TejasAvinashShetty/simple_gate.ipynb

## simple_gate.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Optimization of a State-to-State Transfer in a Two-Level-System"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:33.073705Z",
     "start_time": "2019-10-21T16:05:33.068747Z"
    },
    "attributes": {
     "classes": [],
     "id": "",
     "n": "1"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The watermark extension is already loaded. To reload it, use:\n",
      "  %reload_ext watermark\n",
      "qutip            4.4.1\n",
      "matplotlib       3.1.1\n",
      "scipy            1.3.1\n",
      "numpy            1.17.2\n",
      "matplotlib.pylab 1.17.2\n",
      "krotov           0.3.0\n",
      "CPython 3.7.4\n",
      "IPython 7.8.0\n"
     ]
    }
   ],
   "source": [
    "# NBVAL_IGNORE_OUTPUT\n",
    "%load_ext watermark\n",
    "import qutip\n",
    "import numpy as np\n",
    "import scipy\n",
    "import matplotlib\n",
    "import matplotlib.pylab as plt\n",
    "import krotov\n",
    "%watermark -v --iversions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\newcommand{tr}[0]{\\operatorname{tr}}\n",
    "\\newcommand{diag}[0]{\\operatorname{diag}}\n",
    "\\newcommand{abs}[0]{\\operatorname{abs}}\n",
    "\\newcommand{pop}[0]{\\operatorname{pop}}\n",
    "\\newcommand{aux}[0]{\\text{aux}}\n",
    "\\newcommand{opt}[0]{\\text{opt}}\n",
    "\\newcommand{tgt}[0]{\\text{tgt}}\n",
    "\\newcommand{init}[0]{\\text{init}}\n",
    "\\newcommand{lab}[0]{\\text{lab}}\n",
    "\\newcommand{rwa}[0]{\\text{rwa}}\n",
    "\\newcommand{bra}[1]{\\langle#1\\vert}\n",
    "\\newcommand{ket}[1]{\\vert#1\\rangle}\n",
    "\\newcommand{Bra}[1]{\\left\\langle#1\\right\\vert}\n",
    "\\newcommand{Ket}[1]{\\left\\vert#1\\right\\rangle}\n",
    "\\newcommand{Braket}[2]{\\left\\langle #1\\vphantom{#2} \\mid\n",
    "#2\\vphantom{#1}\\right\\rangle}\n",
    "\\newcommand{op}[1]{\\hat{#1}}\n",
    "\\newcommand{Op}[1]{\\hat{#1}}\n",
    "\\newcommand{dd}[0]{\\,\\text{d}}\n",
    "\\newcommand{Liouville}[0]{\\mathcal{L}}\n",
    "\\newcommand{DynMap}[0]{\\mathcal{E}}\n",
    "\\newcommand{identity}[0]{\\mathbf{1}}\n",
    "\\newcommand{Norm}[1]{\\lVert#1\\rVert}\n",
    "\\newcommand{Abs}[1]{\\left\\vert#1\\right\\vert}\n",
    "\\newcommand{avg}[1]{\\langle#1\\rangle}\n",
    "\\newcommand{Avg}[1]{\\left\\langle#1\\right\\rangle}\n",
    "\\newcommand{AbsSq}[1]{\\left\\vert#1\\right\\vert^2}\n",
    "\\newcommand{Re}[0]{\\operatorname{Re}}\n",
    "\\newcommand{Im}[0]{\\operatorname{Im}}$\n",
    "\n",
    "This first example illustrates the basic use of the `krotov` package by solving\n",
    "a simple canonical optimization problem: the transfer of population in a two\n",
    "level system."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Two-level-Hamiltonian"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We consider the Hamiltonian $\\op{H}_{0} = - \\frac{\\omega}{2} \\op{\\sigma}_{z}$, representing\n",
    "a simple qubit with energy level splitting $\\omega$ in the basis\n",
    "$\\{\\ket{0},\\ket{1}\\}$. The control field $\\epsilon(t)$ is assumed to couple via\n",
    "the Hamiltonian $\\op{H}_{1}(t) = \\epsilon(t) \\op{\\sigma}_{x}$ to the qubit,\n",
    "i.e., the control field effectively drives transitions between both qubit\n",
    "states."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:33.216759Z",
     "start_time": "2019-10-21T16:05:33.074989Z"
    }
   },
   "outputs": [],
   "source": [
    "def hamiltonian(omega=1.0, ampl0=0.2):\n",
    "    \"\"\"Two-level-system Hamiltonian\n",
    "\n",
    "    Args:\n",
    "        omega (float): energy separation of the qubit levels\n",
    "        ampl0 (float): constant amplitude of the driving field\n",
    "    \"\"\"\n",
    "    H0 = -0.5 * omega * qutip.operators.sigmaz()\n",
    "    H1 = qutip.operators.sigmax()\n",
    "\n",
    "    def guess_control(t, args):\n",
    "        return ampl0 * krotov.shapes.flattop(\n",
    "            t, t_start=0, t_stop=5, t_rise=0.3, func=\"sinsq\"\n",
    "        )\n",
    "\n",
    "    return [H0, [H1, guess_control]]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:33.335745Z",
     "start_time": "2019-10-21T16:05:33.222461Z"
    }
   },
   "outputs": [],
   "source": [
    "H = hamiltonian()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The control field here switches on from zero at $t=0$ to it's maximum amplitude\n",
    "0.2 within the time period 0.3 (the switch-on shape is half the first period of\n",
    "a $\\sin^2$ function). It switches off again in the time period 0.3 before the\n",
    "final time $T=5$). We use a time grid with 500 time steps between 0 and $T$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:33.423025Z",
     "start_time": "2019-10-21T16:05:33.341994Z"
    },
    "lines_to_next_cell": 2
   },
   "outputs": [],
   "source": [
    "tlist = np.linspace(0, 5, 500)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:33.497819Z",
     "start_time": "2019-10-21T16:05:33.424352Z"
    },
    "attributes": {
     "classes": [],
     "id": "",
     "n": "10"
    }
   },
   "outputs": [],
   "source": [
    "def plot_pulse(pulse, tlist):\n",
    "    fig, ax = plt.subplots()\n",
    "    if callable(pulse):\n",
    "        pulse = np.array([pulse(t, args=None) for t in tlist])\n",
    "    ax.plot(tlist, pulse)\n",
    "    ax.set_xlabel('time')\n",
    "    ax.set_ylabel('pulse amplitude')\n",
    "    plt.show(fig)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:33.722409Z",
     "start_time": "2019-10-21T16:05:33.498933Z"
    },
    "attributes": {
     "classes": [],
     "id": "",
     "n": "11"
    }
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_pulse(H[1][1], tlist)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Optimization target"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `krotov` package requires the goal of the optimization to be described by a\n",
    "list of `Objective` instances. In this example, there is only a single\n",
    "objective: the state-to-state transfer from initial state $\\ket{\\Psi_{\\init}} =\n",
    "\\ket{0}$ to the target state $\\ket{\\Psi_{\\tgt}} = \\ket{1}$, under the dynamics\n",
    "of the Hamiltonian $\\op{H}(t)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:33.758953Z",
     "start_time": "2019-10-21T16:05:33.723765Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Objective[|(2)⟩ - {[Herm[2,2], [Herm[2,2], u2(t)]]} - |(2)⟩],\n",
       " Objective[|(2)⟩ - {[Herm[2,2], [Herm[2,2], u2(t)]]} - |(2)⟩]]"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "objectives = krotov.gate_objectives(\n",
    "    basis_states=[qutip.ket('0'), qutip.ket('1')],\n",
    "    gate=qutip.operators.sigmay(),\n",
    "    H=H)\n",
    "objectives"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In addition, we would like to maintain the property of the control field to be\n",
    "zero at $t=0$ and $t=T$, with a smooth switch-on and switch-off. We can define\n",
    "an \"update shape\" $S(t) \\in [0, 1]$ for this purpose: Krotov's method will\n",
    "update the field at each point in time proportionally to $S(t)$; wherever\n",
    "$S(t)$ is zero, the optimization will not change the value of the control from\n",
    "the original guess."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:33.831300Z",
     "start_time": "2019-10-21T16:05:33.760402Z"
    },
    "lines_to_next_cell": 2
   },
   "outputs": [],
   "source": [
    "def S(t):\n",
    "    \"\"\"Shape function for the field update\"\"\"\n",
    "    return krotov.shapes.flattop(\n",
    "        t, t_start=0, t_stop=5, t_rise=0.3, t_fall=0.3, func='sinsq'\n",
    "    )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Beyond the shape, Krotov's method uses a parameter $\\lambda_a$  for each control\n",
    "field that determines the overall magnitude of the respective field in each\n",
    "iteration (the smaller $\\lambda_a$, the larger the update; specifically, the\n",
    "update is proportional to $\\frac{S(t)}{\\lambda_a}$). Both the update-shape\n",
    "$S(t)$ and the $\\lambda_a$ parameter must be passed to the optimization routine\n",
    "as \"pulse options\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:33.906866Z",
     "start_time": "2019-10-21T16:05:33.832550Z"
    },
    "lines_to_next_cell": 0
   },
   "outputs": [],
   "source": [
    "pulse_options = {\n",
    "    H[1][1]: dict(lambda_a=5, shape=S)\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulate dynamics under the guess field"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before running the optimization procedure, we first simulate the dynamics under the\n",
    "guess field $\\epsilon_{0}(t)$. The following solves equation of motion for the\n",
    "defined objective, which contains the initial state $\\ket{\\Psi_{\\init}}$ and\n",
    "the Hamiltonian $\\op{H}(t)$ defining its evolution. This delegates to QuTiP's\n",
    "usual `mesolve` function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We use the projectors $\\op{P}_0 = \\ket{0}\\bra{0}$ and $\\op{P}_1 = \\ket{1}\\bra{1}$ for calculating the population:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:33.983684Z",
     "start_time": "2019-10-21T16:05:33.908843Z"
    }
   },
   "outputs": [],
   "source": [
    "proj0 = qutip.ket2dm(qutip.ket(\"0\"))\n",
    "proj1 = qutip.ket2dm(qutip.ket(\"1\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:34.123913Z",
     "start_time": "2019-10-21T16:05:33.985806Z"
    },
    "attributes": {
     "classes": [],
     "id": "",
     "n": "12"
    }
   },
   "outputs": [],
   "source": [
    "guess_dynamics = objectives[0].mesolve(tlist, e_ops=[proj0, proj1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The plot of the population dynamics shows that the guess field does not transfer\n",
    "the initial state $\\ket{\\Psi_{\\init}} = \\ket{0}$ to the desired target state\n",
    "$\\ket{\\Psi_{\\tgt}} = \\ket{1}$ (so the optimization will have something to do)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:34.324997Z",
     "start_time": "2019-10-21T16:05:34.124899Z"
    },
    "attributes": {
     "classes": [],
     "id": "",
     "n": "13"
    }
   },
   "outputs": [],
   "source": [
    "def plot_population(result):\n",
    "    fig, ax = plt.subplots()\n",
    "    ax.plot(result.times, result.expect[0], label='0')\n",
    "    ax.plot(result.times, result.expect[1], label='1')\n",
    "    ax.legend()\n",
    "    ax.set_xlabel('time')\n",
    "    ax.set_ylabel('population')\n",
    "    plt.show(fig)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:34.559859Z",
     "start_time": "2019-10-21T16:05:34.329672Z"
    },
    "attributes": {
     "classes": [],
     "id": "",
     "n": "14"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_population(guess_dynamics)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Optimize"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the following we optimize the guess field $\\epsilon_{0}(t)$ such\n",
    "that the intended state-to-state transfer $\\ket{\\Psi_{\\init}} \\rightarrow\n",
    "\\ket{\\Psi_{\\tgt}}$ is solved, via the `krotov` package's central\n",
    "`optimize_pulses` routine.  It requires, besides the previously defined\n",
    "`objectives`, information about the optimization functional $J_T$ (implicitly,\n",
    "via `chi_constructor`, which calculates the states $\\ket{\\chi} =\n",
    "\\frac{J_T}{\\bra{\\Psi}}$).\n",
    "\n",
    "Here, we choose $J_T = J_{T,\\text{re}} = 1 - F_{\\text{re}}$ with $F_{\\text{re}}\n",
    "= \\Re\\Braket{\\Psi(T)}{\\Psi_{\\tgt}}$, with $\\ket{\\Psi(T)}$ the forward\n",
    "propagated state of $\\ket{\\Psi_{\\init}}$. Even though $J_T$ is not explicitly\n",
    "required for the optimization, it is nonetheless useful to be able to calculate\n",
    "and print it as a way to provide some feedback about the optimization progress.\n",
    "Here, we pass as an `info_hook` the function `krotov.info_hooks.print_table`,\n",
    "using `krotov.functionals.J_T_re` (which implements the above functional; the\n",
    "`krotov` library contains implementations of all the \"standard\" functionals used in\n",
    "quantum control). This `info_hook` prints a tabular overview after each\n",
    "iteration, containing the value of $J_T$, the magnitude of the integrated pulse\n",
    "update, and information on how much $J_T$ (and the full Krotov functional $J$)\n",
    "changes between iterations. It also stores the value of $J_T$ internally in the\n",
    "`Result.info_vals` attribute.\n",
    "\n",
    "The value of $J_T$ can also be used to check the convergence. In this example,\n",
    "we limit the number of total iterations to 10, but more generally, we could use\n",
    "the `check_convergence` parameter to stop the optimization when $J_T$ falls below\n",
    "some threshold. Here, we only pass a function that checks that the value of\n",
    "$J_T$ is monotonically decreasing. The\n",
    "`krotov.convergence.check_monotonic_error` relies on\n",
    "`krotov.info_hooks.print_table` internally having stored the value of $J_T$ to\n",
    "the `Result.info_vals` in each iteration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:39.516811Z",
     "start_time": "2019-10-21T16:05:34.560837Z"
    },
    "attributes": {
     "classes": [],
     "id": "",
     "n": "15"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  iter.        J_T    ∫gₐ(t)dt          J       ΔJ_T         ΔJ  secs\n",
      "      0   1.00e+00    0.00e+00   1.00e+00        n/a        n/a     0\n",
      "      1   1.00e+00    4.23e-31   1.00e+00   0.00e+00   4.23e-31     2 *\n",
      "      2   1.00e+00    4.19e-31   1.00e+00   2.22e-16   2.22e-16     1 **\n"
     ]
    }
   ],
   "source": [
    "oct_result = krotov.optimize_pulses(\n",
    "    objectives,\n",
    "    pulse_options=pulse_options,\n",
    "    tlist=tlist,\n",
    "    propagator=krotov.propagators.expm,\n",
    "    chi_constructor=krotov.functionals.chis_re,\n",
    "    info_hook=krotov.info_hooks.print_table(J_T=krotov.functionals.J_T_re),\n",
    "    check_convergence=krotov.convergence.check_monotonic_error,\n",
    "    iter_stop=1000,\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:39.520961Z",
     "start_time": "2019-10-21T16:05:39.518433Z"
    },
    "attributes": {
     "classes": [],
     "id": "",
     "n": "16"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Krotov Optimization Result\n",
       "--------------------------\n",
       "- Started at 2019-10-21 21:35:34\n",
       "- Number of objectives: 2\n",
       "- Number of iterations: 2\n",
       "- Reason for termination: Reached convergence: Loss of monotonic convergence; error decrease < 0\n",
       "- Ended at 2019-10-21 21:35:39"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "oct_result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulate the dynamics under the optimized field"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Having obtained the optimized control field, we can now plot it and calculate\n",
    "the population dynamics under this field."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:39.832786Z",
     "start_time": "2019-10-21T16:05:39.521982Z"
    },
    "attributes": {
     "classes": [],
     "id": "",
     "n": "17"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_pulse(oct_result.optimized_controls[0], tlist)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In contrast to the dynamics under the guess field, the optimized field indeed\n",
    "drives the initial state $\\ket{\\Psi_{\\init}} = \\ket{0}$ to the desired target\n",
    "state $\\ket{\\Psi_{\\tgt}} = \\ket{1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:40.039082Z",
     "start_time": "2019-10-21T16:05:39.834053Z"
    },
    "attributes": {
     "classes": [],
     "id": "",
     "n": "18"
    }
   },
   "outputs": [],
   "source": [
    "opt_dynamics = oct_result.optimized_objectives[0].mesolve(\n",
    "    tlist, e_ops=[proj0, proj1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-21T16:05:40.321806Z",
     "start_time": "2019-10-21T16:05:40.040331Z"
    },
    "attributes": {
     "classes": [],
     "id": "",
     "n": "19"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_population(opt_dynamics)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "hide_input": false,
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   "display_name": "Python 3",
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   "name": "python3"
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   "title_cell": "Table of Contents",
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   "toc_position": {},
   "toc_section_display": true,
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  "varInspector": {
   "cols": {
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     "delete_cmd_postfix": ") ",
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   "types_to_exclude": [
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 "nbformat": 4,
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