03.many_body_localization.ipynb

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    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Importing standard Qiskit libraries\n",
    "from qiskit import QuantumCircuit, QuantumRegister, transpile, Aer, IBMQ, execute\n",
    "from qiskit.tools.jupyter import *\n",
    "from qiskit.visualization import *\n",
    "from qiskit.providers.aer import QasmSimulator\n",
    "from tqdm.notebook import tqdm\n",
    "\n",
    "from qiskit.providers.aer import QasmSimulator\n",
    "from qiskit.tools.monitor import job_monitor\n",
    "from qiskit.circuit import Parameter\n",
    "import qiskit.quantum_info as qi\n",
    "\n",
    "from qc_grader.challenges.spring_2022.helpers import generate_disordered_tb_instruction\n",
    "\n",
    "# Suppress warnings\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fe9e4c6f",
   "metadata": {},
   "source": [
    "## Many-body quantum dynamics\n",
    "\n",
    "In this exercise we examine lattice disorder and particle-particle interaction. A closed quantum many-body system initialized in a non-equilibrium will reach the equilibrium state, refered to as thermalization, under its own dynamics. This behavior is as a result of the laws of statistical mechanics, and analogous to a hot cup of coffee cooling down to the surronding tempreture if left unattended. \n",
    "\n",
    "However, the presence of lattice disorder prevents the system from evolving into an egodic thermalized state. This interplay between disorder and particle interaction results in many-body localization (MBL), and it allows the system to retain a memory of its initial condition at all times. For more information about MBL refer to [1]\n",
    "\n",
    "Let's again consider the tight-binding Hamiltonian with arbitrary site energies:\n",
    "\n",
    "$$H_{\\rm tb}/\\hbar = \\sum^{3}_{i=0} (X_i X_{i+1} + Y_i Y_{i+1}) + \\sum_i \\epsilon_i Z_i $$\n",
    "\n",
    "As we explored in exercise 2, we can emulate disorder in this system by setting $\\epsilon_i=W \\rm{cos}(2\\pi\\beta i)$, where $W$ is the disorder strength, and $\\beta$ determines the quasicrystal periodicity. We can add multiple particles to the system by initalizing multiple qubits to the $|1\\rangle$ state prior to the unitary time evolution. Under $H_{\\rm tb}$, each site can only be occupied by a single particle, resulting in particle repulsion interaction.\n",
    "\n",
    "[1] https://arxiv.org/abs/1804.11065"
   ]
  },
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   "id": "324e267a",
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   "source": [
    "One of the signatures of the breakdown of thermalization is the emergence of **imbalance**. In a thermalized system, we expect each site of the lattice to be occupied by the same average number of particles after reaching steady state. However, adding disorder to the interacting system will result in deviations from the thermal value observed on each site. The system imbalance can be quantified using the following expression:\n",
    "\n",
    "$$\\mathcal{I}= \\langle (N_e-N_o)/(N_e+N_o) \\rangle$$\n",
    "\n",
    "where $N_e$ and $N_o$ are the even and odd site populations of the system. For a thermalized system this $\\mathcal{I}=0$, and system disorder will cause deviations from this value."
   ]
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   "source": [
    "<div class=\"alert alert-block alert-danger\">\n",
    "    \n",
    "<b>Challenge question 3a</b> \n",
    "\n",
    "Write a function that outputs the imbalance of a quantum state\n",
    "    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
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   "source": [
    "def get_imbalance(state):\n",
    "    ###EDIT CODE BELOW\n",
    "    ### HINT: MAKE SURE TO SKIP CALCULATING IMBALANCE OF THE |00...0> STATE\n",
    "    imbalance_val=0\n",
    "    probs = state.probabilities()\n",
    "    qubits = np.log2(len(probs))\n",
    "    for i in range(1,len(probs)):\n",
    "        if(probs[i]>0):\n",
    "            curr_state = list(bin(i)[2:])\n",
    "            while(len(curr_state)!=qubits):\n",
    "                curr_state.insert(0,'0')\n",
    "            even,odd = 0,0\n",
    "            for j,s in enumerate(curr_state):\n",
    "                if (j%2==0 and s=='1'):\n",
    "                    even+=1\n",
    "                elif(j%2==1 and s=='1'):\n",
    "                    odd+=1\n",
    "            imbalance_val += probs[i]*((even-odd)/(even+odd))\n",
    "    ###DO NOT EDIT BELOW\n",
    "    \n",
    "    return imbalance_val"
   ]
  },
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   "cell_type": "code",
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    {
     "name": "stdout",
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     "text": [
      "-0.3333333333333333\n",
      "0.3333333333333333\n",
      "-0.1111111111111111\n"
     ]
    }
   ],
   "source": [
    "from qiskit.quantum_info import Statevector\n",
    "print(get_imbalance(Statevector.from_label('0111')))\n",
    "print(get_imbalance(Statevector.from_label('1011')))\n",
    "print(get_imbalance(np.sqrt(1/3) * Statevector.from_label('1011') + np.sqrt(2/3) * Statevector.from_label('0111')))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "id": "5f6fdd44",
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Submitting your answer. Please wait...\n",
      "Congratulations 🎉! Your answer is correct and has been submitted.\n"
     ]
    }
   ],
   "source": [
    "## Grade and submit your solution\n",
    "from qc_grader.challenges.spring_2022 import grade_ex3a\n",
    "\n",
    "grade_ex3a(get_imbalance) \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "276339f9",
   "metadata": {},
   "source": [
    "Next, we consider how quantum information and entanglement entropy grows in the system. In an entangled system, ignoring information about the remaining system puts the subsystem into a mixed state of different number states. We can probe the **von Neumann entropy** to gain insight into the degree of entanglement between subsystem $A$ and the remaining of the system:\n",
    "\n",
    "$$\\mathcal{S}_{\\rm vn}(\\rho_A)= -\\rm{tr}(\\rho_A \\rm{ln} \\rho_A)$$\n",
    "\n",
    "Here, $\\rho_A= \\rm{tr}_{\\bar{A}} \\rho$ is the reduced density matrix describing subsytem $A$ when we trace out the rest of the system. If a subsytem $A$ is fully entangled with the rest of the system, $\\mathcal{S}_{\\rm vn}(\\rho_A) = \\rm{ln}2$, whereas if the subsytem is completely separable (in a product state) with respect to its environment $\\mathcal{S}_{\\rm vn}(\\rho_A)=0$. Consider the example below:"
   ]
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   "cell_type": "code",
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   "id": "b0e069e1",
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     "text": [
      "QB0 vn entropy:  0.6931471805599454\n",
      "QB1 vn entropy:  0.6931471805599454\n"
     ]
    }
   ],
   "source": [
    "bell_state = qi.Statevector(np.array([0,1,1,0])/np.sqrt(2))\n",
    "\n",
    "rho_0 = qi.partial_trace(bell_state,[1]) # We trace out qubit 1\n",
    "rho_1 = qi.partial_trace(bell_state,[0]) # We trace out qubit 0\n",
    "\n",
    "print('QB0 vn entropy: ', qi.entropy(rho_0, base=np.exp(1)))\n",
    "print('QB1 vn entropy: ', qi.entropy(rho_1, base=np.exp(1)))"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "## Probing the many-body quantum dynamics in a 12 qubit chain\n",
    "\n",
    "For this part of the exercise, we use the same lattice disorder pattern as we explored in exercise 2."
   ]
  },
  {
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   "source": [
    "t = Parameter('t')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "id": "0ab29437",
   "metadata": {},
   "outputs": [],
   "source": [
    "num_qubits=12\n",
    "deltas=[Parameter('delta_{:d}'.format(idx)) for idx in range(num_qubits)]\n",
    "disorder_trot_step=generate_disordered_tb_instruction(t, deltas, num_qubits)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "id": "e1ae8b4e",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Here we define the disorder pattern\n",
    "\n",
    "beta=(np.sqrt(5)-1)/2 # DO NOT EDIT\n",
    "AA_pattern=np.cos(2*np.pi*beta*np.arange(num_qubits)) # DO NOT EDIT"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "73f306b7",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-danger\">\n",
    "    \n",
    "<b>Challenge question 3b</b> \n",
    "\n",
    "Initialize the system by setting qubits 0, 4, and 8 to $|1\\rangle$.\n",
    "    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "id": "1941aa83",
   "metadata": {},
   "outputs": [],
   "source": [
    "delta_t=0.1\n",
    "time_steps=np.arange(0,21,2)\n",
    "\n",
    "circuits={}\n",
    "Ws=[1,4,10]\n",
    "\n",
    "for W in Ws:\n",
    "    disorders=W*AA_pattern\n",
    "    \n",
    "    circuits[W]=[]\n",
    "\n",
    "    for n_steps in time_steps:\n",
    "\n",
    "        qr = QuantumRegister(num_qubits)\n",
    "        qc = QuantumCircuit(qr)\n",
    "\n",
    "        ###EDIT CODE BELOW\n",
    "        \n",
    "        qc.x(qr[0])\n",
    "        qc.x(qr[4])\n",
    "        qc.x(qr[8])\n",
    "        ###DO NOT EDIT BELOW     \n",
    "\n",
    "        for _ in range(n_steps):\n",
    "            qc.append(disorder_trot_step, [i for i in range(num_qubits)])\n",
    "        \n",
    "        if n_steps!=0:\n",
    "            qc = qc.bind_parameters({t: delta_t})\n",
    "            qc = qc.bind_parameters({deltas[idx]: disorders[idx] for idx in range(num_qubits)})\n",
    "\n",
    "        circuits[W].append(qc)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68d6bb10",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-danger\">\n",
    "    \n",
    "<b>Challenge question 3c</b> \n",
    "\n",
    "Extract the von Neumann entropy of qubit 0 at different evolution time steps for different disorder strengths.\n",
    "    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7daa18a2",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-danger\">\n",
    "    \n",
    "<b>Challenge question 3d</b> \n",
    "\n",
    "Extract the imbalance of the lattice at different evolution time steps for different disorder strengths.\n",
    "    \n",
    "</div>"
   ]
  },
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   "source": [
    "from qiskit import transpile\n",
    "\n",
    "# Use Aer's statevector simulator\n",
    "from qiskit import Aer\n",
    "\n",
    "# Run the quantum circuit on a statevector simulator backend\n",
    "backend_sim = Aer.get_backend('statevector_simulator')\n",
    "\n",
    "probability_densities={}\n",
    "state_vector_imbalances={}\n",
    "vn_entropies={}\n",
    "\n",
    "for W in tqdm(Ws):\n",
    "    probability_densities[W]=[]\n",
    "    state_vector_imbalances[W]=[]\n",
    "    vn_entropies[W]=[]\n",
    "    \n",
    "    for circ in circuits[W]:\n",
    "\n",
    "        transpiled_circ=transpile(circ, backend_sim, optimization_level=3)\n",
    "\n",
    "        job_sim = backend_sim.run(transpiled_circ)\n",
    "\n",
    "        # Grab the results from the job.\n",
    "        result_sim = job_sim.result()\n",
    "        outputstate = result_sim.get_statevector(transpiled_circ, decimals=6)\n",
    "        ps=[]\n",
    "        for idx in range(num_qubits):\n",
    "            ps.append(np.abs(qi.partial_trace(outputstate,[i for i in range(num_qubits) if i!=idx]))[1,1]**2)\n",
    "        \n",
    "        entropy=0\n",
    "        ### EDIT CODE BELOW (extract the density matrix of qubit 0 by tracing out all other qubits)\n",
    "        entropy = qi.entropy(qi.partial_trace(outputstate,[i for i in range(1,num_qubits)]), base=np.exp(1))\n",
    "        \n",
    "        ###DO NOT EDIT BELOW \n",
    "        \n",
    "        imbalance=0\n",
    "        ### EDIT CODE BELOW \n",
    "        imbalance = get_imbalance(outputstate)\n",
    "        \n",
    "        ###DO NOT EDIT BELOW\n",
    "        \n",
    "        \n",
    "        vn_entropies[W].append(entropy)\n",
    "        probability_densities[W].append(ps)\n",
    "        state_vector_imbalances[W].append(imbalance)"
   ]
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\n",
      "text/plain": [
       "<Figure size 1080x360 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axs = plt.subplots(1,3,figsize=(15,5), facecolor='white', sharey=True)\n",
    "\n",
    "for i,W in enumerate(Ws):\n",
    "    ax=axs[i]\n",
    "    ax.pcolormesh(np.arange(0,num_qubits,1), time_steps*delta_t ,probability_densities[W])\n",
    "    ax.set_xlabel('Qubit index')\n",
    "    ax.set_xticks(np.arange(1,num_qubits+1,1))\n",
    "\n",
    "axs[0].set_ylabel('Time (1/J)')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "id": "cff7aa4e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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DF3h9ZQpHMvO5dlC0u0MxDKMFMC0GD3coI4/kA6bT2TAM1zGJwcPNWX8Ui49wU6Ipr20YhmuYxODBikutzN+QxhV9THltwzBcx6WJQUQmisheEUkRkafs7L9HRE6LyBbbnwdcGZ+nOZFTSLtWgdw61Mx0NgzDdVzW+SwiFuANYByQBqwXkYVKqV1VDv1EKTXdVXF5stioYBbPGOnuMAzDaGFc2WIYCqQopQ4qpYqBOcA1Lnx9r3Imr5jcwhJExKzpbBiGS7kyMUQDqRUep9m2VXWDiGwTkfkiYrfHVUSmicgGEdlw+vRpZ8Tqdv9efYCkF1aQV1Tq7lAMw2hhPK3z+Uugq1IqAVgGvGvvIKXUbKVUolIqsW3b5ldQrrzTOal7FCEBZqqJYRiu5crEkA5UbAHE2Ladp5TKVEoV2R7+Fxjiotg8yje7TpCZV2w6nQ3DcAtXJob1QA8R6SYi/sBUYGHFA0SkY4WHU4DdLozPY3z801FTXtswDLdx2X0KpVSpiEwHlgIW4C2l1E4ReQ7YoJRaCMwQkSlAKZAF3OOq+DzFsewCfkjJ5InxPc1MZ8Mw3EKU8u61gxMTE9WGDRvcHYZD7TlxlrahAUSFBrg7FMMwmikR2aiUSrS3z/RseqDeHVq5OwTDMFowTxuV1KJ9veM4M+dsJie/xN2hGIbRgpkWgwd5b+0RjmTmExZo/lkMw3Af02LwEBXLa/uYTmfDMNzIJAYPYcprG4bhKUxi8ACmvLZhGJ7E3Mz2AIWlZdyYGMNlPc2ENsMw3M8kBg/QKtCPp6/s4+4wDMMwAHMrye3SzuTz3d5TWK3ePdHQMIzmwyQGN3t/7RHuf3cDGeeK6j7YMAzDBUxicKOi0jLmbdSdzu1Mp7NhGB7CJAY3WrbrJFmmvLZhGB7GJAY3+uhHXV77UlNe2zAMD2ISg5ucKyrlcEaemelsGIbHMcNV3SQ0wJfVvxpNSZkZjWQYzlBSUkJaWhqFhYXuDsWtAgMDiYmJwc/Pr97PMYnBxRZsTufFpXs4nl1Ip4ggnpzQi2sHRbs7LMNodtLS0ggLC6Nr166ItMxWuVKKzMxM0tLS6NatW72fZ24ludCCzek8/dl2jmUXooD07AKe/mw7Czan1/lcwzAaprCwkKioqBabFABEhKioqAa3mkxicKG/Lt1LQUlZpW0FJWX8deleN0VkGM1bS04K5RrzHpjE4ELHsgsatN0wDMMdTGJwoU4RQQ3abhiG93rsscd45ZVXzj+eMGECDzzwwPnHv/zlL3n55ZfrPM/rr79OfHw8IkJGRoYzQq3GpYlBRCaKyF4RSRGRp2o57gYRUSJid6Fqb/XkhF5YqgxNDfKz8OSEXm6KyDCMcgs2pzPihRV0e2oRI15Y0eS+vxEjRpCcnAyA1WolIyODnTt3nt+fnJxMUlJSvc7z7bff0qVLlybF0xAuSwwiYgHeAK4E+gK3ikhfO8eFATOBH10Vm6tcOyiahy+LIyLIDwGiI4L48/UDzKgkw3Cz8oEh6dkFDhsYkpSUxNq1awHYuXMn/fv3JywsjDNnzlBUVMTu3bsZPHhwnecZNGgQXbt2bXQcjeHK4apDgRSl1EEAEZkDXAPsqnLcH4G/AE+6MDaXeXJCb56c0NvdYRhGi3PLm2urbbsqoSN3Du/Ki1/vsTswZNaXO7l2UDRZecU88sHGSvs/eWh4ra/XqVMnfH19OXr0KMnJyQwfPpz09HTWrl1LeHg4AwYMoKioiKFDh9p9/kcffUTfvtWunV2iwYlBREKAQqVUWZ0HVxYNpFZ4nAZcUuXcg4HOSqlFIlJjYhCRacA0gNhY76kztPv4WfKLSxnUubWZ7WwYHuR4jv3hnNn5JU06b1JSEsnJySQnJ/P444+Tnp5OcnIy4eHhjBgxgrCwMLZs2dKk13CGOhODiPgAU4HbgYuBIiBARDKARcCbSqmUpgZie52XgXvqOlYpNRuYDZCYmOg1U4f/u+YQK/acZOPvxrk7FMNocWq7wu8UEUS6ndGB0baBIZEh/nW2EOwp72fYvn07/fv3p3Pnzvztb3+jVatW3HvvveTm5jJq1Ci7z/X0FsNK4FvgaWCHUsoKICKRwGjgLyLyuVLqgzrOkw5UXOk+xratXBjQH/jONu62A7BQRKYopTbU55fxZEop1h7IYHj3KNNaMAwP8+SEXjz92fZKt5McMTAkKSmJl156ibi4OCwWC5GRkWRnZ7Nz507+85//eGyLoT6dz1copf6olNpWnhQAlFJZSqlPlVI3AJ/U4zzrgR4i0k1E/NGtkIUVzpejlGqjlOqqlOoKrAOaRVIAOJqVz7GcQoZ3b+PuUAzDqOLaQdH8+foBREcEOXRgyIABA8jIyGDYsGGVtoWHh9OmTf2+C1577TViYmJIS0sjISGh0pBXZ6mzxaCUqvMmWz2PKRWR6cBSwAK8pZTaKSLPARuUUgtrP4N3Sz6QCcDwuCg3R2IYhj3XDop2+AhBi8XC2bNnK2175513GnSOGTNmMGPGDAdGVbfGdD7fBkwBygABvlRKfVyf5yqlFgOLq2x7poZjL29obJ7sx4OZtAsLoHvbEHeHYhiGUavGDFe9TCk1tfyBiLwB1CsxtGQv3JBA2pkCU7vFMAyP15jEECAik9FDT2MAU8+hHgL9LMS3C3V3GIZhGHVq0Mxn25DSnwMR6BnMkcB0x4fVvCzZfpwXluyhuNRa98GGYRhuVu/EYOs4PgnsAX4NZCqlPlBK5TsruOZiwZZ0vtp2DH9fU7PQMAzP15Bvql8CA5RS0cAEYISI/ME5YTUfVqti3cEsMxrJMAyv0ZDEcA44BaCUOg7cD9zgjKCak13Hz5JTUEJSvEkMhtGSOKrsdrkZM2YQGuqafsqGJIZ/AfNEJN72OBYwt5HqsPb8/AUzsc0wPNq2ufD3/jArQv+9bW6TTueostsAGzZs4MyZM02KpyHqnRiUUv8EPgT+KyJngBRgr4jcJCI9nBWgtysqLeOizhF0CA90dyiGYdRk21z4cgbkpAJK//3ljCYlB0eV3S4rK+PJJ5/kxRdfbHQsDdWg4apKqc+Az0TEF72mwiAgCXgEGOP48Lzf9DE9mD7G5E3DcLu3J1ff1u9aGPogfPsslFQpoldSAEt+DQk3Q14mzL2r8v57F9X6co4qu/36668zZcoUOnbs2IBftmnqU11VlFKVKpgqpUqBbbY/79qGsRpVWK3KFMwzDG9wtoYFeQqymnTappbdPnbsGPPmzeO7775rUhwNVa/qqiLyKfCFUupo+UZbIbyRwN3oCqzvOCVCL/avVQdYsDmdL38+kkA/i7vDMYyWrbYr/PAY222kqtttBaFDoupsIdjT1LLbhw4dIiUlhfh43bWbn59PfHw8KSlNXumgVvVJDBOB+4CPRaQbkA0EogvhfQO8opTa7LQIvdjaA5lYfMQkBcPwdGOf0X0KFW8n+QXp7U3Q1LLbffv25cSJE+cfh4aGOj0pQP2qqxYC/wT+KSJ+QBugQCmV7eTYvFpRaRnrD2dx2yXes8KcYbRYCTfrv5c/BzlpugUx9pkL2xupvOz2bbfdVmnbuXPn6l122x0a2vlcAhx3UizNyuaj2RSVWkky6y8YhndIuLnJiaAqR5TdrujcuXNNjKh+GlNEDxH5DXq1NQGUUuq2Op7S4qw9kImPwNBuke4OxTAMo0EalRgAMcmgdhfFRvDo6HjCg/zcHYphGEaDNDYxdBeRm4A8OL8Aj1HB6F7tGN2rnbvDMAzDaLDGJoZV6HUYgh0YS7NxLLuAolIrXaOCzcI8hmF4ncZOTJsI+AMLlFLvOjCeZuH9dUcY9/IqCkrK3B2KYRhGgzU2MdwGHAL+JCL1TgwiMlFE9opIiog8ZWf/wyKyXUS2iMj3ItK3kfG5VfKBTC7qHEGwv50GmYMLdXmFlvg7G4YXa2xiiAB6oFdwq9fwVRGxAG+gV37rC9xq54v/I6XUAKXURcCLQP1r0nqIs4UlbE/LJqm7nTLbTijU5fFa4u9sGDiu7Pbrr79OfHw8IkJGRsb57UopZsyYQXx8PAkJCWzatMlhsdcrMYhIkoi8apvgBvA8eiW325RS1a78azAUSFFKHVRKFQNzgGsqHqCUqjjgNwSoVKPJG6w/lIVVwXB78xeWP2e/UNfy51wTnDu0xN/ZtJC80qKDixg/fzwJ7yYwfv54Fh1seAmMihxVdnvEiBF8++23dOnSpdL2JUuWsH//fvbv38/s2bN55JFHmhRvRfVtMdwL7AWsItIeGAE8CgQ04LWigYrFSNJs2yoRkUdF5AC6xTDD3olEZJqIbBCRDadPn25ACM6XfCATf18fBsVGVN+Zk2b/STVtbw5a2u9sWkheadHBRcxKnsXxvOMoFMfzjjMreVaTkoOjym4PGjSIrl27Vtv+xRdfcNdddyEiDBs2jOzsbI4fd8z84/qOSiqzrceAiDwHLACKgGfR6z87jFLqDeANEbkN+B26SF/VY2YDswESExM9qlXx6Oh4rujT3n59pBoLdcU4PzB3CYqAAjsLjDTX37m2FpKDZ9UaDXPv1/dW2zah6wSm9p7KKxtfobCssNK+wrJC/vzTn5kcN5kzhWd4/LvHK+1/e+Lbtb6eo8pu1yQ9PZ3OnTuffxwTE0N6erpDynPXNzEcF5Hh6Kv8RKXUEAAR+bgBr5UOdK7wOMa2rSZz0KvGeZXIEH+G2+tfABj9G1hQpbnngEJdHqvwLJSVgviAsl7Y3px/55bWQmomTuaftLs9pyinSedtatltd6lvYngR+ByIR1daLdeQpT3XAz1sFVrTgano0U3niUgPpdR+28PJwH68yIbDWWw+ms3tw2Ltj0jqMwX2fg1HkyEvA8I6Qs8JzfdKcs1LUJyrE+Km9y+0lgbd2Xx/55pahf7BYLWCTzNdumTbXIcXoHO02q7wO4R04Hhe9dswHUP01XfrwNZ1thDsaWrZ7dpaDNHR0aSmXvispaWlER1d7e58o9TrU6qUKlBKTVRKxSulVgOISDC6VlK92Bb3mQ4sBXYDc5VSO0XkORGZYjtsuojsFJEtwOPYuY3kyb7Ycoy/f7sPP0sNb2tAKNzyHjyZArOyIeEm2PgOHNviwihdpKwUUlbAwNvgsl/DYzvgmSxo0xMO/6C/JJujsc+ApUrXm48vxFwCzXWyYzPoV5k5eCaBlsrL7wZaApk5eGaTzpuUlMRXX31FZGRkpbLba9euJSkp6XyLwd6f2pICwJQpU3jvvfdQSrFu3TrCw8Mdtspboy5fbENPbwXqOyIJ0KUzlFI9lVLdlVLP27Y9o5RaaPt5plKqn1LqIqXUaKXUztrP6FmSD2QwtFuk/cRQdA52fQElFe5jjnwcgqP08oHKo7pKms7iC9NWwpUvXNjmY4FRv4RTO+HACvfF5kx+wVBWBMFtANELvVz7L7jzM50Ysg5V74Pwds1g5NnkuMnMSppFx5COCELHkI7MSprF5Dg7y4E2QHnZ7WHDhlXaFh4eXu+y26+99hoxMTGkpaWRkJBwfsjrpEmTiIuLIz4+ngcffJB//vOfTYq1IlFe/oWUmJioNmzY4O4wOHW2kKF/Ws5vJvVm2qXdqx+w9RP4fBrc+zV0GX5h+8Z39dXVDf+DATe6LmBnytgPYR0gIKz6vrJS2L8Uel7Z/G6rKAVvjoLiPHh0vU6OFRXnw+sXQ2Q3uPVj+++PN5oVgf2R5aJbxm6ye/du+vTp47bX9yT23gsR2aiUSrR3fDP7n+k+aw9mAjA8roargO1zITwWOl9SefugO6BDAix7Rn+heLuyUph7N7x/vf39Fl/oPbn5JQWAvYvhxHa49FfVkwLofoYrZsGRZHh3CuQ3bT1hj1HTCLPmOvKsBWiG/zvdI+1MAVEh/vTt1Kr6znOn4cBKGHBD9S9EHwtc+SJ0GQGlRa4J1pk2vaNvFQ1/tPbjfvoPzLndJSG5hFLw3QsQGQcDbqr5uISbYOqHcHInvD0JznrxuldlJZB5QPer+AVV2SlwWYPuNBsexCQGB3l0dDxrnx6LxcdOB+POz0GVwYAaRml0GQ43/AeCvXxRn4IzsOJ5neT6XlP7sdZS2PMVHFnrmticLfsInD1Wc2uhol5Xwh3zdSetF92HryT3BLx7NbxzFfSaBFe/pvtTEGgVAxP/AoPvcHeUePutckdozHvQ2LLbhh3+vjXk2SM/QPv+0L6OmoAndsC+JXDpk44PzhVWvaiTw8QX6h6BM/huWP2SHtLa5VPXxOdMrbvCL7ZVH5FUk26Xwr1L9PO8zZFkmHcPFOXClH/o0Xb2lsW0WmHZ7/VFQmf7k7icKTAwkMzMTKKiolps+XulFJmZmQQGBtZ9cAUmMTjA55vT+PinVN68YwitQ/yrH3DTO3reQl32LoaVz0PnYdDN/thmj2W1QmYKDL4LOibUfbx/sL7dtPxZSN8E0XWXBvBYOekQ2h78Qxr2vPL3qTgPPn0QLn3Cs98HpWDdv+Cb3+mEdueC2i92CrP1Z3rzBzoJ1nVh5GDlI3k8rWyOqwUGBhIT07D+HjMqyQEe+2QLq/edZsPvrqh+ZaJU/cevlxTA60MhsBVMW1X3LQlPoxSUFYNvPa+aC8/CK/2h6yh9390bKQWzL4OQtnBHI1s+2UfhncmQfwZumwNdRzo2RkexWuGjm3Sr6Lp/QWB43c85cxjemqjfp/uXemcLqZkyo5KcSClF8oEMhne301xVCt6aAGvrOb7YLwjG/xFO7oBNXrT+0bHNkJ2qE2B9kwLoBDj5ZRjRtElEbrXvazi+FfrVMAqrPiJi4b6l0KoTfHAD7PvGcfE5QsZ+3X/i4wM3vwe3fFC/pAA6EdzxGZQWwnvXwrlTzoy05XByBV+TGJroUEYeJ88W2a+PdHwrpP5oZ8RGLfpeA11Gwor/s198ztOUlcBn0+CjWxo3SW/AjW65/+wQ5SORWneFhFuadq5WnfTtlra9Yc6tsPsrh4TYZLu/hNmj4StbATn/kIYPNW7fF26fD0Vn4dQux8fY0rhgprlJDE2UfEDPX0iyt/7C9nng41f3CJ2KRPRs4YvvB4ud/gpPs/5/kLEPxv6+8SUfck/Cl7/QQx+9yb6lcHyLHizgiNt+IVFw95d6nkc7N0/MKiuFZX+AT+6Atj1h8ktNO1/ni2HmNoi7XD/28lvYbpOXCUt/4/SZ5l52E9vztG8VyJSBnegaFVx5h7UMdnwKPcY1fBhqhwH6j6fLz4Lv/gxxo6HnxKada8tHegjrNa87JjZX2D7PMa2FigJb6ds1oL88D6yA+LGOO3995GXC/Hvg0GoYci9c+ZeG3SKsSUCo/nvzB7BnMdz8Llj8an+OJ2ts4cDSIv1/pyAL8jP1n+gh+pbiqd3w/d8vbM/P1H1Pt7wH3cfA4TWQV0NnugMr+JrE0ETj+rZnXN/21Xcc/h5yj8OAPzX+5Cnf6qb8Va94ZgG2lX/SQxYn/rlp8YW116OZNr6jC+5FdK7zKR7h+tm6Ge+sL7ft8+GzB2DUEzDmd677DFh89ZfRNf+EQU6YhFhWAnsXwYKfwXVveucs+PLbOeVX7uW3c0oKwFqi37/yL/aCLJ1g+1ylby+/eWn18137L7joNijJh6PrdA214ChddDI4SldiBj1HKKSt/eTgwJnmJjE0QU5BCUopIoLt3PIJ6whDH9KTmRorI0V/WfYYr28veBKlAAVDH3TMbY8RM2Hj25D8Gkz6a9PP50xK6SGmAaHOHWXT/3o4vFrP9Sg6qyeNOetLVCndwu09WXcsP7RKz8p3hsR79Zfl8ucgqLVukXjihU9taioc+N0LkHtMP/YP03cLgqN0YUXQkwBH/+7C9uAo/XNErN4fPUTPh6lJaFuY8KfKSQkcvsaJGa7aBG99f4g/LtrFj0+PpV2rhk0gqZeyEvj3SD2i49GfHNOcd7SGDMetyxeP6qvkmdt0K8JT7ftGF0S8+0vn3/JTSs8bWPs6JEyFa95w/DDmkgLdubz1I5jwZxj+M8ee356Kv9flv4HLHboQpHMpBc9G1LBT4Jd7ICgSfJ3YR+iA9S9qG65qWgxNkHwgk9jI4OpJ4fhWXV6789CmfWla/PRtmvevg7VvwKjH636OKxz+HnwDISbRsVd6Ix8HsWC/UqeHUEr3qwS00iOInE0Exv8fBEbAqr/A0GkQM8Rx5886BHPv1LPuL3sKLnnYceeuTfnvVXBGr/DnTdb8reZ94TG6srCz2Ztp7kAmMTRSmVXx46FMJg+wszDG6r/C0R/1lYM0sTnefQz0mqzLR1x0u/uvpEuL4IvpuvXyyFrH3tqI6g5TXnPc+Zxh/zI4tkmXgnBVx6kIXPakvrUUZSvp7ojV4A6t0aOOUHDbXOg5vsmhNoiIbgGVX1wU5zV89rirVLx9OOhOyDqga6A58XaOO3lZqvYcO4/lkFtYWn3+QkG2HsbY/wbH3aOd8H/6PmxIW8ecryl+/DecOQQTnnfe/e7U9fqWkqdRCla9oO8HD7zV9a9fnhR2fApvX9n0eS6tOukaXtNWuT4plCtPCie2w6sDPW9yH+jqyJ/cAR/epEcbhrXXncUVCweGd9aPPWw508YyLYZGWmubv1AtMez+UpeFSKil9HJDRcbpP+DYe/oNde4UrPor9JgA8Vc473V+eEUPy+sxXg/f9BRpGyB9o/4CcOcwS4u/brW8c5WeVdyQVmR+Fmz5EIZP14nm3kXOi7MhIrpAq2iYexfc+XnlxazcaecCWPS4XoFxzG8r73Py7Rx3Mi2GRpo0oCMv3zyQdmFV+he2z9Vf4p2cUAxty0fw4Y3uWy95xR+htEC3Fpxp1C+hMAfW/9e5r9NQnS+G+75xT2uhoj5Xw22fQNZBeHuirrVUH8e36rpO3z7reTOQA1vpWlPhMXoW/Ynt7o2nIBvm3w/z7tYtxIdW65Fzzhqp5WFMYmikzpHBXD+4yrjhonO6E2/AzU66qhc9t2HbJ044dz1EdtcdxG16OPd1ogfrFsnaN/RymJ6gPBnHXuLc0Sb11X2Mrm6an6mL1NW1GtzmD+B/4/WtkPu+hvb9XBJmg4S00a2FgFC9AuCZI+6LxcdXJ9LRv4P7l0E7Fww08CAuTQwiMlFE9opIiohUW95JRB4XkV0isk1ElotIF1fGV18HTp/j041p5BWVVt4REKo7nOtavayxEm7R45y//YOeWOZqI39RvTntLKOegPwMzygmqBS8e5Veb8KTxF4C9yzS8wKCWtd83LJn9FDgzkP1lW+M3RGKniGis054PSfoUuauVJhzYX5CQCg8kqw7/b15dnYjuSwxiIgFeAO4EugL3CoiVQu0bwYSlVIJwHzAw/4naou3HeeJ+VspLq1yS0cpPVrHWffFfXz0MqDnTtY+ZM7R9n2jOzxdOeely3DofZVnDGVMWa4XWwpt5+5IquswQNdqEtG3X1b+qXrVzW6XwohfwB2f66tyT9e2py6N4heoW0KuWBv7wEr4Z5IuR3Fojd7mCS1DN3Hl/7qhQIpS6qBSqhiYA1SqLqeUWqmUKr93sA7wyNXEkw9k0rdjq8qL8pzeB/8YokfUOFNMor7HvfYNXXzO2UoKYNEvYc3LoFzctzH1Q7jkIde+ZlXlI5HCO8PA29wbS22Ugnn36rkOVatu5mfBuGe9b30Pq1XP4fnoFj1U1BmKzunP9/vX6uGm9y9z3wgtD+LKxBANpFZ4nGbbVpP7gSX2dojINBHZICIbXL06U2FJGRuPniGp6mik7XP1ME5X1Pm5YpbuqHPFnIa1r0POUT3Rzh0db1Yr7F2iq326w4HlkLZed4h78hWkCJTY+fJ0cNVNl/Lx0avapW/Qw0VLix3/Ggun6wrBwx6Fh9d49m02F/KAdnp1InIHkAjYLZqjlJqtlEpUSiW2bevasf2bjp6huNRaucy2UrrSZrdLXTPrMayDfi3QZTOc5exxWPN3fUun/PVc7eBK+Hgq7HDTvIbVf9OthYucUEzO0c4et7/dgVU3Xa7P1Xp48IEVugyJtazp5ywp0KOOQJfjuGcRTPxTw9ZNaeZcmRjSgYqX0zG2bZWIyBXAb4EpSqkiF8VWbzvSc7D4CBd3q1BKO229XsJwgIvHNP/wql5ExVnJYfmzulLk+P9zzvnrI260noS15m/uGaZ77Rt6MpMntxbK1VRd04FVN91i8J0w7o96pvF3LzTtXGkbdXXTr36hH7ftCV1HNDnE5saViWE90ENEuomIPzAVWFjxABEZBLyJTgoeuQbgtEu78+NvxhIaUOF+7fZ5unZQn6tdG0xUPJzcDhvecs75e4yHsX+AyG7OOX99+PjoGlEZ+2D3wrqPd7TIOOg2yvWv2xhjn6l+1dtcyjSMmKGryybe17jnlxbpW2r/G6f7Kwbf5dj4mhmXJQalVCkwHVgK7AbmKqV2ishzIjLFdthfgVBgnohsERE3fBPUrU1olSqn8eN0vXxXz9LtNUmviLXyeb24iqP1vx6Spjv+vA3V91qdBFe/5LqRUSnLdaenKzr4HSXh5mZdpoFhD0Orjrq/KWV5/Z+XsR/+M0a3OgfeCj9bq+eBGDUyZbcbIPlABu8lH2HWlH50CHdCme3GOLUb/jUChtwDV73smHPuXKBn1Sb93HPGcG/+EFa/CPcshvDaxiw4gFJ6MtjZYzBjk2eWO2/J1r6hl7csX9ymLrkn9aijsX+AXk1cabAZqa3stkd2PnuqVXtPs2LPKcKDKnxZ7lmkSxe7S7s+cPEDuv7NOQeM0CrOg6+fhl1fNL0yrCMl3ALTNzo/KYDu8E77Sd/CMknB81z8gO57+mK6/v9nz6k9ehhqedG7h38wSaEBvGxgs3slH8jkotgIgvxtX5jFefDpgzDgRveWix79tK7TH+qAEVo/vKpXoLrxLc9acrF8DH5JAeRl1GtY8KKDi3h106ucyDtBh5AOzBw8k8lxdayEp5Tu4GwVDYPuaFSojXpdo/58A+CWD+C9KXruRtJ0PZEvJ01fOHQepotZ+ofo9SXa9PCsz7IXMImhnnLyS9hxLIeZYyvUCdq7RI8dH+DASqqNEdT6QkmE/Cy9VGBjZKfqxNDves+pbllR+S2eoAi9elotFh1cxKzkWRSWFQJwPO84s5JnATCk/RBKrCUIgo/44CM++Pn4ERUUBQdXkpO+Huu45/ApK4SyQnzEB18fX4J8dcducVkxgiAi588httpYtb2uSQ4OFBAKt8+HfyVVrgKQkwY586HjRXD7PM+cre4FTGKop3WHMlGKyvMXts/TV5ZdPGS42ze/10P6pq9v3Jjsb/+g/x7noROiRPQtpW9+C6k/6do/dqScSeH5dc+f/3IuV1hWyKubXqVdcDu2nt5aaV9C2wQ+nPQhdLyIe+L7k7JvNuybfX7/8I7DmT1eP77686s5lnes0vPHxo7lldGv8OqmV+2+7l/X/9UkBkcLjgRqKFaZn2mSQhOYxNAAg2MjuKhzhH6Ql6krnQ77mec0U3uMh+TX4IfXGreG7rCf6dEarpi93ViJ9+orxNUvwe1zAcgtzmX9ifWM7jwaEeG9Xe+RW2K/yOCJvBM8M/wZsgqzsCorSikUioiACH1AcCQPDnua7KJsFOr8MR1CLkxcvLf/veQW52JVVqxYQUG38G7nz29PZqEeNVZcVszvvv8d/dv0J6FtAn2i+hBgMf0YjZbbDCf1eQCTGOppQr8OTOhXYVZzum0klCcNBew2CvpeowuBDbq94RObYhI9vySAfwhq2CPs//5Fvk9+njVnU9hyagulqpT5V8+nV2QvHhr4EMnHkjmZX32oaYeQDoyMHln9vErBlzOhzxQmxU+qNYSpvafWuK9DSAeO51X/smofrMuXnMg7wdbTW1lyWFd78fPxo09kH6YPms7wTsOxKuv521RGPYTH2GpD2dluNJqHXOp6tqLSsuqVVHtOgCf261m5nmTcHwGlSy3X15aPYeEMz1n7wI68kjxyinIAWBPTnxtiOvL3/XPILc7l7n53887Ed4iL0KvcRYdG89iQxwi0VB5SHGgJZObgmfZf4NAq2PiOHqbbBDMHz7T7uo8NeQyA2FaxLL1xKStuWsEro1/hjr534GfxI9BXP2d12mpGzx3NjBUz+O/2/7L+xHrySzz338XtmvOkPjcyLYZ6WLbrJE/O28aXPx9JfLvQC8trNraT15lad4GkGXr1s7yMusssF+XqvoWIWI+qFaOU4lDOIdakr2FN+ho2ntzIIwMfYVrCNBJjL+fZgTMY0eNq2ofYr01Vfj+/XqODykcihXXSC703QX1ft21wW8bGjmVs7NhK2yMDIxkRPYJtp7exMnUlAD7iw5fXfklsq1iOnj1KmSqjS6su+NgpSd7iRkSVt9iXP2cblRSjk4InteS9kJngVg+/+Xw7C7ccY8sz4/C1+OhbNfuX6fV2/TxkoltFxflQWli/xPXtLP37PLACYoY4PbTaKKUQEazKyrVfXMuhHD0/JD4inlHRo5jQbQL9oqqsPFacp4clNsXBVXro46SXYOiDTTuXA2UXZrM9Yzs7MnfwUMJD+IgPz659lvn75tPKvxUD2g5gYJuBJLRNIKlTEosPLa40Igp0a2VW0qzmnRyMRqltgptpMdTD2gOZXNItUicFpWDrJxAY7plJAcA/WP+xlulbIzUtxZl1SM8iHXirU5NCTVexSimOnD3C9+nfsyZ9DUVlRbwz8R18xIcru15JVFAUo6JH0TG0o/0T71kEnz2kyyU3tp7T+dZCxya3FhwtIjCCUTGjGBVzoVbTvf3uJaFNAltPb2Vbxjb+tfVfRIdGs+SGJTWOiHp106smMRgNYhJDHY5lF3AoI4/bL4nVG07ugNO7YbILV1BrrEWPw+6vdFmHwPDq+5c/Bz5+ulSAs0KoYVz/T8d/Yv3J9aTm6o7DbuHduDT60vOthkcueqTuk3caDGVF8MMrcPWrjQtQKRh4C/gFe26iryC2VSyxrWK5rsd1gO57OX5Od3bXNCKqpu2GUROTGOqw9oAeZnh+/sK2uXqh8L7XuTGqehpyL2x8V69VPOH56vvHPQv9b9CFyZykpqvY5anLGdh2IHf1vYuR0SOJCWvEKJJWtqv8Te/Bpb9qXLkMHx9dZ8pLhfiFEN86Hqh5RFQrfxcXdzS8nhmVVIeBncN5YnxPencI0+sB7PgUuo+FkKi6n+xunS7Stex//LeuMFnOatVXyhGx0Ocqp718dmG23S8qgLNFZ3lj7BtM7T21cUmh3IiZgNLzNxrq6Dr4cbYuydwM2BsRZRELDwx4AIBDOYdIOZPijtAML2MSQx3i24UxfUwPfHwErKV6UfVhD7s7rPob84y+TfL10xe2bXpHr6VbmOO0l80uzGbiZzUXLetQw2iiBmvdBRKm6lZD4dmGPXf5H/VkOS8fgFFuctxkZiXNomNIRwShY0hHnh/5PPf0vweAN7a8wfULr+fx7x5nb9Ze9wZreDQzKqkWp3OL2HEsh+FxUQT6eVCl0YZKfl2P0b//GxAf+MdgaNML7l2sh906SGpuKj8e/5Ebe94IwAe7PqCorIh/b/23c0fKZKfq0Untetf/OYfWwLtX6cVfvCnRN8GZwjO8v+t9Pt7zMedKznF558t5eODD1Ud6GS2CGZXUSMt3n+Spz7bz7eOXEh8ZANs+0au0BUW4O7SGueQhPXT1zUsvzBKNH+uwpLD/zH7+t+N/fH3oa/x8/BgbO5bWga25o6+uTtohpINzx9ZXLOFRPsekLt+9AKEdYMjdjovDw7UObM2MwTO4p/89fLj7Qz7Y9QHLjyynX1S/853+hgEmMdQq+UAmbcMC6N42FPYuhoXTIawD9Bjn7tAaZufneoRSScGFbWte0n0MTZgIlJabxovrX2Rl6kqCfIO4o88d3NXvLloHtq503OS4yc4fLllWCp89AO36wWVP1n7soTVw5HuY+IJHTepzlVb+rXhk4CPc2efC8NzvUr/jwz0f8lDCQ1zc4WL3BWd4BNPHUAOlFMkHMknqHqWvpLbNheA2eoEQb7P8ucpJAfTj5Q2voqqU4myxvpcf6BvIzoydPDLwEb654RueuPgJ2gW7qaKlxVd3Iq99Xc/mro1vAPSc6NWjkRwh1D+UUP9QAIrKikg5k8J9S+/j7iV3s/bYWrz9NrPReCYx1ODA6XNknCsiqXuU7tTc97VeA9nihY2smipNNqACpVVZWXl0JXcsvoPpy6ejlKJNUBuW3riUn130MyICIxwTa1OMegIKs2HDW7Uf13ko3PZJi2wt1GRit4l8fcPXPDX0KdLOpTFt2TSeXF1Hy8totlyaGERkoojsFZEUEXnKzv5LRWSTiJSKyI2ujK2qtQezANv8hT1f6RIT7l6Qp7FqqjRZjwqUpdZSFh9czA0Lb2DGyhlkFmZyVdxVKPTVpK+PByXKmCG6RZf8evUWUrn1/9M1pIxqAn0Dub3P7Sy5fgm/H/Z7xnQeA+jWxHep35kWRAvissQgIhbgDeBKoC9wq4j0rXLYUeAe4CNXxVWT24bGsmTmKDpHBsOxzdC6K8R46b3XJlSgXJCygF+v+TVKKf408k98dd1X3NzrZrsF3DzCpU9C3ik9fLWqwz/ovpZtc10flxfxt/hzc6+bmRSny48vObSEn6/4OTd+eSNLDy/Fqqx1nMHwdq683BsKpCilDgKIyBzgGmBX+QFKqcO2fW7/5Fl8hD4dbTNGJ/0VCn7r0KGdLtWACpT5JfnM2zeP9iHtmdh1IpPjJhMZGMnlnS/33GRQUdcRuhhe32uq71v1AoS214v9GPV2VdxVWMTC7G2zeWLVE8SFx/FgwoNM6jbJOz4TRoO5MjFEAxVX1EgDLmnMiURkGjANIDY2tumRVbH/ZC7vJB/mkcu7ExMRpBOCtw1RrSrh5lpHIOUU5fDRno/4cPeH5BTlcF38dUzsOpEg3yDGxI5xYaAOYK9C6uEf4NBqmPAn07fQQL4+vlzd/WomdZvEsiPLeHPbm3yw6wMmd9MjzZRSLD60uGWV+27mPOgGcf0ppWYDs0FPcHP0+VftO82HPx5l+ph4eGcyxA6Hsb939Mt4jDl75vD3jX8nvzSfy2Mu5/4B93NRu4vcHVbTHF0HP82G62brAQOrXoCQdrp+lNEoFh8LE7tNZHzX8WQVZiEinCk8w01f3kRWYRYl1hLgQqFEwCQHL+XKdmA6UHEx4RjbNo+z9kAmcW1C6Fh6DI784P2tBXSV0/Hzx5PwbgLj54/n/V3vk1ush3VGBUVxWefLmH/1fP4x9h/enxRAdzDv+BR2fgYlhRAYASMf0+XIjSbxER/aBOmiklmFWZWSQrnyct+Gd3Jli2E90ENEuqETwlTgNhe+fr2Ulln58VAW11zUCbbPA0RXIPVi9kpfv7j+RTaf3MzLo19mXJdxjOviZZP26tJrkl6RbcEj8Nk03a/S52p3R9XsdI/oTqm11O4+U+7be7msxaCUKgWmA0uB3cBcpdROEXlORKYAiMjFIpIG3AS8KSI7XRVfue3pOZwrKiUpLkqPXuk6Elp1cnUYDmWv9DXA1oytbojGRXbMh/wMXfgQpUuBfDnDjEhygpoKInYI6cDSw0vJLsx2bUBGk7l0SIFSarFSqqdSqrtS6nnbtmeUUgttP69XSsUopUKUUlFKKZdX98o8V0x0RBAjgo9C1gGvXju2fNx5TVdup/NPuzIc11r+HJQVV97WyNneRu3slfsOtARyX//7+PXqXzP+0/G8tP4lTuWfclOERkOZsWZVXNG3PT88NYaIdjEw8nGvvP1QYi1h4YGF3PTlTaTlptV6RddsOWC2t1E/9sp9z0qaxdTeU5l/9XzGxo7lg90fMPHTiTy39rnmfUHSTJiy2xVYrQoRvLbKZH5JPp/u/5T3dr3HibwTxEfE82zSs6Tmpra8ReL/3v9CJdmKwjvDYztcH08Ll5qbyts73mbJoSUsvHYhbYPbUlJWgp/Fz92htVi1ld02iaGCdQczmTlnMx9eFUJ8YC50H+M1tZGKyoqY+OlEMgoyGNJ+CPf1v49R0aPOJ7lFBxe1rHHm2+bqPoWKpTH8guDq17z69qC3yy/JJ9hPjwy7b+l9hPmF8WDCg/Rv09/NkbU8Zj2Gelp7IJPTuUXE7psH+5fAE/s8OjGknk1lZepK7up3FwGWAB5OeJjeUb0Z2HZgtWNdUvrakzRgtrfhOuVJocxaxpD2Q/hw94esWLSC4R2H82DCgyS2T/TaFntzYloMFdz877VYS/KZn3s39LsGrnnDIed1tF2Zu3hrx1ssO7IMi1j46rqv6BTq3SOnjJbpXPE55u6by3s73yOzMJM/jfwTV3f3vn49b2RaDPVQUFzG5tQzvNjnIGTmemQl1bTcNJ5d+yzrjq8j1C+Uu/vdzR197nDfGgiG0USh/qHc1/8+but9GwsPLGRs7FgAVqetJr80n3Gx47D4ePGyul7KJAabDUeyKClTjCpYqZd87DrK3SEBuuz1qfxTdArtRERABKfyT/HYkMe4qedNhPmHuTs8w3CIQN9Abu514TbfvH3z+C71O7q06sL9/e/nqrirTEe1C5lbSTZ7Tpzl47UH+cORu/HpcxVMeN4B0TVeYWkhC1IW8O7Odwn0DeTTKZ/iIz5mbV6jRSizlrEidQX/2fYfdmftpkNIB36Z+Esmdp3o7tCaDXMrqR56d2jFs9ddBNbNUJLvtjhyinKYs2cOH+35iKzCLBLaJnBf//vO7zdJwWgJLD4WxnUZxxWxV/DDsR/4z7b/YLXqavz5JfmUqTLTYnYikxiA/OJSDpzKo2/HUCwWCwSEui2W1WmreX3L64yKHsV9/e9jSPshJhkYLZaIMDJ6JCOjR56fyf/Rno94a/tbTO09lTv63sHaY2tb1lBsFzCJAT1M9XfvLmVV6z9iueFfEH+F016r6nyCW3rdwoHsA/SO7M1d/e5iYreJ9I7sTY/WPZwWg2F4o/ILpJHRI9mVuYv/bv8vb+94G4WiTJUBpuS3o5iSGEDygUyu9VuHf8EpiIxz2uuUVzk9nnccheJ43nFe2fQKXx/6+vysZD8fP5MUDKMWvSN78/LlL7PgmgX4WfzOJ4VypuR305nEgE4MtwSu1Ws6OzEx1FTlNDIokmkJ05z2uobRHMVFxFFYWv3/E+iWw7s73yX1rJ2yKEadWnxiOJNXTOmJXXQtOeiUuQupZ1N5a8db3PrVrTVWOTVVJw2jcWoqBOnr48tLG15i0ueTuO6L69idudvFkTlX1YW3Fh1c5NDzt/g+hnUHM7nW8j1KLEi/6x1yzpyiHD7Z+wnLjixjT9YeAPpH9ScyMJLMwsxqxzfrKqeG4UQzB8+ssUDkwLYDWZm6ku9Sv6N9SHsAvjzwJVtObWF07GiGdhiKv8XfTZE3nr2Ftxzdr9Li5zHkF5eyd/1yEtiHZcTPG32eA9kHyCvJI6FtAjlFOYyeO5q+UX31kLsuVxAdGl3tHxRaQJVTw3CyhhSI/PfWf/PWjrcoKC0gxC+EEZ1GMDZ2LJPiJrk46oZJy03j68NfszdrL8uOLKvWrwLQMaQj39z4Tb3PaaqrOoFSin1n9vHNkW9YdmQZh3IOMbjdYN698l1AtxrCA8KrPa/FVTk1DA9TVFbEj8d/ZMXRFXyX+h2dwzrz/qT3AVh+ZDl9ovq4pfaYVVlJz01nd9Zu9mTtYU/WHu7seyfDOw3np+M/cf839xMdGk36uXS7zxeEbXdvq/frmcRQg9O5Raz96m0uGTKY9j2HNui5v179axYfWoyP+HBx+4u5ossVjI0dS9vgto2KxTAM17MqK1mFWbQJakNeSR4j54yk1FpKr9a9GB07mjGdx9A7srfD5xKVlJWQkp1CoG8g3cK7cfzcca5feD3nSs4BYBELcRFxPDrwUcZ2GUtxWTEFpQWEB4Qzfv54jucdr3ZO02KooCmJYeGmw4z8YgQ+3UcTcdcHdo+xKivbM7az7PAyVqau5MNJHxIRGMHKoyvJLMxkTOwYIgMjm/IrGIbhIY6cPcLKoytZmbqSzac2o1D86uJfcWffOykpKwE4X7OpIa1/pRQf7fmI3Zm72XtmLynZKZRaS7mp5008M/wZyqxlvLj+RXq27knvqN7ER8QTYAmwey5H3ZI2JTFqkLllCZFyjrKLb6u279i5Y7y/632WHVnGyfyT+Pr4ktQpiZziHCICIxgdO9oNERuG4UxdWnXhnv73cE//e8gqzGJV6iou7nAxACtTVzIreRYjY0YS4R/B5ymfV+sAzinKISYs5vytoMjASH437HeICO/vep+C0gI9mbXvXfSO7M2ANgMAXQLk6UuerleM5V/+zrwl7dIWg4hMBF4FLMB/lVIvVNkfALwHDAEygVuUUodrO2djWgyvzXucL3KWctpX6FBaxtWtJvDIjS+x6eQmQvxC6NemH6lnU7lu4XUkdUpiXJdxXNb5Mlr5t2rQ6xiG0XzszNzJJ3s+YVXaKrIKs+o8vnNYZ0Z0GsFvh/0WgLPFZwnzC/OYEjcecStJRCzAPmAckAasB25VSu2qcMzPgASl1MMiMhW4Til1S23nbWhieG3e47x/bimFPhemcFiUwlf8KaKEK7teyYuXvQhUXobQMAwDdOXXi96/qMb970x8h16texHq776aa/XhKbeShgIpSqmDtqDmANcAuyoccw0wy/bzfOB1ERHlwOz1Rc5SCv0qz+srE8HXWszLY/7OiE4jzm83ScEwjKosPhY6hnSssQN4SPshbojKsVw58zkaqDg/Pc22ze4xSqlSIAeIqnoiEZkmIhtEZMPp06cbFMRpX/vNuGKBcV3GmWRgGEadZg6eSaAlsNK2QEsgMwfPdFNEjuWVJTGUUrOVUolKqcS2bRs2PLRtqf3GR03bDcMwqpocN5lZSbPoGNIRQegY0rFZTVR15a2kdKBzhccxtm32jkkTEV8gHN0J7TDXhE+o1scQaLVyTfgER76MYRjN3OS4yc0mEVTlyhbDeqCHiHQTEX9gKrCwyjELgbttP98IrHBk/wLAjJte5s7QCbQrsSJK0a7Eyp2hE5hx08uOfBnDMAyv5bIWg1KqVESmA0vRw1XfUkrtFJHngA1KqYXA/4D3RSQFyEInD4ebcdPLzHDGiQ3DMJoBl05wU0otBhZX2fZMhZ8LAcfXvjYMwzDqzSs7nw3DMAznMYnBMAzDqMQkBsMwDKMSkxgMwzCMSry+7LaInAaONPLpbYAMB4bjKCauhjFxNZynxmbiapimxNVFKWV3hrDXJ4amEJENNRWRcicTV8OYuBrOU2MzcTWMs+Iyt5IMwzCMSkxiMAzDMCpp6YlhtrsDqIGJq2FMXA3nqbGZuBrGKXG16D4GwzAMo7qW3mIwDMMwqjCJwTAMw6ik2SYGEZkoIntFJEVEnrKzP0BEPrHt/1FEulbY97Rt+14RcehCDfWI63ER2SUi20RkuYh0qbCvTES22P5ULVnu7LjuEZHTFV7/gQr77haR/bY/d1d9rpPj+nuFmPaJSHaFfc58v94SkVMisqOG/SIir9ni3iYigyvsc8r7VY+YbrfFsl1EkkVkYIV9h23bt4hI/RdRd1xsl4tIToV/r2cq7Kv1M+DkuJ6sENMO22cq0rbPKe+ZiHQWkZW274GdIlJtWTinf76UUs3uD7qs9wEgDvAHtgJ9qxzzM+Dftp+nAp/Yfu5rOz4A6GY7j8WFcY0Ggm0/P1Iel+3xOTe+X/cAr9t5biRw0PZ3a9vPrV0VV5Xjf44u5+7U98t27kuBwcCOGvZPApYAAgwDfnTB+1VXTEnlrwVcWR6T7fFhoI0b36/Lga+a+hlwdFxVjr0avUaMU98zoCMw2PZzGLDPzv9Hp36+mmuLYSiQopQ6qJQqBuYA11Q55hrgXdvP84GxIiK27XOUUkVKqUNAiu18LolLKbVSKZVve7gOvdKds9Xn/arJBGCZUipLKXUGWAZMdFNctwIfO+i1a6WUWo1eM6Qm1wDvKW0dECEiHXHi+1VXTEqpZNtrgus+W+WvXdf7VZOmfDYdHZdLPl9KqeNKqU22n3OB3UB0lcOc+vlqrokhGkit8DiN6m/s+WOUUqVADhBVz+c6M66K7kdfFZQLFJENIrJORK51UEwNiesGW7N1voiUL9PqEe+X7ZZbN2BFhc3Oer/qo6bYnfl+NUTVz5YCvhGRjSIyzQ3xAAwXka0iskRE+tm2ecT7JSLB6C/YTytsdvp7JvoW9yDgxyq7nPr5culCPUb9icgdQCJwWYXNXZRS6SISB6wQke1KqQMuCulL4GOlVJGIPIRubY1x0WvXx1RgvlKqrMI2d75fHktERqMTw8gKm0fa3qt2wDIR2WO7mnaVTeh/r3MiMglYAPRw4evX5WrgB6VUxdaFU98zEQlFJ6JfKKXOOuq89dFcWwzpQOcKj2Ns2+weIyK+QDiQWc/nOjMuROQK4LfAFKVUUfl2pVS67e+DwHfoKwmXxKWUyqwQy3+BIfV9rjPjqmAqVZr5Tny/6qOm2J35ftVJRBLQ/37XKKUyy7dXeK9OAZ/juNun9aKUOquUOmf7eTHgJyJtcPP7VUFtny+Hv2ci4odOCh8qpT6zc4hzP1+O7jjxhD/oltBB9K2F8g6rflWOeZTKnc9zbT/3o3Ln80Ec1/lcn7gGoTvbelTZ3hoIsP3cBtiPgzrh6hlXxwo/XwesUxc6uw7Z4mtt+znSVXHZjuuN7ggUV7xfFV6jKzV3pk6mcufgT85+v+oRUyy6zyypyvYQIKzCz8nAREe+V/WIrUP5vx/6C/ao7b2r12fAWXHZ9oej+yFCXPGe2X7v94BXajnGqZ8vh/7De9IfdK/9PvSX7G9t255DX4UDBALzbP9RfgLiKjz3t7bn7QWudHFc3wIngS22Pwtt25OA7bb/GNuB+10c15+BnbbXXwn0rvDc+2zvYwpwryvjsj2eBbxQ5XnOfr8+Bo4DJej7uPcDDwMP2/YL8IYt7u1AorPfr3rE9F/gTIXP1gbb9jjb+7TV9m/8W0e+V/WMbXqFz9c6KiQve58BV8VlO+Ye9ICUis9z2nuGvsWngG0V/q0mufLzZUpiGIZhGJU01z4GwzAMo5FMYjAMwzAqMYnBMAzDqMQkBsMwDKMSkxgMwzCMSkxiMFosEYmqUDnzhIik234+JyL/dNJr/kJE7rL9fJOteqZVRKot6G4rtRBgq+LZRkT8RWS1bUKmYTiN+YAZLZbSM38vAhCRWehqrC856/VsX+j3oat5AuwArgfetHNsNyBd6RIk5fEWi8hy4BbgQ2fFaRimxWAYVdjWBvjK9vMsEXlXRNaIyBERuV5EXrTV4f/aVroAERkiIqtsV/lLbZUuqxoDbFK6aCNKqd1Kqb01hDER+NrO9gXA7U3+JQ2jFiYxGEbduqO/1KcAHwArlVIDgAJgsi05/AO4USk1BHgLeN7OeUYAG+v5mjUlhh3AxQ0L3zAaxtxKMoy6LVFKlYjIdvTCMeVf2NvRdXZ6Af3RFTaxHXPcznk6omvr10pE/IEYpYv/VaKUKhORYhEJU7pWv2E4nEkMhlG3IgCllFVEStSFOjJW9P8hAXYqpYbXcZ4CdI2uuowCvq9lfwBQWI/zGEajmFtJhtF0e4G2IjIcdMnkCgvNVLQbiK/H+SZSeRGd80QkCshQSpU0NljDqItJDIbRREovOXkj8BcR2Yquhplk59Al6DWGARCR60QkDRgOLBKRpbZdlwOrKjzPF1urBb0m+CJHxm8YVZnqqobhQiLyOfArpdT+GvbHAP9RSl1pe9wW2KKUirY9/gx4Sim1z1UxGy2PaTEYhms9he6EtksplVYhKUwB1gBP2x77AwtMUjCczbQYDMMwjEpMi8EwDMOoxCQGwzAMoxKTGAzDMIxKTGIwDMMwKjGJwTAMw6jk/wHLB1YTSpn7mwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "for W in Ws:\n",
    "    plt.plot(time_steps*delta_t,vn_entropies[W], '--o', label='W={:d}'.format(W))\n",
    "\n",
    "plt.xlabel(r'Time (1/J)')\n",
    "plt.ylabel(r'$\\mathcal{S}_{\\rm vn}(\\rho_0)$')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "id": "c98610e8",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Submitting your answer. Please wait...\n",
      "Congratulations 🎉! Your answer is correct and has been submitted.\n"
     ]
    }
   ],
   "source": [
    "## Grade and submit your solution\n",
    "from qc_grader.challenges.spring_2022 import grade_ex3b\n",
    "\n",
    "grade_ex3b(vn_entropies)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "id": "9ed1e6a0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "for W in Ws:\n",
    "    plt.plot(time_steps*delta_t,state_vector_imbalances[W], '--o', label='W={:d}'.format(W))\n",
    "\n",
    "plt.xlabel(r'Time (1/J)')\n",
    "plt.ylabel(r'$\\mathcal{I}$')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "id": "a141777d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Submitting your answer. Please wait...\n",
      "Congratulations 🎉! Your answer is correct and has been submitted.\n"
     ]
    }
   ],
   "source": [
    "## Grade and submit your solution\n",
    "from qc_grader.challenges.spring_2022 import grade_ex3c\n",
    "\n",
    "grade_ex3c(state_vector_imbalances)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f679af34-b3bf-40e0-81bd-7d261f3983b5",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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