 36.375 47.5 \n",
       "Q 32.28125 48.484375 27.875 48.484375 \n",
       "Q 21.1875 48.484375 17.84375 46.4375 \n",
       "Q 14.5 44.390625 14.5 40.28125 \n",
       "Q 14.5 37.15625 16.890625 35.375 \n",
       "Q 19.28125 33.59375 26.515625 31.984375 \n",
       "L 29.59375 31.296875 \n",
       "Q 39.15625 29.25 43.1875 25.515625 \n",
       "Q 47.21875 21.78125 47.21875 15.09375 \n",
       "Q 47.21875 7.46875 41.1875 3.015625 \n",
       "Q 35.15625 -1.421875 24.609375 -1.421875 \n",
       "Q 20.21875 -1.421875 15.453125 -0.5625 \n",
       "Q 10.6875 0.296875 5.421875 2 \n",
       "L 5.421875 11.28125 \n",
       "Q 10.40625 8.6875 15.234375 7.390625 \n",
       "Q 20.0625 6.109375 24.8125 6.109375 \n",
       "Q 31.15625 6.109375 34.5625 8.28125 \n",
       "Q 37.984375 10.453125 37.984375 14.40625 \n",
       "Q 37.984375 18.0625 35.515625 20.015625 \n",
       "Q 33.0625 21.96875 24.703125 23.78125 \n",
       "L 21.578125 24.515625 \n",
       "Q 13.234375 26.265625 9.515625 29.90625 \n",
       "Q 5.8125 33.546875 5.8125 39.890625 \n",
       "Q 5.8125 47.609375 11.28125 51.796875 \n",
       "Q 16.75 56 26.8125 56 \n",
       "Q 31.78125 56 36.171875 55.265625 \n",
       "Q 40.578125 54.546875 44.28125 53.078125 \n",
       "z\n",
       "\" id=\"DejaVuSans-115\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-71\"/>\n",
       "      <use x=\"77.490234\" xlink:href=\"#DejaVuSans-114\"/>\n",
       "      <use x=\"116.353516\" xlink:href=\"#DejaVuSans-111\"/>\n",
       "      <use x=\"177.535156\" xlink:href=\"#DejaVuSans-117\"/>\n",
       "      <use x=\"240.914062\" xlink:href=\"#DejaVuSans-110\"/>\n",
       "      <use x=\"304.292969\" xlink:href=\"#DejaVuSans-100\"/>\n",
       "      <use x=\"367.769531\" xlink:href=\"#DejaVuSans-45\"/>\n",
       "      <use x=\"403.853516\" xlink:href=\"#DejaVuSans-115\"/>\n",
       "      <use x=\"455.953125\" xlink:href=\"#DejaVuSans-116\"/>\n",
       "      <use x=\"495.162109\" xlink:href=\"#DejaVuSans-97\"/>\n",
       "      <use x=\"556.441406\" xlink:href=\"#DejaVuSans-116\"/>\n",
       "      <use x=\"595.650391\" xlink:href=\"#DejaVuSans-101\"/>\n",
       "      <use x=\"657.173828\" xlink:href=\"#DejaVuSans-32\"/>\n",
       "      <use x=\"688.960938\" xlink:href=\"#DejaVuSans-101\"/>\n",
       "      <use x=\"750.484375\" xlink:href=\"#DejaVuSans-110\"/>\n",
       "      <use x=\"813.863281\" xlink:href=\"#DejaVuSans-101\"/>\n",
       "      <use x=\"875.386719\" xlink:href=\"#DejaVuSans-114\"/>\n",
       "      <use x=\"914.75\" xlink:href=\"#DejaVuSans-103\"/>\n",
       "      <use x=\"978.226562\" xlink:href=\"#DejaVuSans-121\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pd4f85b6dad\">\n",
       "   <rect height=\"217.44\" width=\"334.8\" x=\"52.160938\" y=\"7.2\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "result = numpy.load('./output/summary_data.npz')\n",
    "\n",
    "eig_val, eig_state = numpy.linalg.eig(\n",
    "                     pauli_str_to_matrix(Hamiltonian, N))\n",
    "min_eig_H = numpy.min(eig_val.real)\n",
    "min_loss = numpy.ones([len(result['iter'])]) * min_eig_H\n",
    "\n",
    "plt.figure(1)\n",
    "func1, = plt.plot(result['iter'], result['energy'], \n",
    "                  alpha=0.7, marker='', linestyle=\"-\", color='r')\n",
    "func_min, = plt.plot(result['iter'], min_loss, \n",
    "                  alpha=0.7, marker='', linestyle=\":\", color='b')\n",
    "plt.xlabel('Number of iteration')\n",
    "plt.ylabel('Energy (Ha)')\n",
    "\n",
    "plt.legend(handles=[\n",
    "    func1,\n",
    "    func_min\n",
    "],\n",
    "    labels=[\n",
    "        r'$\\left\\langle {\\psi \\left( {\\theta } \\right)} '\n",
    "        r'\\right|H\\left| {\\psi \\left( {\\theta } \\right)} \\right\\rangle $',\n",
    "        'Ground-state energy',\n",
    "    ], loc='best')\n",
    "\n",
    "#plt.savefig(\"vqe.png\", bbox_inches='tight', dpi=300)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Determining the interatomic distance\n",
    "\n",
    "Recall the above calculation is done with an interatomic distance $d = 74$ pm between two hydrogen atoms. Another interesting aspect we can try with VQE is determining the true interatomic distance by modifying the `h2.xyz` file. The results are summarize in figure below,\n",
    "\n",
    "![VQE-fig-dist](figures/VQE-fig-distance.png)\n",
    "\n",
    "The lowest value is found around $d = 74$ pm (1 pm = $1\\times 10^{-12}$m), which is consistent with the [experimental data](https://cccbdb.nist.gov/exp2x.asp?casno=1333740&charge=0) $d_{exp} (H_2) = 74.14$ pm.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_______\n",
    "\n",
    "## References\n",
    "\n",
    "[1] [Cao, Yudong, et al. Quantum chemistry in the age of quantum computing. Chemical reviews 119.19 (2019): 10856-10915.](https://pubs.acs.org/doi/10.1021/acs.chemrev.8b00803)\n",
    "\n",
    "[2] [McArdle, Sam, et al. Quantum computational chemistry. Reviews of Modern Physics 92.1 (2020): 015003.](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.92.015003)\n",
    "\n",
    "\n",
    "[3] [Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).](https://www.nature.com/articles/ncomms5213)\n",
    "\n",
    "[4] [Moll, Nikolaj, et al. Quantum optimization using variational algorithms on near-term quantum devices. Quantum Science and Technology 3.3 (2018): 030503.](https://iopscience.iop.org/article/10.1088/2058-9565/aab822)\n",
    "\n",
    "[5] Helgaker, Trygve, Poul Jorgensen, and Jeppe Olsen. Molecular electronic-structure theory. John Wiley & Sons, 2014.\n",
    "\n",
    "[6] [Dirac, Paul Adrien Maurice. Quantum mechanics of many-electron systems. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 123.792 (1929): 714-733.](https://royalsocietypublishing.org/doi/10.1098/rspa.1929.0094)\n",
    "\n",
    "[7] Szabo, Attila, and Neil S. Ostlund. Modern quantum chemistry: introduction to advanced electronic structure theory. Courier Corporation, 2012.\n",
    "\n"
   ]
  }
 ],
 "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": "3.8.3"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {
    "height": "calc(100% - 180px)",
    "left": "10px",
    "top": "150px",
    "width": "426.667px"
   },
   "toc_section_display": true,
   "toc_window_display": true
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}