BraketTutorial.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Braket module in Tequila"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hi! This is a tutorial to get to know the new Tequila's function: `braket`. With this new function you will be able to easily compute:\n",
    "\n",
    "1. self overlaps;\n",
    "2. overlaps between two different states;\n",
    "3. the expectation value of an operator;\n",
    "4. the transition element of an operator."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, let's import Tequila (and some of its modules) and see how this function works."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "#tequila importations\n",
    "import tequila as tq\n",
    "from tequila.circuit.gates import H, X, Rx, CX\n",
    "from tequila.hamiltonian.qubit_hamiltonian import QubitHamiltonian\n",
    "\n",
    "#extra importations\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Basic usage"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. Self overlap"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's begin from the simplest (and maybe trivial) possible application of the `braket` function: the computation of the self overlap of a quantum state. One can prepare an aritrary circuit, corresponding to the state $|\\psi\\rangle$ and then calculate the self overlap: $\\langle \\psi|\\psi\\rangle$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can prepare for example a Bell state of two qubits:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "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: ┤ H ├──■──\n",
       "     └───┘┌─┴─┐\n",
       "q_1: ─────┤ X ├\n",
       "          └───┘\n",
       "c: 2/══════════\n",
       "               </pre>"
      ],
      "text/plain": [
       "     ┌───┐     \n",
       "q_0: ┤ H ├──■──\n",
       "     └───┘┌─┴─┐\n",
       "q_1: ─────┤ X ├\n",
       "          └───┘\n",
       "c: 2/══════════\n",
       "               "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bell = tq.QCircuit()\n",
    "bell += H(0)\n",
    "bell += CX(0,1)\n",
    "\n",
    "tq.draw(bell, backend='qiskit')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then, we can simply call the `braket` function passing as unique argument our quantum circuit, this can also be specified as `ket`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "self_overlap_real, self_overlap_im = tq.braket(ket=bell)\n",
    "#joining real and imaginary part\n",
    "self_overlap = self_overlap_real + 1.0j*self_overlap_im"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we have seen the braket function always return two objects, one for the real part and one for the imaginary one. These are `tq.Objecitve` and need to be simulated in order to obtain a numeric value.\n",
    "\n",
    "Of course, from the laws of quantum mechanics it follows that the self overlap must always be equal to 1, namely $\\langle \\psi|\\psi\\rangle=1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "self_overlap_value = tq.simulate(self_overlap)\n",
    "self_overlap_value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this particular case Tequila is aware that the self overlap is always 1, so no circuit needs to be executed. This can be checked by printing the `self_overlap` Objective:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Objective with 0 unique expectation values\n",
      "total measurements = 0\n",
      "variables          = []\n",
      "types              = []\n"
     ]
    }
   ],
   "source": [
    "print(self_overlap)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. Overlap between two different states"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The second step will be to compute the overlap between different states $\\langle\\psi _2|\\psi _1\\rangle$. Even in this case we can exploit the `braket` function.\n",
    "\n",
    "Let's prepare two orthogonal states. In this case the overlap must be equal to 0:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "U0 = X(0) # state |10>\n",
    "U1 = H(0)+CX(0,1) # Bell state 1/sqrt(2)*(|00>+|11>)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can call the `braket` function passing as arguments our quantum circuits, specifying which will be the `ket` and which the `bra`.\n",
    "Otherwise the first will be automatically selected as `ket` and second as `bra`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "br_objective_real, br_objective_im = tq.braket(ket=U0, bra=U1)\n",
    "\n",
    "#joining real and imaginary part\n",
    "br_overlap = br_objective_real + 1.0j*br_objective_im\n",
    "# let's simulate the Objective\n",
    "br_overlap_value = tq.simulate(br_overlap)\n",
    "\n",
    "br_overlap_value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also define a non-orthogonal state with respect to the previous prepared Bell state. This can be obtained for example by preparing a Bell state followed by a rotation around X-axis. \n",
    "\n",
    "One can then check that changing the angle value will result in changing the overlap between the two states:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Overlap')"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 576x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "angle = tq.Variable(\"angle\")\n",
    "U2 = U1 + Rx(angle*np.pi, 1) # Bell state followed by a rotation around X-axis\n",
    "\n",
    "br_objective_real, br_objective_im = tq.braket(ket=U1, bra=U2)\n",
    "br_overlap = br_objective_real + 1.0j*br_objective_im\n",
    "S = tq.compile(br_overlap) # translate to backend\n",
    "\n",
    "overlaps = []\n",
    "for value in np.arange(0,4.1, 0.1): \n",
    "# set the value we want to simulate\n",
    "    variables = {\"angle\" : value}\n",
    "    # let's simulate the Objective\n",
    "    br_overlap_value = S(variables=variables)\n",
    "    overlaps.append(round(br_overlap_value,3))\n",
    "\n",
    "#plotting the trend\n",
    "fig = plt.figure(figsize=(8,5))\n",
    "\n",
    "plt.plot(np.arange(0,4.1, 0.1), overlaps, marker = 'o')\n",
    "plt.xlabel(r'Angle[Units of $\\pi$]')\n",
    "plt.ylabel('Overlap')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. Expectation value of an Hamiltonian"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "At this stage, we can see how to use the `braket` function to calculate the expectation value of an Hamiltonian $H$ with respect to a given state $|\\psi\\rangle$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "# state definition\n",
    "psi = H(0)+CX(0,1) # Bell state 1/sqrt(2)*(|00>+|11>)\n",
    "\n",
    "#Hamiltonian definition\n",
    "operator = QubitHamiltonian.from_string(\"-1.0*X(0)X(1)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, one needs to pass to the `braket` function as `ket` argument our state and as `operator` argument the Hamiltonian.\n",
    "As already seen before, always keep in mind that the `braket` function returns two Objecitves that need to be compiled afterwards. \n",
    "\n",
    "With previous defined state and Hamiltonian we need to obtain an expectation value equal to $-1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.0"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#applying braket function\n",
    "br_exp_value_real, br_exp_value_im = tq.braket(ket=psi, operator=operator)\n",
    "#joining real and imaginary part\n",
    "br_exp_value_tmp = br_exp_value_real + 1.0j*br_exp_value_im\n",
    "#simulate the expectation value\n",
    "br_exp_value = tq.simulate(br_exp_value_tmp)\n",
    "br_exp_value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tequila is aware that an `ExpectationValue` is always real, so for the imaginary part no circuit needs to be executed. One can see this by printting `br_exp_value_im`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Objective with 0 unique expectation values\n",
      "total measurements = 0\n",
      "variables          = []\n",
      "types              = []\n"
     ]
    }
   ],
   "source": [
    "print(br_exp_value_im)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4. Transition element of an Hamiltonian"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As last application, the `braket` function can be used to calculate transition elements of an operator $H$ with respect to two different states $|\\alpha\\rangle, \\ |\\beta\\rangle$: $\\langle\\beta|H|\\alpha\\rangle$.\n",
    "\n",
    "In order to see how things can change, we can procede similarly to section 2 and vary the angle of a rotation:  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "#states definition\n",
    "alpha = H(0)+CX(0,1) # Bell state 1/sqrt(2)*(|00>+|11>)\n",
    "angle = tq.Variable(\"angle\")\n",
    "beta = alpha + Rx(angle*np.pi, 1) # Bell state followed by a rotation around X-axis\n",
    "\n",
    "#Hamiltonian definition\n",
    "operator = QubitHamiltonian(\"-1.0*X(0)X(1)\")\n",
    "\n",
    "br_trans_real, br_trans_im = tq.braket(ket=alpha, bra=beta, operator=operator)\n",
    "br_trans_el = br_trans_real + 1.0j*br_trans_im\n",
    "H_ab = tq.compile(br_trans_el) # transalte to backend\n",
    "\n",
    "transitions = []\n",
    "for value in np.arange(0,4.1, 0.1): \n",
    "    # set the value we want to simulate\n",
    "    variables = {\"angle\" : value}\n",
    "    br_trans_value = H_ab(variables=variables)\n",
    "    transitions.append(round(br_trans_value,3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Transitions')"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAgAAAAFBCAYAAAABlBeuAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/YYfK9AAAACXBIWXMAAAsTAAALEwEAmpwYAAA97ElEQVR4nO3deXhU5fn/8fedlWENEGQJIHsARUAjWrFqUQFtK0itRX9t1bbSze5ipV/bWmvrkrp1Ly61rVutUqRWjbhjLUog7BD2LeyEsGbP8/tjJnRIJskAmTmzfF7XNVdmnnNm5j45M+fc85xnMeccIiIiklxSvA5AREREok8JgIiISBJSAiAiIpKElACIiIgkISUAIiIiSUgJgIiISBJK8zqAaMrOznb9+vXzOgwREZGoWLhw4V7nXLdQy5IqAejXrx+FhYVehyEiIhIVZra5qWW6BCAiIpKElACIiIgkISUAIiIiSUgJgIiISBJSAiAiIpKElACIiIgkISUAIiIiScjTBMDMnjCz3Wa2vInlZma/NrN1ZrbUzM4OWnaDma0N3G6IXtQiEq7ZRSWMvfct+t/+b8be+xazi0q8DklEAryuAXgSmNjM8iuAwYHbNOAPAGbWBfgpcB4wBvipmXWOaKQickJmF5UwY9YySsrKcUBJWTkzZi1TEiASIzwdCdA5956Z9WtmlUnAX51zDphvZllm1hO4BJjrnCsFMLO5+BOJZyMcsojgP7nnFxSzvaycXlk+pk/IZfLonOPWue+11ZRX1x5XVl5dS35BcaN1w3k9EWldsT4UcA6wNejxtkBZU+WNmNk0/LUH9O3bNzJRiiSR+l/29Sf3krJybnthCW+t3kW7zDQ27j3Cpr1H2XmwIuTzS8rKufZP/2VAdjv6Zbdj18EKnvlwC5U1dceWz5i1DEBJgEgExXoCcMqcczOBmQB5eXnO43BE4t79IX7ZV9U65izZQZd2GfTr2paxg7KZu3InBytqGj2/bUYqdXWON1btYu/hqpDv0VRNgYi0nlhPAEqAPkGPewfKSvBfBggufydqUYkkoQNHq3n6o81sPxD6l70Bi358+bHHs4uyj6spAPClp/LLq0ccO7EfrKhm5J2vEyozLykrp/RIFV3aZbTmZohIQKwnAHOAW8zsOfwN/g4453aYWQHwy6CGf+OBGV4FKZIoQl2LP+f0zjz+/kaeL9zK0apaMtNSjlXXB+uV5Tvucf1Jvrlr+x3bpNMry0dJWXnIeC64900+c3ZvvnxhfwZ0a6+2AiKtyPzt6zx6c7Nn8f+SzwZ24W/Znw7gnPujmRnwW/wN/I4CNznnCgPP/RLwo8BL/cI59+eW3i8vL89pOmCR0Bpe2wdIMahzkJZiXDWqF1+5cABrdh0K+cv+nikjTupkHOp9fempfOfSQWzce5R/FpVQXVfHGT07smb3YaqCko9TeV+RZGBmC51zeSGXeZkARJsSAJGmjb33rZC/xNtnpvHG9y+mR6c2x8pa+5d4c6+3+1AFf/vvZn779jpCHa5ysnz85/ZxJ/3eIomsuQQg1i8BiEiUbG+iGv5IZc1xJ3/wV++35q/u5l7vtA5t+MH4XH771rqQy5uKW0Sa5/VAQCISA5aXHCA1xUIua3ht3ytNxZHdPjPKkYgkBiUAIknMOccT729kyu8/oG1GKhlpxx8SfOmpTJ+Q61F0x5s+IRdfeupxZQbsPVzJH95ZT11d8lzOFGkNugQgkqRKj1Qx/R9LeHP1bi4bdhr3XzOS99bsidlW9qF6FXxz3ED+s3Yf9722mg/W7+WBa0dyWoc2LbySiIAaAYokpf+u38d3/17E/iPVzLhyKDde0A9/p5v445zjuQVb+dm/VtA+M40Hrh3FxUO6eR2WSExQI0CRJBfcyr59ZhqHKmsYkN2Ox284lzNzOnkd3ikxM64b05e80ztzyzNF3PDER3witxvFOw+x40BFzNVkiMQKtQEQSXANZ+U7VFlDqhlfvWhA3J/8gw3u3oGXbhnL2IFdeLt4D9sPVGgWQpFmKAEQSXD5BcWNxu6vdY5fN9GtLp61SU9l077G3QLr5xYQkf9RAiCS4JrqJ5+o/eeTbXtFTpYSAJEEtmjL/iaXxUr//tbW1HZ1ba9JhUSCKQEQSVCLtuzni49/RNd26bSJ4f79ra2p8QIOHK3mv+v3eROUSAxSAiCSgOpP/tntM/jXtz/OvZ85i5wsH4Z/7PxEnkBn8ugc7pky4rjt/dmkM+iX3Y6bnvxISYBIgMYBEEkwwSf/Z6edT89OiVnVf6L2Hq7kupnz2br/KH++cQwfG9jV65BEIq65cQBUAyCSQIJP/s9N+5hO/kGy22fy7LTz6dO5rWoCRFACIJIwFm4+/uTfcAY/aZwEfLB+r9chiXhGlwBE4lTw6H5d22dyqLyKXp3b8uzN5+vk34K9hyu5/tH5bCk9ypcu7M9LRdtjcv4DkVOlSwAiCabh6H57D1dSVeu44YLTdfIPQ3b7TJ65+XyyfOn8/u31x/6PGjVQkokSAJE4FGp0Pwc8+t5GbwKKQ9ntMyHEBEgaNVCShRIAkTik0e5ax64DFSHL9X+UZKAEQCQO9cwKXc2fqKP7RUpT/y/9HyUZKAEQiUMDs9s1Kkvk0f0iJdSogalm3Dp+iEcRiUSPpwmAmU00s2IzW2dmt4dY/pCZLQ7c1phZWdCy2qBlc6IauIiHnl+wlXnr9vGJ3OykGd0vUhqOGtihTRq1znGgvNrr0EQiLs2rNzazVOB3wOXANmCBmc1xzq2sX8c5972g9b8FjA56iXLn3KgohSsSE4q27OeO2cu5cFA2j37xXNJSVYl3qiaPzjmWONXVOab9bSE///cqcnt01GiBktC8PHqMAdY55zY456qA54BJzax/HfBsVCITiUG7D1bwtacW0r1TJr+5brRO/hGQkmI89LmR9Ovalm8+s4ht+496HZJIxHh5BMkBtgY93hYoa8TMTgf6A28FFbcxs0Izm29mkyMWpUgMqKyp5WtPLeRgeQ0zv5BH53aa2jZSOrRJ59Ev5lFdU8dX/7aQ8qralp8kEofi5SfEVOAF51zwN/H0wOhG1wMPm9nAUE80s2mBRKFwz5490YhVpNXdOWcli7aU8avPjmRYz45eh5PwBnRrzyPXjWLljoPcPmspyTRiqiQPLxOAEqBP0OPegbJQptKg+t85VxL4uwF4h+PbBwSvN9M5l+ecy+vWrdupxiwSdU9/uJlnP9rCNy4ZyCfP6ul1OElj3NDu3Do+l5cWb+exeRpgSRKPlwnAAmCwmfU3swz8J/lGrfnNbCjQGfhvUFlnM8sM3M8GxgIrGz5XJN4t2FTKT19awSdyu/GD8eriF23fuGQgV47owT2vruK9NapBlMTiWS8A51yNmd0CFACpwBPOuRVmdhdQ6JyrTwamAs+54+vghgF/MrM6/EnMvcG9B0TiWfAkP2bQpV0GD08dTWpK42FrJbLMjPxrRrJ+9xG++rdCOvrS2X2wUpMGSULQbIAiMaR+kp/gcf4z01K47zNn6WTjocfmbeDuf686rsyXnqqxFyTmaTZAkTgRapKfypo6TU7jsT//Z1OjMk0aJPFOCYBIDNEkP7FJ+0USkRIAkRjSo5Mm+YlFmjRIEpESAJEY0rNj4wRAk/x4L9SkQSmGJg2SuKYEQCRG/GvJdhZtLePKM3tokp8Y03DSoE6+NOocVNXWeR2ayEnzrBugiPzPzgMV3DF7OaP6ZPFrjfMfkxpOGvT/HvuQu/61ko8NyKZv17YeRydy4nSUEfGYc47bXlxKZU0tD147Uif/OJCSYvzq2pGkmHHrP5ZQW5c83aklcehII+Kxpz7cwntr9vB/Vw5jQLf2XocjYcrJ8nHnVWfw0aZSHpu3wetwRE6YEgARD23ce4Rf/nsVHx+czefPP93rcOQETTk7hwlndOeB19ewasdBr8MROSFKAEQ8UlNbx/efX0xGWgr514zETEP9xhsz45dXj6CjL53v/X0xlTWaOljihxIAEY/88d31FG0p4+eTz2yy/7/Evq7tM7nvMyNYvfMQD7+x1utwRMKmBEDEA8tLDvDwG2v59MheXDWyl9fhyCm6dFh3pp7bhz+9u57CTaVehyMSFnUDFImS4Fn+UlOMthkp/HzSGV6HJa3kjk8N5z/r9zLtb4VkpqWy80CFZg2UmKYaAJEoqJ/lr6SsHAfU1Dkqqh3vFGuO+UTRPjONyaNyKD1SzY4DFTigpKycGbOWMbuoxOvwRBpRAiASBaFm+auq1Sx/iWbWosYnes0aKLFKCYBIFGg2ueSg/SzxRAmASBRoNrnkoP0s8UQJgEgU3HhBv0ZlmuUv8YSaNTAzLUX7WWKSegGIRJhzjnnr9pKZanRul8mug2odnqjq92d9bw8z6JXVRl09JSYpARCJsFeW7eS9NXv46aeHc9PY/l6HIxEWPGvg8wu2ctuLS3lh0TauzevjcWQix9MlAJEIOlRRzV0vr+CMXh35gsb6TzrXnNObvNM7c88rq9h/pMrrcESOowRAJIIemruW3Ycq+cXVIzTNbxJKSTHuvvpMDlbUcH/Baq/DETmOp0ckM5toZsVmts7Mbg+x/EYz22NmiwO3rwQtu8HM1gZuN0Q3cpGWrdh+gCc/2Mj1Y/oyqk+W1+GIR4b26MiXL+zPsx9tZeHm/V6HI3KMZwmAmaUCvwOuAIYD15nZ8BCr/t05Nypweyzw3C7AT4HzgDHAT82sc5RCF2lRXZ3jjtnL6dw2g9smDPU6HPHYdy4dTM9Obbhj9nJqauu8DkcE8LYGYAywzjm3wTlXBTwHTArzuROAuc65UufcfmAuMDFCcYqcsOcWbKVoSxk/unIYndqmex2OeKxdZho//fRwVu04yJMfbPI6HBHA2wQgB9ga9HhboKyhz5jZUjN7wczqm9GG+1yRqNt3uJL7XlvNef27MOVsfSzFb8IZPfhEbjcemruGHQc0MqB4L9ZbJf0L6OecOwv/r/y/nOgLmNk0Mys0s8I9ezTxikTePa+u5khlDXdPPhMz8zociRFmxs+uOpOaOsfPX17pdTginiYAJUBwx9jegbJjnHP7nHOVgYePAeeE+9yg15jpnMtzzuV169atVQIXacpHG0t5YeE2br5oAIO7d/A6HIkxfbu25VvjBvHKsp28U7zb63AkyXmZACwABptZfzPLAKYCc4JXMLOeQQ+vAlYF7hcA482sc6Dx3/hAmYhnqmvruGP2MnKyfHx73GCvw5EYdfNFAxjQrR0/eWkFFQ1miBSJJs9GAnTO1ZjZLfhP3KnAE865FWZ2F1DonJsDfNvMrgJqgFLgxsBzS83s5/iTCIC7nHOlUd8IEWB2UQn5BcWUBGZ8+8qF/fFlpLbwLElWmWmp3D3pTK5/7EPO/cUbHK6o0dDQ4glPhwJ2zr0CvNKg7CdB92cAM5p47hPAExENUKQFs4tKmDFrGeVBv+Se/nALZ+Z00sFcmrT7UCWpZhyqqAGgpKycGbOWAehzI1ET640ARWJafkHxcSd/gPLqWvILij2KSOJBfkExtc4dV6bPjUSbEgCRU7C9LHR3rqbKRUCfG4kNSgBETkHPrDYhy3tl+aIcicSTpj4f+txINCkBEDkFFw1u3LXUl57K9Am5HkQj8WL6hFx86cc3FE1PNX1uJKo8bQQoEs8OV9bw5urdnN7FR02dY3tZhVpzS1jqPx/5BcVsLysnIy2FFINP5J7mcWSSTJQAiJykP76znj2HKvnnNy5gdF/NRSUnZvLonGOJwOqdB7nykXk88uZafvLpUHOiibQ+XQIQOQklZeU8Om8Dk0b10slfTtnQHh353Ll9+et/N7Fhz2Gvw5EkoQRA5CTc9+pqAG6bqKl+pXV8//IhtElP5ZevrPY6FEkSSgBETtCiLfuZs2Q7N398ADlqtS2tpFuHTL7xiYG8sWoXH6zb63U4kgSUAIicAOf8M7l165DJ1y8Z6HU4kmC+NLY/vTv7uOvlldTWuZafIHIKlACInIA5S7ZTtKWM6eNzaZepNrTSutqkp3L7FUNZvfMQ/yjc6nU4kuCUAIiEqaK6lvteXc3wnh35zDm9vQ5HEtQnR/Qk7/TO/Or1NRyurPE6HElgSgBEwvTYvA1sP1DBjz81nNQU8zocSVBmxo8/NZy9hyv5/dvrvA5HEpgSAJEw7D5Ywe/fWc/44d352MCuXocjCW5knyyuHp3DY+9vZGvpUa/DkQSlBEAkDL96vZjq2jp+dOUwr0ORJHHbxFxSDO57Td0CJTKUAIi0YHnJAf6xcBs3fKwf/bLbeR2OJImenXxMu2ggLy/dwcLNpV6HIwlIzZhFmjC7qIT8gtWUlFWQYjDgNJ38Jbq+dvEAnvzPBqbOnE9NrdNcE9KqlACIhDC7qIQZs5ZRXl0LQJ2Dn/9rFW3T03Twlah5fcUuyqvrqK71jwlQUlbOjFnLAPQ5lFOmSwAiIeQXFB87+dcrr64lv6DYo4gkGeUXFB87+dfT51BaixIAkRC2l5WfULlIJOhzKJGkBEAkhB6d2oQs76Wx/yWKmvq86XMorUEJgEgIo/tkNSrzpacyfUJu9IORpDV9Qi6+9NTjytJSTJ9DaRWeJgBmNtHMis1snZndHmL5981spZktNbM3zez0oGW1ZrY4cJsT3cglkZUeqeK9tXs5s1dHcrJ8GJCT5eOeKSPU8EqiavLoHO6ZMuLY59CfDDjOH6DBqOTUedYLwMxSgd8BlwPbgAVmNsc5tzJotSIgzzl31My+DtwPfC6wrNw5NyqaMUty+O1b6zhaVcNDnxvF4O4dvA5Hktzk0TnHEs+tpUcZ98A7PPLmGu6ZcpbHkUm887IGYAywzjm3wTlXBTwHTApewTn3tnOufhzM+YBmYJGI2rb/KE/N38w15/TWyV9iTp8ubfl/553O84XbWLf7sNfhSJzzMgHIAYLnu9wWKGvKl4FXgx63MbNCM5tvZpMjEJ8koQfnrgGD7142xOtQREK6Zdwg2qSl8Ct1BZRTFBeNAM3s80AekB9UfLpzLg+4HnjYzAY28dxpgUShcM+ePVGIVuLVqh0H+WdRCTdd0E+trCVmZbfPZNpFA3ltxU4WbdnvdTgSx7xMAEqAPkGPewfKjmNmlwH/B1zlnKusL3fOlQT+bgDeAUaHehPn3EznXJ5zLq9bt26tF70knPyCYjpkpvH1S0LmkiIx4ysf7092+wzue3U1zrmWnyASgpcJwAJgsJn1N7MMYCpwXGt+MxsN/An/yX93UHlnM8sM3M8GxgLBjQdFTsiHG/bx1urdfP2SQWS1zfA6HJFmtctM49uXDubDjaW8s0Y1m3JyPEsAnHM1wC1AAbAKeN45t8LM7jKzqwKr5QPtgX806O43DCg0syXA28C9DXoPiITNOce9r62me8dMbrygn9fhiIRl6rl96dulLfe9upq6OtUCyInzdDIg59wrwCsNyn4SdP+yJp73ATAistFJsnh95S6KtpRxz5QR+DJSW36CSAzISEvhB+OH8J3nFvPSkhKuHq1OUnJiTqgGwMxSzKxjpIIRibaa2jruf201A7q147Pn6AAq8eXTZ/XijF4deeD1NVTW1Lb8BJEgLSYAZvaMmXU0s3bAcmClmU2PfGgikffiom2s33OE2ybkkpYaF51iRI5JSTF+OHEo2/aX8/T8LV6HI3EmnCPecOfcQWAy/n74/YEvRDIokWioqK7loblrGdUniwln9PA6HJGT8vHB2Ywd1JXfvr2OQxXVXocjcSScBCDdzNLxJwBznHPVgFqcSNx78oNN7DxYwe1XDMXMvA5H5KSY+WsBSo9U8eh7G7wOR+JIOI0A/wRsApYA7wUm5DkYyaBEIml2UQn3vbaaHQcqyExLYeeBCq9DEjklZ/XOYlTvTvzmrXX85q119MryMX1Criavkma1mAA4534N/DqoaLOZfSJyIYlEzuyiEmbMWkZ5tb/BVGVNHTNmLQPQwVLi1uyiElbtPHSsarakrFyfa2lRiwlAYMCdzwD9Gqx/V4RiEomY/ILiYyf/euXVteQXFOtAKXErv6CYypq648r0uZaWhHMJ4CXgALAQqGxhXZGYtr2s/ITKReKBPtdyMsJJAHo75yZGPBKRKDitYya7DjbOYzX5j8SzXlk+SkKc7PW5luaE0wvgAzPTqHuSEHI6NT4g+tJTmT4h14NoRFrH9Am5+NKPH8UyxdDnWpoVTgJwIbDQzIrNbKmZLTOzpZEOTKS1rdx+kEVby7h02GnkZPkwICfLxz1TRug6qcS1yaNzuGfKiGOf645t0qhz0KdLW69DkxgWziWAKyIehUgUPPB6MR3bpPHgZ0fRqW261+GItKrJo3OOJbJHKmu4OP9t8gtW8+zN52ucCwmpxRoA59xmIAv4dOCWFSgTiRuFm0p5c/VuvnrxQJ38JeG1y0zjlk8MYv6GUuat3et1OBKjwpkL4DvA08BpgdtTZvatSAcm0lqcc9xfUEx2+0xuGtvP63BEouK68/qSk+Ujv6AY5zR4qzQWThuALwPnOed+Epiq93zg5siGJdJ63l2zh482lvLtSwfRNsPTGbBFoiYzLZXvXT6EZSUHeHX5Tq/DkRgUTgJgQPDIKbWBMpGYV1fnyC8opndnH1PP7et1OCJRdfXoHAad1p4HXi+mprau5SdIUgknAfgz8KGZ3WlmdwLzgccjGpVIK3l1+U5WbD/I9y4bQkaapvuV5JKaYtw6fgjr9xxhVlGJ1+FIjAmnEeCDwE1AaeB2k3Pu4QjHJXLKamrreGBuMUO6t1c3P0laE87owcjenXjkjbVU1tS2/ARJGk0mAGbWMfC3C/7ZAJ8K3DYHykRi2ouLtrFhzxF+MD6X1BRdtZLkZGZMnzCUkrJynp6/xetwJIY0VwPwTODvQqAw6Fb/WCRmVVTX8sgbaxnZJ4vxw7t7HY6Ip8YO6srHBnTld2+v40hljdfhSIxoMgFwzn0q8Le/c25A0K2/c25A9EIUOXFPf7iF7QcquG1CrgZBkaRnZkyfmMu+I1U88f5Gr8ORGBHOOABvhlMmEisOV9bwu7fXMXZQV8YOyvY6HJGYcHbfzlw+vDsz39vA/iNVXocjMaC5NgBtAtf6s82ss5l1Cdz6Aa3SosrMJgbmGFhnZreHWJ5pZn8PLP8w8N71y2YEyovNbEJrxCOJ4fF5Gyk9UsX0CUO9DkUkptw6PpfDVTX88d31XociMaC5UVG+CnwX6AUsCio/CPz2VN/YzFKB3wGXA9uABWY2xzm3Mmi1LwP7nXODzGwqcB/wOTMbDkwFzgjE94aZDXHOqYlrEptdVMJ9r61mx4EK2qSlsGnvEUb1yfI6LJGYkdujA+f0zeJP721g5nsb6JXlY/qEXPWSSVLNtQF4xDnXH7g1cN2//jbSOXfKCQAwBljnnNvgnKsCngMmNVhnEvCXwP0XgEvNf0F3EvCcc67SObcRWBd4PUlSs4tKmDFrGTsOVABQUVPHjFnLmK2+zyLHzC4qYVnJQQAcUFJWru9JEmvuEsC4wN0SM5vS8NYK750DbA16vI3GlxaOreOcqwEOAF3DfK4kkfyCYsqrj68AKq+uJb+g2KOIRGJPfkExlTXHjwio70nyau4SwMXAW/hnAGzIAbMiElErM7NpwDSAvn01FGyi2l5WfkLlIslI3xMJ1mQC4Jz7aeDvTRF67xKgT9Dj3oGyUOtsM7M0oBOwL8znAuCcmwnMBMjLy9OUWAnqtI6Z7DpY2ai8V5bPg2hEYlOvLB8lIU72+p4kp7CmAzazjub3mJktMrPxrfDeC4DBZtbfzDLwN+qb02CdOcANgfvXAG85/7yWc4CpgV4C/YHBwEetEJPEqZxOjQ9gvvRUpk/I9SAakdg0fUIuvvTU48pSDH1PklQ4s6N8yTl3EBiP//r7F4B7T/WNA9f0bwEKgFXA8865FWZ2l5ldFVjtcaCrma0Dvg/cHnjuCuB5YCXwGvBN9QBIXiu3H2TR1jIuG3YaOVk+DMjJ8nHPlBFq3SwSZPLoHO6ZMuLY96RjmzTqHPTpohqAZGT+H9TNrGC21Dl3lpk9ArzjnPunmRU550ZHJ8TWk5eX5woLNYpxovnSkwso3FTKvNvG0altutfhiMSNo1U1XHT/Owzs1o7npp2vUTMTkJktdM7lhVoWTg3AQjN7HbgSKDCzDoAmlpaYULiplLdW7+ZrlwzUyV/kBLXNSONb4wbx4cZS3lu71+twJMrCSQC+jL/q/Vzn3FEgA//0wCKecs5x/2vFdOuQyY0X9PM6HJG4NHVMH3KyfOQXrKalGmFJLC0mAM65OmAXMNzMLsI/+l5WhOMSadG7a/bw0aZSvjVuEG0zmuvRKiJNyUxL5XuXD2F5yUFeXb7T63Akilo8aprZfcDn8De4q29o54D3IhiXSLPq6hz5BcX06eJj6rka30HkVFw9Ooc/vbueX71ezPjh3UlLDadyWOJdOHt5MpDrnLvSOffpwO2qlp4kEkmvLN/Biu0H+d5lQ8hI08FK5FSkphg/GJ/Lhj1HmLVIwwIni3COnBsAta6SmFFTW8eDr69hSPf2TBqlbn4irWHCGd0Z2bsTD7+xhopq9apOBuEkAEeBxWb2JzP7df0t0oGJNOXFRdvYsPcIt47PJTVF3ZZEWoOZMX3CULYfqOCZD7d4HY5EQTgtp+bQeIQ+EU9UVNfy8BtrGdUni8uHd/c6HJGEcuHgbC4Y2JXfvb2Oa8/tQ/tMNa5NZC3uXefcX1paRyRanpq/mR0HKnjgsyM1aIlIBNw6IZcpv/+AJ97fyLcvHex1OBJB4cwFMNjMXjCzlWa2of4WjeBEgh2urOH376znwkHZXDAo2+twRBLS2X07c/nw7jz63gb2H6nyOhyJoHDaAPwZ+ANQA3wC+CvwVCSDEgnl8XkbKT1SpYlLRCLs1vG5HK6q4Y/vrvc6FImgcC7w+Jxzb5qZOec2A3ea2ULgJxGOTYTZRSXkFxQfm6/8rJyOjOyT5W1QIgkut0cHrh6Vw2PzNvDS4u3sOlhBrywf0yfkaoKtBBJODUClmaUAa83sFjO7Gmgf4bhEmF1UwoxZyygpK8fhH32qeNdhZhepn7JIpJ3RqyO1DnYerMABJWXlzJi1TN+/BBJOAvAdoC3wbeAc4PPADZEMSgQgv6CY8gb9kStr6sgvKPYoIpHk8cR/NjUqK6+u1fcvgTR7CcDMUoHPOeduBQ6jSYAkiuqr/cMtF5HWo+9f4muyBsDM0pxztcCFUYxH5JheWb4TKheR1qPvX+Jr7hLAR4G/RWY2x8y+YGZT6m/RCE6S2/QJuTQc6M+XnqpeACJRMH1CLr701OPK9P1LLOH0AmgD7APG4W+HZYG/syIYlwh9u7alzkGHNmkcrqhRK2SRKKr/nuUXFFMSqPa/8YJ++v4lkOYSgNPM7PvAcv534q/nIhqVJD3nHPe+uprs9pm8O/0S2mlIUpGomzw6h8mjc6ioruXSB97l/XV7mV7nSNEcHAmhuUsAqfi7+7UHOgTdr7+JRMw7xXv4aGMp37l0kE7+Ih5rk57K9y4fwrKSA7yyfIfX4Ugrae7IusM5d1fUIhEJqK1z3Pfaak7v2papY/p6HY6IAFePzuHR9zbwq4JiJpzRg/TUcHqRSyxrbg+qjkc88dLiElbvPMQPxufqICMSI1JTjNsm5rJp31GeW7DV63CkFTR3dL00Um9qZl3MbK6ZrQ387RxinVFm9l8zW2FmS83sc0HLnjSzjWa2OHAbFalYJboqa2p54PU1nJnTkU+N6Ol1OCISZNzQ0zi3X2d+/eZajlbVeB2OnKImEwDnXGkE3/d24E3n3GDgzcDjho4CX3TOnQFMBB42s6yg5dOdc6MCt8URjFWi6Kn5WygpK+eHE4eqoZFIjDEzbr9iKHsOVfLE+xu9DkdOkVf1q5OAvwTu/wWY3HAF59wa59zawP3twG6gW7QClOg7WFHNb99ay4WDsvn4YO1qkVh0zulduHx4d/747gZKNV1wXPMqAejunKtvSroT6N7cymY2BsgAguem/EXg0sBDZpYZoTglih59bwP7j1bzw4lDvQ5FRJpx24RcjlbV8Lu313kdipyCiCUAZvaGmS0PcZsUvJ5zrn6it6ZepyfwN+Am51xdoHgGMBQ4F+gC/LCZ508zs0IzK9yzZ8+pbpZEyO5DFTw2byOfPKsnI3p38jocEWnG4O4duOac3vztv5vZtv+o1+HISYpYAuCcu8w5d2aI20vArsCJvf4EvzvUa5hZR+DfwP855+YHvfYO51cJ/BkY00wcM51zec65vG7dVK0cq37z5jqqa+u4dbyGGRWJB9+9bAgYPDh3jdehyEny6hLAHP43pfANwEsNVzCzDOCfwF+dcy80WFafPBj+9gPLIxmsRNamvUd49qMtTB3Th/7Z7bwOR0TC0CvLx00X9OOfRSWs3nnQ63DkJHiVANwLXG5ma4HLAo8xszwzeyywzrXARcCNIbr7PW1my4BlQDZwd1Sjl1b1q9eLSU9N4duXDvY6FBE5AV+/ZCAdMtPIf63Y61DkJJj/EnxyyMvLc4WFhV6HIcDsohLyC4rZXlaOA8YPP42ZXzzX67BE5AT94Z313PfaarLbZ7DvcJUm7YoxZrbQOZcXapmGWZOom11UwoxZyygJnPwB3lu7l9lFJZ7GJSInrmu7DAD2Hq7CASVl5cyYtUzf5zigBECiLr+gmPLq2uPKKqrryC9QNaJIvHnkzbWNysqra/V9jgNKACTqtgfmFg+3XERil77P8UsJgERdryzfCZWLSOzS9zl+KQGQqPvWuIGNynzpqUyfoDEAROLN9Am5+NJTjytrk56i73McSPM6AEk+m0v9VYPd2mey93ClWg2LxLH6721wr55z+3XR9zkOKAGQqNpaepTH39/I1aNzeOhzo7wOR0RaweTROcdO+P/3z2U8t2Ar63YfZtBp7T2OTJqjSwASVfe+tpoUg9smqnpQJBF9//IhtE1P5ZevrPI6FGmBEgCJmsJNpfx76Q6mXTSQnp3UQEgkEXVtn8kt4wbx1urdzFurCdhimRIAiYq6OsfPX15J946ZfO3iAV6HIyIRdOPYfvTt0pa7X15FTW1dy08QTygBkKiYs2Q7S7YdYPqEobTNUNMTkUSWmZbKjCuGUrzrEH8v3Op1ONIEJQASceVVtdz32mpG5HRiiloGiySFiWf2YEz/Ljz4+hoOVlR7HY6EoARAIu7ReRvYcaCCH39qOCkp5nU4IhIFZsaPPzmc0qNV/O7tdV6HIyEoAZCI2nWwgj+8s54rAr8GRCR5jOjdiSmje/Pn9zexZd9Rr8ORBpQASETlFxRTW+eYccUwr0MREQ/cNjGX1BTj3tfULTDWKAGQiFlecoAXF23jprH96Nu1rdfhiIgHundsw9cuHsgry3by0cZSr8ORIEoAJCKcc9z18kq6tM3gm+MGeR2OiHho2kUD6NmpDT9/eSV1dc7rcCRA/bGkVc0uKiG/oJiSwFSg15zTm45t0j2OSkS85MtI5baJuXzv70s45+65lB2t1hwgMUA1ANJqZheVMGPWsmMnf4CXl25ndlGJh1GJSEyoAzPYf7QaB5SUlTNj1jIdHzykBEBaTX5BMeXVtceVVVTXkV9Q7FFEIhIrfjV3Da5B7X95da2ODx5SAiCtZnvQL/9wykUkeej4EHuUAEir6ZUVeoKfpspFJHno+BB7PEkAzKyLmc01s7WBv52bWK/WzBYHbnOCyvub2Ydmts7M/m5mGdGLXpryyRE9GpX50lOZPkFT/4oku+kTcvGlpx5Xlp5qOj54yKsagNuBN51zg4E3A49DKXfOjQrcrgoqvw94yDk3CNgPfDmy4UpLKqpreW3FLk7rkEGvTm0wICfLxz1TRqiVr4gweXQO90wZQU6WD8N/8velp3DpsNO8Di1pmWvYKiMab2pWDFzinNthZj2Bd5xzjdJAMzvsnGvfoMyAPUAP51yNmX0MuNM5N6Gl983Ly3OFhYWttBUS7IHXi/nNW+t45ubzuGBgttfhiEiMW7y1jKt//x9uuqA/P/n0cK/DSVhmttA5lxdqmVc1AN2dczsC93cC3ZtYr42ZFZrZfDObHCjrCpQ552oCj7cB+onpofV7DvPHd9dz9egcnfxFJCyj+mRx/Zi+PPnBRlZsP+B1OEkpYgmAmb1hZstD3CYFr+f8VRBNVUOcHshcrgceNrOBJxHHtEASUbhnz54T3xBplnOOH89eji89lR9dqfH+RSR8t00YSpd2Gdwxe7lGCPRAxBIA59xlzrkzQ9xeAnYFqv4J/N3dxGuUBP5uAN4BRgP7gCwzqx/FsDfQ5EgSzrmZzrk851xet27dWm37xG/Oku18sH4f0ycOpVuHTK/DEZE40qltOj+6chhFW8p4bsFWr8NJOl5dApgD3BC4fwPwUsMVzKyzmWUG7mcDY4GVgRqDt4Frmnu+RN6B8mp+/vIqRvbuxPVj+nodjojEoatH53D+gC7c99pq9h6u9DqcpOJVAnAvcLmZrQUuCzzGzPLM7LHAOsOAQjNbgv+Ef69zbmVg2Q+B75vZOvxtAh6PavQC+Bv+lR6p5O7JI0hNMa/DEZE4ZGbcPflMjlTWcM8rq70OJ6l4MhmQc24fcGmI8kLgK4H7HwAjmnj+BmBMJGOU5i3dVsbf5m/mi+efzojenbwOR0Ti2KDTOnDzRQP4wzvruTavN+cN6Op1SElBIwHKCautc9wxeznZ7TP5gQbxEJFW8O1xg8nJ8nHH7OVU19Z5HU5SUAIgJ+yZDzezdNsB7vjkME31KyKtwpeRys+uOoO1uw/z+PsbvQ4nKSgBkBOy+1AF9xcUM3ZQV64a2cvrcEQkgVw2vDuXD+/OI2+sZdv+o16Hk/A8aQMg8Wd2UQn5BcWUBGbuunhIN/yDMoqItJ47rzqDS+5/m8seeJfKmjp6ZfmYPiFXQ4pHgGoApEWzi0qYMWvZsZM/wENz1zK7qMnhF0RETsqCjaU4oKKmDgeUlJUzY9YyHW8iQAmAtCi/oJjy6trjysqra8kvKPYoIhFJVPkFxdQ0GBVQx5vIUAIgLdoe9Ms/nHIRkZOl4030KAGQFnVplxGyvFeWL8qRiEiia+q4ouNN61MCIM0qO1pFVU0tDZv7+dJTma4xAESklU2fkIsvPbVR+Vcv7u9BNIlNCYA0647ZyymvruP744eQk+XDgJwsH/dMGaFWuSLS6iaPzuGeKSOOHW9O65BJqsE7xXvxTwUjrUXdAKVJc5Zs5+WlO7h1/BBuGTeYb40b7HVIIpIEJo/OOe4HxhPvb+Sul1fy9wVbmaqJx1qNagAkpJ0HKrjjn8sY3TeLr1080OtwRCSJ3XhBP8YO6srPX17Jln0aIKi1KAGQRpxzTH9hCdW1jgevHUVaqj4mIuKdlBQj/5qRpKQY339+MbV1uhTQGnRkl0aemr+ZeWv38qNPDqN/djuvwxERoVeWj59ddQaFm/fz6LwNXoeTEJQAyHE27DnML15ZxcVDuvH583StTURix9Wjc7jizB48+PoaVu046HU4cU8JgBxTU1vH955fQmZaKvdfc5bG+heRmGJm/OLqEXT0pfO9vy+msqa25SdJk5QAyDF/eGc9S7aWcffkM+nesY3X4YiINNKlXQb3XzOC1TsP8dDctV6HE9fUDTDJ1c/yt72sHAec3SeLT2uaXxGJYeOGdue6MX3447vreWHhVvYdrtKsgSdBNQBJLHiWv/o2tSt3HtSsWyIS80b2zsKAvYerNGvgSVICkMRCzfJXUV2nWbdEJOb95q11NOwMqFkDT4wSgCSmWbdEJF7p+HXqlAAksW4dMkOWa9YtEYl1mjXw1HmSAJhZFzOba2ZrA387h1jnE2a2OOhWYWaTA8ueNLONQctGRXsb4l3Z0Spq6uoalWuWPxGJB5o18NR5VQNwO/Cmc24w8Gbg8XGcc28750Y550YB44CjwOtBq0yvX+6cWxyFmBNGTW0dtzxTxOGKWr572WDN8icicSfUrIHpKcacxTuoqmn840Ya86ob4CTgksD9vwDvAD9sZv1rgFedc5oFohXc99pq3l+3l/uvOYtr8/rw3cuGeB2SiMgJazhr4MtLt3PLM0Xc+a8V/PLqER5GFh+8qgHo7pzbEbi/E+jewvpTgWcblP3CzJaa2UNmFvpitjQyu6iER+dt5IaPnc61eX28DkdEpNV86qxefP2SgTzz4Rae/nCz1+HEvIglAGb2hpktD3GbFLyec85Bo94cwa/TExgBFAQVzwCGAucCXWim9sDMpplZoZkV7tmz51Q2Ke4tLznAD19cypj+XbjjU8O9DkdEpNXdOj6Xi4d04845KyjcVOp1ODEtYgmAc+4y59yZIW4vAbsCJ/b6E/zuZl7qWuCfzrnqoNfe4fwqgT8DY5qJY6ZzLs85l9etW7fW2bg4tPdwJdP+WkjXdhn8/v+dTbqm+BWRBJSaYvx66mhysnx87alF7DxQ4XVIMcurs8Ac4IbA/RuAl5pZ9zoaVP8HJQ8GTAaWt36IiaO6to5vPr2IfUeq+NMX8shurysmIpK4OrVNZ+YX8yivquGrTy2kolqTBoXiVQJwL3C5ma0FLgs8xszyzOyx+pXMrB/QB3i3wfOfNrNlwDIgG7g7GkHHq1/8exUfbizl3s+MYETvTl6HIyIScUO6d+CBa0exZGsZd8xejv9qswSzZPqn5OXlucLCQq/DiIr6SX5KAqNiXTIkmye/dJ7HUYmIRNeDc9fw6zfX0smXzsHy6qSbNMjMFjrn8kIt04XgBBQ8yU+9+RtLNUmGiCSd/l3akmJwoLxakwY1oAQgAWmSHxERv1/NXUNdg4puTRrkpwQgAWmSDBERPx0Pm6YEIMFs3ncEs9DLNEmGiCSbpo57TU2GlkyUACSQzfuOMHXmfNqkpZCZdvyu1SQ/IpKMmpo06GhVDWt3HfIgotihBCBBbNrrP/lXVNfy4jfGct9nztIkPyKS9BpOGpST5WPGlUPxZaRx3aPzkzoJUDfABLBp7xGue9R/8n/m5vMZ1rOj1yGJiMS09XsOM3XmfJxzPHvz+Qzu3sHrkCJC3QATWP0v/8qaOp38RUTCNLBbe56bdj4pZlz36HzWJGFNgBKAOFZ/8q+qrePpr5ynk7+IyAkY2K09zwaSgOuTMAnQJYA4Uz/C3/aycsz8jfte/MYFDO2hk7+IyMlYv+cw182cz5HKGtplprHnUGXCjBioSwAJIniEPwfUOaipc6zekVxZq4hIaxrYrT1f+Xh/jlTVsvtQZdKMGKgEII6EGuGvskYj/ImInKq/fLC5UVmijxioBCBO1NW548b2D6YRrURETk0yjhioBCAO7DlUyY1PLmhyuUb4ExE5NU0dR9NTU9hxIDGTACUAMe69NXu44pF5fLhhH9ec0xtfukb4ExFpbaFGDExPNRyOKx6Zx9yVuzyKLHKUAMSo6to67nl1FV984iM6t01nzi0X8qvPjuSeKRrhT0SktYUaMTD/mpG89t2L6NXJx81/LeTOOSuoaNAOK56pG2AMCO7a1yvLx5fG9mPO0h0s2VrGdWP68pNPDceX0XgsaxERibzKmlrueWU1T36wieE9O/Kb60ezbNuB447bsdplsLlugEoAPFbfta9h6/7MNOPBa0fzybN6ehSZiIgEe2PlLqa/sITDlTUAVNf+7/zpS0+NyRpZjQMQw0J17QPIapuhk7+ISAy5bHh3Xv3ORRh23Mkf4rPLoBIAjzXVxWT3wcooRyIiIi3p0akN1bV1IZfFW5dBJQAeWbi5lK/9bSFNXYBR1z4RkdjU1PE5My2F+Rv2ES+X1tO8DiCRNWzc94PLh+DLSOXReRtYtKWMTr50Lht2Gu+v3UtFzf8ySnXtExGJXdMn5DZqu5WWYqSmGFNnzmdETiduvmgAV57Zg5eX7ojZxoKeNAI0s88CdwLDgDHOuZAt88xsIvAIkAo85py7N1DeH3gO6AosBL7gnKtq6X1bqxFgwxN7qB0aqnGfAQ7o08XHVy4cwDXn9KZdZlpYryciIrEj1HF74pk9eHHRNh6ft5ENe4+Q5UvnSFVNWI0FI3UeiLleAGY2DKgD/gTcGioBMLNUYA1wObANWABc55xbaWbPA7Occ8+Z2R+BJc65P7T0vq2RAIQ6sdfv0EmjerH7UCUb9hzh608tpKy8utHzu7RNZ8Edl5OaYqcUh4iIxKa6Osebq3fzzacXURWivUB2+wz+9a0L6d6hDSkp1ux55VSTgOYSAE8uATjnVgGYNXsSHAOsc85tCKz7HDDJzFYB44DrA+v9BX9tQosJQGsI1Wq/vLqWW/+xhB/9cxlHq5ofJGL/0Wqd/EVEElhKinH58O5NNhbce7iKj93zFm3SU+jXtR2b9h2hovr4det7FUSyNjiW2wDkAFuDHm8DzsNf7V/mnKsJKm/yP2Rm04BpAH379j3loJpq5VlT5/jCuX0YkN2OftntuPUfS9gVoiW/GveJiCSHXlm+kJO4ZbfP4LuXDWHj3iNs2nuE1TtDT+ke6V4FEUsAzOwNoEeIRf/nnHspUu/bkHNuJjAT/JcATvX1mtqhOVk+fvrpM449nnHFsJBVOmrcJyKSHEI1FvSlp3LHJ4cf98t+7L1vhTyvRPoHY8S6ATrnLnPOnRniFu7JvwToE/S4d6BsH5BlZmkNyqMi1IQRoU7socaVjsVRokREJDLCPQ+Ee15pbbF8CWABMDjQ4r8EmApc75xzZvY2cA3+ngA3AFGrUajfceG01pw8OkcnfBGRJBbOeeBEziutyateAFcDvwG6AWXAYufcBDPrhb+735WB9a4EHsbfDfAJ59wvAuUD8J/8uwBFwOedcy0OnReLcwGIiIhESsx1A/SKEgAREUkmmgxIREREjqMEQEREJAkpARAREUlCSgBERESSkBIAERGRJKQEQEREJAkpARAREUlCSTUOgJntATa34ktmA3tb8fW8pG2JPYmyHaBtiUWJsh2gbWnO6c65bqEWJFUC0NrMrLCpARbijbYl9iTKdoC2JRYlynaAtuVk6RKAiIhIElICICIikoSUAJyamV4H0Iq0LbEnUbYDtC2xKFG2A7QtJ0VtAERERJKQagBERESSkBKAMJjZRDMrNrN1ZnZ7iOWZZvb3wPIPzayfB2GGJYxtudHM9pjZ4sDtK17E2RIze8LMdpvZ8iaWm5n9OrCdS83s7GjHGI4wtuMSMzsQtD9+Eu0Yw2VmfczsbTNbaWYrzOw7IdaJl/0SzrbE/L4xszZm9pGZLQlsx89CrBMXx68wtyUujl8AZpZqZkVm9nKIZdHZJ8453Zq5AanAemAAkAEsAYY3WOcbwB8D96cCf/c67lPYlhuB33odaxjbchFwNrC8ieVXAq8CBpwPfOh1zCe5HZcAL3sdZ5jb0hM4O3C/A7AmxOcrXvZLONsS8/sm8H9uH7ifDnwInN9gnXg5foWzLXFx/ArE+n3gmVCfoWjtE9UAtGwMsM45t8E5VwU8B0xqsM4k4C+B+y8Al5qZRTHGcIWzLXHBOfceUNrMKpOAvzq/+UCWmfWMTnThC2M74oZzbodzblHg/iFgFZDTYLV42S/hbEvMC/yfDwcepgduDRt+xcXxK8xtiQtm1hv4JPBYE6tEZZ8oAWhZDrA16PE2Gh8Ijq3jnKsBDgBdoxLdiQlnWwA+E6iefcHM+kQntFYX7rbGg48Fqj1fNbMzvA4mHIEqy9H4f6UFi7v90sy2QBzsm0BV82JgNzDXOdfkPonx41c42wLxcfx6GLgNqGtieVT2iRIAaehfQD/n3FnAXP6XhYo3FuEfynMk8BtgtrfhtMzM2gMvAt91zh30Op5T0cK2xMW+cc7VOudGAb2BMWZ2pschnbQwtiXmj19m9ilgt3NuodexKAFoWQkQnEX2DpSFXMfM0oBOwL6oRHdiWtwW59w+51xl4OFjwDlRiq21hbPfYp5z7mB9tadz7hUg3cyyPQ6rSWaWjv+E+bRzblaIVeJmv7S0LfG2b5xzZcDbwMQGi+Ll+HVMU9sSJ8evscBVZrYJ/2XYcWb2VIN1orJPlAC0bAEw2Mz6m1kG/gYZcxqsMwe4IXD/GuAtF2i9EWNa3JYG12Ovwn/tMx7NAb4YaHV+PnDAObfD66BOlJn1qL/2Z2Zj8H9nY/LgHIjzcWCVc+7BJlaLi/0SzrbEw74xs25mlhW47wMuB1Y3WC0ujl/hbEs8HL+cczOcc72dc/3wH4Pfcs59vsFqUdknaa39gonGOVdjZrcABfhb0T/hnFthZncBhc65OfgPFH8zs3X4G3RN9S7ipoW5Ld82s6uAGvzbcqNnATfDzJ7F3wo728y2AT/F3ygI59wfgVfwtzhfBxwFbvIm0uaFsR3XAF83sxqgHJgaiwfngLHAF4Blgeu0AD8C+kJ87RfC25Z42Dc9gb+YWSr+BOV559zL8Xj8IrxtiYvjVyhe7BONBCgiIpKEdAlAREQkCSkBEBERSUJKAERERJKQEgAREZEkpARAREQkCSkBEBERSUJKAERERJKQEgARCcnMJpuZM7Ohp/g6h1tY3s/MyoMG3GlqPV9gjveqWB5yVyReKAEQkaZcB7wf+Btp6wOTvDTJOVceWGd7FOIRSXhKAESkkcAseBcCXyYwDGngl/oqM3vUzFaY2euBMdnrn/NjMys2s/fN7FkzuzXE637ezD4K/JL/U2BY14brDDSzPWa2KbBeqZmtN7OOEdxkkaSjBEBEQpkEvOacWwPsM7P6WdUGA79zzp0BlAGfATCzcwP3RwJXAHkNX9DMhgGfA8YGfsnXAv+v4XrOufX4ax6+EFhvKTA53qcWFok1SgBEJJTr8E9VSuBv/WWAjc65xYH7C4F+gftjgZeccxXOuUP452Vv6FL807MuCFzvvxQY0MT7nwEsD9wfBhSf1FaISJM0G6CIHMfMugDjgBFm5vDPHOmA3wGVQavWAr7Gr9D0SwN/cc7NaPB+/Ro89gFtnHP7zawPsNc5V3XCGyIizVINgIg0dA3wN+fc6c65fs65PsBGoE8zz/kP8GkzaxNoP/CpEOu8CVxjZqeBP9Ews9NDrDec/83jPowYnNNdJBEoARCRhq4D/tmg7EVgRoh1AXDOLQDm4L9e/yqwDDjQYJ2VwB3A62a2FJiLf473hoKr/8uBs0+1K6KINGbOOa9jEJEEYGbtnXOHzawt8B4wzTm3KIzn9QNeds6dGeb7bALynHN7TyVekWSnGgARaS0zA437FgEvhnPyD6gFOoU7EBCQDtSdQpwigmoAREREkpJqAERERJKQEgAREZEkpARAREQkCSkBEBERSUJKAERERJKQEgAREZEkpARAREQkCSkBEBERSUL/H9xszCAkZUB3AAAAAElFTkSuQmCC",
      "text/plain": [
       "<Figure size 576x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plotting the trend\n",
    "fig = plt.figure(figsize=(8,5))\n",
    "plt.plot(np.arange(0,4.1, 0.1), transitions, marker = 'o')\n",
    "plt.xlabel(r'Angle[$\\pi$]')\n",
    "plt.ylabel('Transitions')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Even in this particular case the imaginary part is 0 (as for an `ExpectationValue`), but in this case this happens because the wave functions are real. The difference can be noticed again by printing `br_trans_im` where in this case there is one circuit to be executed:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Objective with 1 unique expectation values\n",
      "total measurements = 1\n",
      "variables          = [angle]\n",
      "types              = not compiled\n"
     ]
    }
   ],
   "source": [
    "print(br_trans_im)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the end, we can address one small example in which the imaginary part is not 0. We have:\n",
    "\n",
    "$$|\\psi_0\\rangle=R_x(a)\\rvert  0 \\rangle = \\cos(a/2) \\rvert 0 \\rangle - i\\sin(a/2)\\rvert 1\\rangle \\quad , \\quad |\\psi_1\\rangle=R_y(b)\\rvert  0 \\rangle = \\cos(b/2) \\rvert 0 \\rangle + \\sin(b/2)\\rvert 1\\rangle$$ \n",
    "and so the imaginary part of the transition amplitude comes down to $\\text{Im}(\\langle\\psi_1|Z|\\psi_0\\rangle ) = -\\sin(a/2)\\sin(b/2)$ ."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "#states definition\n",
    "a = tq.Variable(\"a\")\n",
    "b = 1\n",
    "\n",
    "U0 = tq.gates.Rx(a*np.pi,0)\n",
    "U1 = tq.gates.Ry(b*np.pi,0)\n",
    "\n",
    "#Hamiltonian definition\n",
    "operator = QubitHamiltonian(\"1.0*Z(0)\")\n",
    "\n",
    "br_trans_real, br_trans_im = tq.braket(ket=U0, bra=U1, operator=operator)\n",
    "\n",
    "H_i = tq.compile(br_trans_im) # transalte to backend\n",
    "\n",
    "transitions = []\n",
    "for a in np.arange(0,4.1, 0.1): \n",
    "    # set the value we want to simulate\n",
    "    variables = {\"a\" : a}\n",
    "    br_trans_value = H_i(variables=variables)\n",
    "    transitions.append(round(br_trans_value,3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Transitions')"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 576x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plotting the trend\n",
    "fig = plt.figure(figsize=(8,5))\n",
    "plt.plot(np.arange(0,4.1, 0.1), transitions, marker = 'o')\n",
    "plt.xlabel(r'Angle a [$\\pi$]')\n",
    "plt.ylabel('Transitions')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Detailed explanation of internal `braket` structure"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's give some insight about the underlying structure of `braket`.\n",
    "This function in general just calls other functions to do all the different tasks one may need.\n",
    "\n",
    "More in detail, the function acts as follows:\n",
    "\n",
    "- If only ket is passed, `braket` directly returns the real and imaginary part of the self-overlap as `Objective()`. This is required since one may combine the different `braket` modalities and this will require the compilation of an `Objective()`.\n",
    "- If ket and operator are passed, `braket` returns the `ExpectationValue` of the operator for the given state.\n",
    "- If ket and bra are passed, `braket` returns the overlap between the two states. This is done by calling a new Tequila's function called `make_overlap`. Here below, we describe also how this function works.\n",
    "- If ket, bra and operator are passed, `braket` returns the transition element of the operator by calling another new Tequila's function called `make_transition`. Even in this case we provide a description of the internal structure."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### `Make_overlap` and `make_transition`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `make_overlap` function constructs the following circuit that is able to calculate the overlap between two states of $n$ qubits. \n",
    "\n",
    "<p align='center'>\n",
    "<img src=\"pics/overlap_circuit.png\" alt=\"overlap circ\" width=\"500\"/>\n",
    "</p>\n",
    "\n",
    "Here, $U_0$ and $U_1$ are two unitaries such that: \n",
    "\n",
    "$$U_0 |0\\rangle^n=|\\psi _ 0 \\rangle \\quad , \\quad U_1 |0\\rangle^n=|\\psi _ 1 \\rangle$$ \n",
    "\n",
    "so the circuit prepares the state:\n",
    "\n",
    "$$ \\frac{1}{\\sqrt{2}}(|\\psi _ 0 \\rangle|0\\rangle+|\\psi _ 1 \\rangle|1\\rangle) \\quad .$$\n",
    "\n",
    "In the end measuring the ancilla in the $2\\sigma_+$ base, gives as output:\n",
    "\n",
    "$$^n\\langle 0|U_ 0 ^\\dagger U_1 |0\\rangle^n=\\langle\\psi _ 0 |\\psi _ 1 \\rangle \\quad .$$\n",
    "\n",
    "We can check the circuit build by the `braket` function with the code contained in the next cell:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Objective with 1 unique expectation values\n",
      "total measurements = 1\n",
      "variables          = []\n",
      "types              = not compiled\n",
      "\n",
      "Expectation Value 0:\n",
      "total measurements = 1\n",
      "variables          = 0\n",
      "circuit            = 5588029959664042738_0.png\n",
      "0: ───────X───────H───@───\n",
      "          │       │   │\n",
      "1: ───────┼───────┼───X───\n",
      "          │       │   │\n",
      "2: ───H───@───X───@───@───\n"
     ]
    }
   ],
   "source": [
    "U0 = X(0) # state |10>\n",
    "U1 = H(0)+CX(0,1) # Bell state 1/sqrt(2)*(|00>+|11>)\n",
    "\n",
    "br_objective_real, br_objective_im = tq.braket(ket=U1, bra=U0) #consistently with the explained notation\n",
    "\n",
    "tq.draw(br_objective_real)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, one ancilla qubit has been added to our quantum register to be used as control qubit. The two circuits have been controlled and applied with the correct structure: the ancilla has an $H$ gate allowing superposition, the $U_0$ is performed when the ancilla is in state $|0\\rangle$ and the $U_1$ when the ancilla is in state $|1\\rangle$. This circuit will be then measured in the $X$ and $Y$ basis to obtain, respectively, the real and the imaginary part $(2\\sigma_+=X+iY)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For what concerns `make_transition` function, this exploits `make_overlap` to calculate multiple contributes of Pauli strings within an Hamiltonian operator. More in detail, suppose we have an Hamiltonian $H$ that can be decomposed as:\n",
    "\n",
    "$$H = \\sum_k c_k P_k$$\n",
    "\n",
    "where $P_k$ are Pauli strings (tensor product of Pauli operators acting on different qubits). Then, in order to calcuate the transition element, we can compute the single contributions and sum them up:\n",
    "\n",
    "$$\\langle\\psi _ 0 | H |\\psi _ 1 \\rangle = \\sum_k c_k \\langle\\psi _ 0 | P_k |\\psi _ 1 \\rangle \\quad .$$\n",
    "\n",
    "Each of these contributions can be obtained from `make_overlap` by appending to the $U_1$ circuit the one that implements the Pauli string:\n",
    "$$\\langle\\psi _ 0 | H |\\psi _ 1 \\rangle = \\sum_k c_k \\langle\\psi _ 0 | P_k \\psi _ 1 \\rangle \\quad .$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Objective with 2 unique expectation values\n",
      "total measurements = 2\n",
      "variables          = []\n",
      "types              = not compiled\n",
      "\n",
      "Expectation Value 0:\n",
      "total measurements = 1\n",
      "variables          = 0\n",
      "circuit            = 5588029959664042738_0.png\n",
      "0: ───────X───────H───@───X───────\n",
      "          │       │   │   │\n",
      "1: ───────┼───────┼───X───┼───X───\n",
      "          │       │   │   │   │\n",
      "2: ───H───@───X───@───@───@───@───\n",
      "\n",
      "Expectation Value 1:\n",
      "total measurements = 1\n",
      "variables          = 0\n",
      "circuit            = 5588029959664042738_1.png\n",
      "0: ───────X───────H───@───@───\n",
      "          │       │   │   │\n",
      "1: ───────┼───────┼───X───┼───\n",
      "          │       │   │   │\n",
      "2: ───H───@───X───@───@───@───\n"
     ]
    }
   ],
   "source": [
    "U0 = X(0) # state |10>\n",
    "U1 = H(0)+CX(0,1) # Bell state 1/sqrt(2)*(|00>+|11>)\n",
    "\n",
    "#Hamiltonian definition\n",
    "operator = QubitHamiltonian(\"-1.0*X(0)X(1)+0.5*Z(0)\")\n",
    "\n",
    "br_trans_real, br_trans_im = tq.braket(ket=U1, bra=U0, operator=operator) #consistently with the explained notation\n",
    "\n",
    "tq.draw(br_trans_real)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that both `br_trans_real` and ` br_trans_im` contain in this case 2 circuits with the same gates in the first part of the circuit, but different gates corresponding to the Pauli operators in the Pauli strings."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These two functions were designed according to \"N. H. Stair *et al.* - A Multireference Quantum Krylov Algorithm for Strongly Correlated Electrons\" - [arXiv:1911.05163v1](https://arxiv.org/abs/1911.05163), [published paper](https://pubs.acs.org/doi/10.1021/acs.jctc.9b01125).\n",
    "\n",
    "**ATTENTION**: In the attached ArXiv paper an $X$ gate before the control of $U_\\alpha$ in Figure 2 is missing. This has been fixed in the published paper."
   ]
  }
 ],
 "metadata": {
  "interpreter": {
   "hash": "ee5daa585c8ec561a96d5711013c1c24469ecacbf4bca7c4925c7fd376e166ab"
  },
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.8.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}