qaoa_vs_ed_tfim.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# In this note book we will analyze the transeverse field Ising model (TFIM) by using ED and QAOA\n",
    "\n",
    "The many-body spin Hamiltnonian in 1-d is given by:\n",
    "\n",
    "$$ \\mathcal{H}= -(1-t)\\sum_{i}\\sigma^z_i\\sigma^z_{i+1}-t\\sum_i \\sigma^x_i $$\n",
    "\n",
    "This is an exactly solvable model, the sollution can be found in the second chapter of <a href=\"https://arxiv.org/pdf/0904.2771.pdf\">this lecture note</a>.\n",
    "\n",
    "For the limit $t=0$, this has a ferromagnetic (FM) ground state (GS) with two fold degeneracy, which spontaneously breaks the global $Z_2$ symmetry of the Hamiltonian. In the limit $t=1$, the GS is unique (called paramagetic, i.e. PM) and doesn't break any symmetry.\n",
    "\n",
    "In this notebook we will do the following things:\n",
    "\n",
    "* <a href='#ed'>Solve it using ED</a>\n",
    "* <a href='#qaoa'>Solve it using QAOA</a>\n",
    "* <a href='#compare'>Comapre the results</a>\n",
    "* <a href='#expect_value'>Magnetization and the correlations.</a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exact diagonalization <a id='ed'></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# basic python operations\n",
    "import numpy as np\n",
    "\n",
    "# scientific coputing\n",
    "from functools import reduce\n",
    "import scipy.sparse as sps\n",
    "import scipy.sparse.linalg as spslin\n",
    "from scipy.optimize import curve_fit\n",
    "\n",
    "# visualization\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Loading the Pauli matrices\n",
    "\n",
    "Id = sps.eye(2)\n",
    "Sx = sps.csr_matrix(np.array([[0,1.+0.j],[1+0.j,0]]))\n",
    "Sy = sps.csr_matrix(np.array([[0,-1.j],[1.j,0]]))\n",
    "Sz = sps.csr_matrix(np.array([[1+0.j,0.],[0,-1+0.j]]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Building the many-body Hamiltonian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def ham_general(L,static):\n",
    "    ham_gen=np.zeros(shape=(2**L,2**L),dtype=np.complex)\n",
    "    for item in static:\n",
    "        string=item[0]\n",
    "        for inside in item[1:len(item)]:\n",
    "            for j in range(len(inside)):\n",
    "                op_list=[Id]*L\n",
    "                for k in range(len(string)):\n",
    "                    if string[k]=='x':\n",
    "                        op_list[inside[j][k+1]]=Sx\n",
    "                    elif string[k]=='y':\n",
    "                        op_list[inside[j][k+1]]=Sy\n",
    "                    elif string[k]=='z':\n",
    "                        op_list[inside[j][k+1]]=Sz\n",
    "                #print(reduce(sps.kron,op_list).toarray())\n",
    "                ham_gen += inside[j][0]*reduce(sps.kron,op_list).toarray()\n",
    "    #            \n",
    "    return ham_gen"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### System size (SS): L=6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "L=6\n",
    "t_arr=np.linspace(0,1,21)\n",
    "evalu=8\n",
    "\n",
    "energy_ed=np.ndarray(shape=(len(t_arr),evalu))\n",
    "gs_vec=np.ndarray(shape=(2**L,len(t_arr)),dtype=np.complex)\n",
    "\n",
    "for j in range(len(t_arr)):\n",
    "    ## defining the parameter\n",
    "    g=t_arr[j]\n",
    "    J=(1-t_arr[j])\n",
    "    ## making the adjacent matrix\n",
    "    x_field=[[-g,i] for i in range(L)]\n",
    "    J_nn=[[-J,i,(i+1)%L] for i in range(L)]\n",
    "    static=[[\"zz\",J_nn],[\"x\",x_field]]\n",
    "    ## diagonalization of the Hamiltonian\n",
    "    hamiltonian=ham_general(L,static)\n",
    "    (e2,u) = spslin.eigsh(hamiltonian, k=evalu, which='SA',return_eigenvectors=True)\n",
    "    energy_ed[j,:]=np.sort(e2)\n",
    "    gs_vec[:,j]=u[:,0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solving the problem with QAOA <a id='qaoa'></a>\n",
    "\n",
    "QAOA method is a <a href='https://arxiv.org/abs/1411.4028'>variational method</a>, which is in turn inspired by the <a href='https://arxiv.org/abs/quant-ph/0001106'>adiabatic quantum computation</a>. We alternatively apply rotation with respect to the Hamiltonoan (any) and a mixer term (uniform x-field) with the angles $\\gamma$ and $\\beta$ respectively. After computing the energy of the final state at the end of the rotation, using the Hamiltonian, we get \n",
    "\n",
    "$$F(\\overrightarrow{\\beta},\\overrightarrow{\\gamma})\\equiv<\\psi{(\\overrightarrow{\\beta},\\overrightarrow{\\gamma})}|\\mathcal{H}|\\psi{(\\overrightarrow{\\beta},\\overrightarrow{\\gamma})}>$$\n",
    "\n",
    "With $|\\psi{(\\overrightarrow{\\beta},\\overrightarrow{\\gamma})}> =  \\left(e^{-i\\beta_p B}e^{-i\\gamma_p H}\\right)\\cdots \\left(e^{-i\\beta_1 B}e^{-i\\gamma_1 C}\\right) H^{\\otimes n} |0>$. Next we optimize the rotation angle so that the energy of the final state is minimized, and we get a optimal set of $(\\overrightarrow{\\beta}^*,\\overrightarrow{\\gamma}^*)$, and the supposedly GS of the system. Since the optimization is done in a classical computer, while the rotation is performed in quantum computer, this is called hybrid solver, which is claimed to be useful for the current noisy quantum computers available today. Thus the current era of quantum computing is called the era of <a href='https://quantum-journal.org/papers/q-2018-08-06-79/'>noisy intermediate-scale quantum (NISQ)</a> computing.\n",
    "\n",
    "The claim is if we increase this alternate layer, we get more closer to the GS.\n",
    "\n",
    "This is the full circuit, with we will now choose, $p=1~;~\\beta=\\text{x-field angle} (B);~\\gamma=\\text{Hamiltonian angle}(\\mathcal{H})$\n",
    "\n",
    "\n",
    "<img src=\"./qaoa.jpeg\" width=\"700\" height=\"700\">\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'qiskit-terra': '0.17.1', 'qiskit-aer': '0.8.2', 'qiskit-ignis': '0.6.0', 'qiskit-ibmq-provider': '0.12.3', 'qiskit-aqua': '0.9.1', 'qiskit': '0.25.3', 'qiskit-nature': None, 'qiskit-finance': None, 'qiskit-optimization': None, 'qiskit-machine-learning': None}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import qiskit\n",
    "qiskit.__qiskit_version__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# to set up the problem\n",
    "import networkx as nx\n",
    "\n",
    "# for classical optimization\n",
    "from scipy.optimize import minimize\n",
    "\n",
    "# for quantum computataion\n",
    "from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister, execute, Aer\n",
    "from qiskit.visualization import plot_histogram"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Creating the graph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "G = nx.Graph()\n",
    "G.add_nodes_from([i for i in range(L)])\n",
    "G.add_edges_from([[i,i+1] for i in range(L-1)])\n",
    "G.add_edges_from([[L-1,0]])\n",
    "options = {\n",
    "    'node_color': 'y',\n",
    "    'node_size': 1000,\n",
    "    'width': 3,\n",
    "}\n",
    "nx.draw(G, with_labels=True,pos=nx.circular_layout(G),**options)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[(0, 1), (0, 5), (1, 2), (2, 3), (3, 4), (4, 5)]\n",
      "6\n"
     ]
    }
   ],
   "source": [
    "print(list(G.edges()))\n",
    "print(G.number_of_nodes())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Defining the rotation circuit corresponding to the Hamiltonian\n",
    "\n",
    "Bellow we define the rotation Hamiltonian in a quantum circuit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def ham_rot(G,gamma):\n",
    "    L=G.number_of_nodes()\n",
    "    qc=QuantumCircuit(L,L)\n",
    "    for i,j in list(G.edges()):\n",
    "        qc.cx(i,j)\n",
    "        qc.rz(2*gamma,j)\n",
    "        qc.cx(i,j)\n",
    "    for n in G.nodes():\n",
    "        qc.rx(2*gamma, n)\n",
    "    return qc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[(0, 1), (0, 5), (1, 2), (2, 3), (3, 4), (4, 5)]\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: ──■────────────────■────■────────────────────────────■──┤ RX(2π/3) ├»\n",
       "     ┌─┴─┐┌──────────┐┌─┴─┐  │                            │  └──────────┘»\n",
       "q_1: ┤ X ├┤ RZ(2π/3) ├┤ X ├──┼───────■────────────────────┼───────■──────»\n",
       "     └───┘└──────────┘└───┘  │     ┌─┴─┐    ┌──────────┐  │     ┌─┴─┐    »\n",
       "q_2: ────────────────────────┼─────┤ X ├────┤ RZ(2π/3) ├──┼─────┤ X ├────»\n",
       "                             │     └───┘    └──────────┘  │     └───┘    »\n",
       "q_3: ────────────────────────┼────────────────────────────┼──────────────»\n",
       "                             │                            │              »\n",
       "q_4: ────────────────────────┼────────────────────────────┼──────────────»\n",
       "                           ┌─┴─┐┌──────────┐            ┌─┴─┐            »\n",
       "q_5: ──────────────────────┤ X ├┤ RZ(2π/3) ├────────────┤ X ├────────────»\n",
       "                           └───┘└──────────┘            └───┘            »\n",
       "c: 6/════════════════════════════════════════════════════════════════════»\n",
       "                                                                         »\n",
       "«                                                                           »\n",
       "«q_0: ──────────────────────────────────────────────────────────────────────»\n",
       "«     ┌──────────┐                                                          »\n",
       "«q_1: ┤ RX(2π/3) ├──────────────────────────────────────────────────────────»\n",
       "«     └──────────┘                 ┌──────────┐                             »\n",
       "«q_2: ─────■────────────────────■──┤ RX(2π/3) ├─────────────────────────────»\n",
       "«        ┌─┴─┐    ┌──────────┐┌─┴─┐└──────────┘                 ┌──────────┐»\n",
       "«q_3: ───┤ X ├────┤ RZ(2π/3) ├┤ X ├─────■────────────────────■──┤ RX(2π/3) ├»\n",
       "«        └───┘    └──────────┘└───┘   ┌─┴─┐    ┌──────────┐┌─┴─┐└──────────┘»\n",
       "«q_4: ────────────────────────────────┤ X ├────┤ RZ(2π/3) ├┤ X ├─────■──────»\n",
       "«                                     └───┘    └──────────┘└───┘   ┌─┴─┐    »\n",
       "«q_5: ─────────────────────────────────────────────────────────────┤ X ├────»\n",
       "«                                                                  └───┘    »\n",
       "«c: 6/══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                           »\n",
       "«                                  \n",
       "«q_0: ─────────────────────────────\n",
       "«                                  \n",
       "«q_1: ─────────────────────────────\n",
       "«                                  \n",
       "«q_2: ─────────────────────────────\n",
       "«                                  \n",
       "«q_3: ─────────────────────────────\n",
       "«                      ┌──────────┐\n",
       "«q_4: ──────────────■──┤ RX(2π/3) ├\n",
       "«     ┌──────────┐┌─┴─┐├──────────┤\n",
       "«q_5: ┤ RZ(2π/3) ├┤ X ├┤ RX(2π/3) ├\n",
       "«     └──────────┘└───┘└──────────┘\n",
       "«c: 6/═════════════════════════════\n",
       "«                                  </pre>"
      ],
      "text/plain": [
       "                                                             ┌──────────┐»\n",
       "q_0: ──■────────────────■────■────────────────────────────■──┤ RX(2π/3) ├»\n",
       "     ┌─┴─┐┌──────────┐┌─┴─┐  │                            │  └──────────┘»\n",
       "q_1: ┤ X ├┤ RZ(2π/3) ├┤ X ├──┼───────■────────────────────┼───────■──────»\n",
       "     └───┘└──────────┘└───┘  │     ┌─┴─┐    ┌──────────┐  │     ┌─┴─┐    »\n",
       "q_2: ────────────────────────┼─────┤ X ├────┤ RZ(2π/3) ├──┼─────┤ X ├────»\n",
       "                             │     └───┘    └──────────┘  │     └───┘    »\n",
       "q_3: ────────────────────────┼────────────────────────────┼──────────────»\n",
       "                             │                            │              »\n",
       "q_4: ────────────────────────┼────────────────────────────┼──────────────»\n",
       "                           ┌─┴─┐┌──────────┐            ┌─┴─┐            »\n",
       "q_5: ──────────────────────┤ X ├┤ RZ(2π/3) ├────────────┤ X ├────────────»\n",
       "                           └───┘└──────────┘            └───┘            »\n",
       "c: 6/════════════════════════════════════════════════════════════════════»\n",
       "                                                                         »\n",
       "«                                                                           »\n",
       "«q_0: ──────────────────────────────────────────────────────────────────────»\n",
       "«     ┌──────────┐                                                          »\n",
       "«q_1: ┤ RX(2π/3) ├──────────────────────────────────────────────────────────»\n",
       "«     └──────────┘                 ┌──────────┐                             »\n",
       "«q_2: ─────■────────────────────■──┤ RX(2π/3) ├─────────────────────────────»\n",
       "«        ┌─┴─┐    ┌──────────┐┌─┴─┐└──────────┘                 ┌──────────┐»\n",
       "«q_3: ───┤ X ├────┤ RZ(2π/3) ├┤ X ├─────■────────────────────■──┤ RX(2π/3) ├»\n",
       "«        └───┘    └──────────┘└───┘   ┌─┴─┐    ┌──────────┐┌─┴─┐└──────────┘»\n",
       "«q_4: ────────────────────────────────┤ X ├────┤ RZ(2π/3) ├┤ X ├─────■──────»\n",
       "«                                     └───┘    └──────────┘└───┘   ┌─┴─┐    »\n",
       "«q_5: ─────────────────────────────────────────────────────────────┤ X ├────»\n",
       "«                                                                  └───┘    »\n",
       "«c: 6/══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                           »\n",
       "«                                  \n",
       "«q_0: ─────────────────────────────\n",
       "«                                  \n",
       "«q_1: ─────────────────────────────\n",
       "«                                  \n",
       "«q_2: ─────────────────────────────\n",
       "«                                  \n",
       "«q_3: ─────────────────────────────\n",
       "«                      ┌──────────┐\n",
       "«q_4: ──────────────■──┤ RX(2π/3) ├\n",
       "«     ┌──────────┐┌─┴─┐├──────────┤\n",
       "«q_5: ┤ RZ(2π/3) ├┤ X ├┤ RX(2π/3) ├\n",
       "«     └──────────┘└───┘└──────────┘\n",
       "«c: 6/═════════════════════════════\n",
       "«                                  "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qc = ham_rot(G, np.pi/3)\n",
    "print(G.edges())\n",
    "qc.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def x_field_rot(G, beta):\n",
    "    N = G.number_of_nodes()\n",
    "    qc = QuantumCircuit(N,N)\n",
    "    for n in G.nodes():\n",
    "        qc.rx(2*beta, n)\n",
    "    return qc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "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: ┤ RX(2π/3) ├\n",
       "     ├──────────┤\n",
       "q_1: ┤ RX(2π/3) ├\n",
       "     ├──────────┤\n",
       "q_2: ┤ RX(2π/3) ├\n",
       "     ├──────────┤\n",
       "q_3: ┤ RX(2π/3) ├\n",
       "     ├──────────┤\n",
       "q_4: ┤ RX(2π/3) ├\n",
       "     ├──────────┤\n",
       "q_5: ┤ RX(2π/3) ├\n",
       "     └──────────┘\n",
       "c: 6/════════════\n",
       "                 </pre>"
      ],
      "text/plain": [
       "     ┌──────────┐\n",
       "q_0: ┤ RX(2π/3) ├\n",
       "     ├──────────┤\n",
       "q_1: ┤ RX(2π/3) ├\n",
       "     ├──────────┤\n",
       "q_2: ┤ RX(2π/3) ├\n",
       "     ├──────────┤\n",
       "q_3: ┤ RX(2π/3) ├\n",
       "     ├──────────┤\n",
       "q_4: ┤ RX(2π/3) ├\n",
       "     ├──────────┤\n",
       "q_5: ┤ RX(2π/3) ├\n",
       "     └──────────┘\n",
       "c: 6/════════════\n",
       "                 "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qc = x_field_rot(G, np.pi/3)\n",
    "qc.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Defining the QAOA circuit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_qaoa_circuit_sv(G, beta, gamma):\n",
    "    assert(len(beta) == len(gamma))\n",
    "    p = len(beta) # infering number of QAOA steps from the parameters passed\n",
    "    N = G.number_of_nodes()\n",
    "    qc = QuantumCircuit(N,N)\n",
    "    # first, apply a layer of Hadamards\n",
    "    qc.h(range(N))\n",
    "    # second, apply p alternating operators\n",
    "    for k in range(p):\n",
    "        qc=qc+ham_rot(G,gamma[k])\n",
    "        qc=qc+x_field_rot(G,beta[k])\n",
    "    # no measurement in the end!\n",
    "    return qc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compute_ising_energy_sv(sv_old,hamilton):\n",
    "    \"\"\"Compute objective from statevector\n",
    "    For large number of qubits, this is slow. \n",
    "    \"\"\"\n",
    "    sv=np.flip(sv_old)\n",
    "    #sv=np.array(sv_old)\n",
    "    adjsv=np.conjugate(sv)\n",
    "    mult=np.matmul(hamilton, sv)\n",
    "    ener=np.matmul(adjsv,mult)\n",
    "    return ener.real"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_black_box_objective_sv(G, p, hamilton, energy_sv):\n",
    "    backend = Aer.get_backend('statevector_simulator')\n",
    "    def f(theta):\n",
    "        # let's assume first half is betas, second half is gammas\n",
    "        beta = theta[:p]\n",
    "        gamma = theta[p:]\n",
    "        qc = get_qaoa_circuit_sv(G,beta, gamma)\n",
    "        sv = execute(qc, backend).result().get_statevector()  # the sv is np.array with complex entries\n",
    "        # return the energy\n",
    "        ener=compute_ising_energy_sv(sv, hamilton)\n",
    "        energy_sv.append(ener)\n",
    "        return ener, energy_sv\n",
    "    return f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/animeshnanda/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:10: DeprecationWarning: The QuantumCircuit.__add__() method is being deprecated.Use the compose() method which is more flexible w.r.t circuit register compatibility.\n",
      "  # Remove the CWD from sys.path while we load stuff.\n",
      "/Users/animeshnanda/anaconda3/lib/python3.7/site-packages/qiskit/circuit/quantumcircuit.py:869: DeprecationWarning: The QuantumCircuit.combine() method is being deprecated. Use the compose() method which is more flexible w.r.t circuit register compatibility.\n",
      "  return self.combine(rhs)\n"
     ]
    }
   ],
   "source": [
    "# we are choosing p number of layers\n",
    "\n",
    "p=8\n",
    "angle_arr=np.ndarray(shape=(len(t_arr),2*p))\n",
    "gs_energy_qaoa=np.ndarray(shape=(len(t_arr),))\n",
    "n_iter=np.ndarray(shape=(len(t_arr),))\n",
    "\n",
    "for j in range(len(t_arr)):\n",
    "    energy_sv=[]\n",
    "    np.random.seed(123)\n",
    "    init_point = np.random.rand(2*p)\n",
    "    # define Hamiltonian\n",
    "    g=t_arr[j]\n",
    "    J=(1-t_arr[j])\n",
    "    ## making the adjacent matrix\n",
    "    x_field=[[-g,i] for i in range(L)]\n",
    "    J_nn=[[-J,i,(i+1)%L] for i in range(L)]\n",
    "    static=[[\"zz\",J_nn],[\"x\",x_field]]\n",
    "    hamiltonian=ham_general(L,static)\n",
    "    obj = get_black_box_objective_sv(G, p, hamiltonian,energy_sv)\n",
    "    # We are going to limit the number of iterations to 2500\n",
    "    res_sv = minimize(obj, init_point, method='COBYLA', options={'maxiter':2500, 'disp': False})\n",
    "    angle_arr[j,:]=res_sv['x']\n",
    "    gs_energy_qaoa[j]=res_sv['fun']\n",
    "    n_iter[j]=res_sv['nfev']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-5.99999749, -5.70393983, -5.41667129, -5.13978302, -4.87534144,\n",
       "        -4.62612836, -4.39616219, -4.19162725, -4.02277284, -3.90631713,\n",
       "        -3.86370077, -3.90630706, -4.02277377, -4.19162994, -4.39615888,\n",
       "        -4.62613143, -4.8753406 , -5.13978866, -5.41665853, -5.70394774,\n",
       "        -5.9999978 ]),\n",
       " array([496., 553., 523., 572., 544., 587., 312., 519., 542., 595., 689.,\n",
       "        738., 959., 638., 432., 508., 372., 351., 423., 288., 372.]))"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gs_energy_qaoa,n_iter"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Comparing the results <a id='compare'></a>\n",
    "\n",
    "We will now compare the results from the two methods, by looking at the GS energy and the overlap between the GS obtained in different methods.\n",
    "\n",
    "Run the circuit for one more time for each values of $t$ with the optimal theta."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "new_sv=np.ndarray(shape=(2**L,len(t_arr)),dtype=np.complex)\n",
    "\n",
    "for j in range(len(t_arr)):\n",
    "    optimal_theta = angle_arr[j,:]\n",
    "    qc = get_qaoa_circuit_sv(G, optimal_theta[:p], optimal_theta[p:])\n",
    "    backend = Aer.get_backend('statevector_simulator')\n",
    "    new_sv[:,j] = execute(qc, backend).result().get_statevector()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ms,fs=8,12\n",
    "\n",
    "plt.figure(figsize=(6,5))\n",
    "# plt.ylim(-0.2,4.2)\n",
    "\n",
    "plt.plot(t_arr,energy_ed[:,0],'cs-',markersize=ms+2,label='GS energy ED')\n",
    "plt.plot(t_arr,gs_energy_qaoa,'k*',markersize=ms,label='GS energy QAOA')\n",
    "\n",
    "plt.xlabel('$t$',fontsize=fs)\n",
    "plt.ylabel('GS energy',fontsize=fs)\n",
    "plt.xticks(fontsize=fs)\n",
    "plt.yticks(fontsize=fs)\n",
    "plt.legend(loc='best',fontsize=fs)\n",
    "plt.grid()\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fidelity=np.ndarray(shape=(len(t_arr),))\n",
    "\n",
    "for j in range(len(t_arr)):\n",
    "    fidelity[j]=abs(np.matmul(np.conjugate(new_sv[:,j]),gs_vec[:,j]))\n",
    "    \n",
    "# plotting the data\n",
    "ms,fs=6,12\n",
    "\n",
    "plt.figure(figsize=(6,5))\n",
    "\n",
    "plt.plot(t_arr,fidelity,'b*-',markersize=ms,label='Inner porduct')\n",
    "\n",
    "plt.xlabel('$t$',fontsize=fs)\n",
    "plt.ylabel('Inner product',fontsize=fs)\n",
    "plt.xticks(fontsize=fs)\n",
    "plt.yticks(fontsize=fs)\n",
    "plt.legend(loc='best',fontsize=fs)\n",
    "plt.grid()\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that although the ground state enrgy computed using both the methods mathes pretty well, however the fidelity or the inner product between the GS vectors sometimes are orthogonal to each other, while matching perfectly most of the time.\n",
    "\n",
    "This disagreement can be attributed to reasons \n",
    "\n",
    "* First, it could be due to <a href='https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.18.1049'>Anderson's orthogonalty catastrophe</a>, which says that in the increasing system size, the GS of two Hamiltonians with a arbitraryly small perturbation decreases with the increasing system size, and becomes zero in the infinite system size.\n",
    "\n",
    "* Second, we know that in the FM case there are two fold degenerate GS which are orthogonal to each other. It could well be the two different GS are chosen by two methods, that's why we are getting a zero overlap.\n",
    "\n",
    "Next we will look at the magnetization and the $\\sigma_z$ correlation function as distance to characterize the phases.\n",
    "\n",
    "# Calculating expectation values <a id='expect_value'></a>\n",
    "\n",
    "### Magnetization calculated for both statevector\n",
    "\n",
    "Magnetization is defined as:\n",
    "\n",
    "$$ m(x)=\\frac{1}{L}\\sum_{i}\\langle\\sigma^x_i \\rangle $$, where $\\langle.\\rangle$ denote the ground state expectation value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "# definging the magnetization operator\n",
    "\n",
    "x_field=[[1./L,i] for i in range(L)]\n",
    "static=[[\"x\",x_field]]\n",
    "hamiltonian=ham_general(L,static)\n",
    "\n",
    "magnet_ed=np.ndarray(shape=(len(t_arr),))\n",
    "magnet_qaoa=np.ndarray(shape=(len(t_arr),))\n",
    "\n",
    "for j in range(len(t_arr)):\n",
    "    vec_ed = np.array(gs_vec[:,j])\n",
    "    vec_qaoa = np.array(new_sv[:,j])\n",
    "    magnet_ed[j] = np.matmul(np.conjugate(vec_ed),np.matmul(hamiltonian,vec_ed)).real\n",
    "    magnet_qaoa[j] = np.matmul(np.conjugate(vec_qaoa),np.matmul(hamiltonian,vec_qaoa)).real"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ms,fs=8,12\n",
    "\n",
    "plt.figure(figsize=(6,5))\n",
    "# plt.ylim(-0.2,4.2)\n",
    "\n",
    "plt.plot(t_arr,magnet_ed,'cs-',markersize=ms+2,label='x-magnetization ED')\n",
    "plt.plot(t_arr,magnet_qaoa,'k*',markersize=ms,label='x-magnetization QAOA')\n",
    "\n",
    "plt.xlabel('$t$',fontsize=fs)\n",
    "plt.ylabel('GS energy',fontsize=fs)\n",
    "plt.xticks(fontsize=fs)\n",
    "plt.yticks(fontsize=fs)\n",
    "plt.legend(loc='best',fontsize=fs)\n",
    "plt.grid()\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## $\\langle\\sigma^z\\sigma^z\\rangle $ Correlation function\n",
    "\n",
    "The correlator function is defined as \n",
    "\n",
    "$$ C(r)=\\frac{1}{L}\\langle \\sum_{i}\\sigma^z_{i}\\sigma^z_{i+r}\\rangle $$\n",
    "\n",
    "Sometime, the self correlator is subtracted, thus the $C(r)$ becomes a connected correlator, but here we will use the above definition."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "t_arr_new=t_arr[0:len(t_arr):2]\n",
    "corr_ed=np.ndarray(shape=(len(t_arr_new), len(range(0,L+1))))\n",
    "corr_qaoa=np.ndarray(shape=(len(t_arr_new), len(range(0,L+1))))\n",
    "\n",
    "# setting the self correlator as 1\n",
    "corr_ed[:,0]=1.\n",
    "corr_qaoa[:,0]=1.\n",
    "#\n",
    "corr_ed[:,-1]=1.\n",
    "corr_qaoa[:,-1]=1.\n",
    "\n",
    "s=np.arange(L)\n",
    "\n",
    "for k in range(1,L):\n",
    "    Txk=(s+k)%L\n",
    "    zz_cor=[[1./L,i,Txk[i]] for i in range(L)]\n",
    "    static=[[\"zz\",zz_cor]]\n",
    "    hamiltonian=ham_general(L,static)\n",
    "    for j in range(len(t_arr_new)):\n",
    "        vec_ed = np.array(gs_vec[:,j*2])\n",
    "        vec_qaoa = np.array(new_sv[:,j*2])\n",
    "        corr_ed[j,k] = np.matmul(np.conjugate(vec_ed),np.matmul(hamiltonian,vec_ed)).real\n",
    "        corr_qaoa[j,k] = np.matmul(np.conjugate(vec_qaoa),np.matmul(hamiltonian,vec_qaoa)).real"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "r=[i for i in range(L+1)]\n",
    "\n",
    "ms,fs=8,12\n",
    "\n",
    "plt.figure(figsize=(6,5))\n",
    "plt.xlim(-2.1,6.2)\n",
    "\n",
    "color1=['c','y','m','b','g']\n",
    "color2=['k','m','g','y','r']\n",
    "\n",
    "for j in range(0,10,2):\n",
    "    plt.plot(r,corr_ed[j,:],'%ss-'%(color1[j//2]),markersize=ms+2,label='ED t=%.2f'%(t_arr_new[j]))\n",
    "    \n",
    "for j in range(0,10,2):\n",
    "    plt.plot(r,corr_ed[j,:],'%s*'%(color2[j//2]),markersize=ms,label='QAOA t=%.2f'%(t_arr_new[j]))\n",
    "\n",
    "plt.xlabel('r',fontsize=fs)\n",
    "plt.ylabel('C(r)',fontsize=fs)\n",
    "plt.xticks(fontsize=fs)\n",
    "plt.yticks(fontsize=fs)\n",
    "plt.legend(loc='best',fontsize=fs)\n",
    "plt.grid()\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To conclude, we see that the $\\sigma^x$-magnetization and the $\\sigma^z\\sigma^z$-correlation are in a very good egreement for the two different methods, although there was a initial disagreement in the overlap in the GS vector. Using this result we can say confidently, QAOA is powerfull algorithm to solve complex problems in the next generations of the quantum computers.\n",
    "\n",
    "In passing we should note that, ED is a very bad method, since it is severely limited by the system size. There are many more methods to slove a quantum system in a classical computers, such as tensor netwok or quantum monte carlo, using which we can increase the system size 10-20 times of the ED. But the QAOA method can be usefull when we get a big enough quantum computer.\n",
    "\n",
    "One immediate drawback of the QAOA method is, it can't calculate the excited states (may be it can, but I am not aware of it). Another interesting question: is there a relation between the no of qubits and the circuit depth (`p`), we will explore these questions later\n",
    "\n",
    "${\\color{red}{\\bf [total~runtime: 34 mins]}}$"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.6"
  }
 },
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