2D Quantum Walk_Hadamard walk.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "d2b09d61",
   "metadata": {},
   "source": [
    "\n",
    "==> Hadamard walk in 1D, but represented in a 2D space\n",
    "\n",
    "#### Coin operator\n",
    "\n",
    " $C \\ = \\ \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 && 1 \\\\ -1 && 1 \\end{pmatrix} $\n",
    " \n",
    "#### Translation operators \n",
    " $Tx \\ =  Ix \\otimes Iy \\otimes Ip $\n",
    "  \n",
    "  $Ty \\ = Ix \\otimes \\ \\displaystyle\\sum_{y} \\ \\lvert y \\ + \\ 1\\rangle\\langle y\\lvert  \\ \\otimes \\ \\lvert R\\rangle\\langle R\\lvert  \\ + Ix \\otimes \\ \\displaystyle\\sum_{y} \\ \\lvert y \\ - \\ 1\\rangle\\langle y\\lvert \\otimes \\ \\lvert L \\rangle\\langle L\\lvert  \\ $\n",
    "\n",
    "#### Protocol \n",
    "$U \\ = \\ Ty Tx (C \\otimes Ix \\otimes Iy) $ \n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "d1470c4c",
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy import *\n",
    "import matplotlib as mpl\n",
    "from matplotlib.pyplot import *\n",
    "import pandas as pd\n",
    "from scipy.sparse import csr_matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "0273649f",
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 2  # number of random steps\n",
    "P = 2*N+1   # number of positions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "8536daae",
   "metadata": {},
   "outputs": [],
   "source": [
    "ket0 = array([1, 0])  # |0>\n",
    "ket1 = array([0, 1])  # |1>\n",
    "\n",
    "C00 = outer(ket0, ket0)  # |0><0| \n",
    "C01 = outer(ket0, ket1)  # |0><1| \n",
    "C10 = outer(ket1, ket0)  # |1><0| \n",
    "C11 = outer(ket1, ket1)  # |1><1| \n",
    "\n",
    "I = C00 + C11  #identity"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "3bce9709",
   "metadata": {},
   "outputs": [],
   "source": [
    "#Coin\n",
    "C = (C00 + C01 - C10 + C11)/sqrt(2.) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "4d35ca6e",
   "metadata": {},
   "outputs": [],
   "source": [
    "iden = csr_matrix((P, P),dtype = np.int8).toarray()\n",
    "for i in range(P):\n",
    "    iden[i][i] = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "45afe068",
   "metadata": {},
   "outputs": [],
   "source": [
    "def Plot2D(qw, steps):\n",
    "    figure(1)\n",
    "    ax1 = subplot(111)\n",
    "    sca(ax1)\n",
    "    title('2D distribution of %s steps Quantum Walk with hadamard coin' % steps)\n",
    "    xlabel('X Position (started in center)')\n",
    "    imshow(qw)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d77050e0",
   "metadata": {},
   "source": [
    "### Translation operators"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "30a2c196",
   "metadata": {},
   "source": [
    "#### Tx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "28e43e40",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(50, 50)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LPs = roll(iden, 1, axis=0)\n",
    "LP = kron(iden,kron(LPs, C00))\n",
    "\n",
    "LMs = roll(iden, -1, axis=0)\n",
    "LM = kron(iden, kron(LMs, C11))\n",
    "\n",
    "Tx = LP + LM\n",
    "shape(Tx)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7e88b8f4",
   "metadata": {},
   "source": [
    "#### Ty"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "fa2b7f18",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "50"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "range_ = kron(iden, kron(iden, eye(2)))\n",
    "rangee = shape(range_)\n",
    "rangeee = rangee[0]\n",
    "rangeee"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "b40eaed8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(50, 50)"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Making Ty = Iy\n",
    "Ty = csr_matrix((rangeee, rangeee), dtype = float).toarray()\n",
    "for i in range(rangeee):\n",
    "    Ty[i][i] = 1\n",
    "\n",
    "shape(Ty)    "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "90159d84",
   "metadata": {},
   "source": [
    "### Protocol \n",
    "\n",
    "$Coin = Ix \\otimes Iy \\otimes C $\n",
    "\n",
    "$U = Ty Tx Coin$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "ae2f7681",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(50, 50)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Protocol\n",
    "Coin = kron(eye(P), kron(eye(P),C))\n",
    "U = Ty.dot(Tx.dot(Coin))\n",
    "\n",
    "shape(U)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "31087574",
   "metadata": {},
   "source": [
    "### Initial State"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "eaba2d19",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(50,)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "initialPositionx = zeros(P)\n",
    "initialPositionx[N] = 1\n",
    "initialPositiony = zeros(P)\n",
    "initialPositiony[N] = 1\n",
    "\n",
    "psi0 = kron(initialPositionx,kron(initialPositiony,(ket0+ket1*1j)/sqrt(2.)))\n",
    "    \n",
    "shape(psi0)    "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ac9a6f5",
   "metadata": {},
   "source": [
    "### Final State"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "f7923340",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(50,)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#State after N random quantum steps.\n",
    "psiN = linalg.matrix_power(U, N).dot(psi0) \n",
    "\n",
    "shape(psiN)    "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c48ce0d6",
   "metadata": {},
   "source": [
    "### Measurement operator and position on the map"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "198c0030",
   "metadata": {},
   "outputs": [],
   "source": [
    "xy_probability = csr_matrix((P, P), dtype = float).toarray()\n",
    "for x in range(P):\n",
    "    positionx = zeros(P)\n",
    "    positionx[x] = 1\n",
    "    for y in range(P):\n",
    "        positiony = zeros(P)\n",
    "        positiony[y] = 1\n",
    "        for p in range(2): \n",
    "            polarization = zeros(2)\n",
    "            polarization[p] = 1\n",
    "            posn = kron(positionx,positiony) \n",
    "            M_hat_k = M_hat_k =kron(outer(posn,posn), eye(2)) \n",
    "            proj = M_hat_k.dot(psiN)\n",
    "            xy_probability[x][y] = proj.dot(proj.conjugate()).real\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "a326c226",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.  , 0.  , 0.  , 0.  , 0.  ],\n",
       "       [0.  , 0.  , 0.  , 0.  , 0.  ],\n",
       "       [0.25, 0.  , 0.5 , 0.  , 0.25],\n",
       "       [0.  , 0.  , 0.  , 0.  , 0.  ],\n",
       "       [0.  , 0.  , 0.  , 0.  , 0.  ]])"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xy_probability"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "3eebd42f",
   "metadata": {},
   "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": [
    "Plot2D(xy_probability,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "33a38105",
   "metadata": {},
   "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\n",
    "fig = figure()\n",
    "ax = fig.add_subplot(111)\n",
    "\n",
    "plot(arange(P), xy_probability)\n",
    "title(\"N = 2\")\n",
    "xlim(0, P)\n",
    "\n",
    "show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "c001d419",
   "metadata": {},
   "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": [
    "Plot2D(xy_probability,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "bdca1387",
   "metadata": {},
   "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\n",
    "fig = figure()\n",
    "ax = fig.add_subplot(111)\n",
    "\n",
    "plot(arange(P), xy_probability)\n",
    "title(\"N = 1\")\n",
    "xlim(0, P)\n",
    "\n",
    "show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9dffe6ee",
   "metadata": {},
   "source": [
    "==>  Hadamard walk in 1D, but represented in a 2D space\n",
    "\n",
    "#### Coin operator\n",
    "\n",
    " $C \\ = \\ \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 && 1 \\\\ -1 && 1 \\end{pmatrix} $\n",
    " \n",
    "#### Translation operators \n",
    " $Tx \\ = \\ \\displaystyle\\sum_{x} \\ \\lvert x \\ + \\ 1\\rangle\\langle x\\lvert  \\ \\otimes Iy \\otimes \\lvert R\\rangle\\langle R\\lvert  \\ \\otimes \\ + \\displaystyle\\sum_{x} \\ \\lvert x \\ - \\ 1\\rangle\\langle x\\lvert \\otimes Iy \\otimes\\ \\lvert L \\rangle\\langle L\\lvert  \\  $\n",
    " \n",
    "  $Ty \\ =  Ix \\otimes Iy \\otimes Ip  \\ $\n",
    "#### Protocol \n",
    "$U \\ = \\ Ty Tx (C \\otimes Ix \\otimes Iy) $ \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "189ee950",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(50, 50)"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Translation operator in y\n",
    "\n",
    "LPlus_ = eye(P)\n",
    "LPlus_ = LPlus_.tolist()\n",
    "LPluss = [row[1:] + row[:1] for row in LPlus_]\n",
    "LPlus = kron(LPluss, kron(eye(P), C00))\n",
    "\n",
    "LMinus_ = eye(P)\n",
    "LMinus_ = LMinus_.tolist()\n",
    "LMinuss = [row[-1:] + row[:-1] for row in LMinus_]\n",
    "LMinus = kron(LMinuss, kron(eye(P), C11))\n",
    "\n",
    "#Tx\n",
    "Ty = LPlus + LMinus\n",
    "\n",
    "shape(Ty)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "f340fed1",
   "metadata": {},
   "outputs": [],
   "source": [
    "#Making Tx = Ix\n",
    "Tx = csr_matrix((rangeee, rangeee), dtype = float).toarray()\n",
    "for i in range(rangeee):\n",
    "    Tx[i][i] = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "b86d625a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(50, 50)"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Protocol\n",
    "Coin = kron(eye(P), kron(eye(P),C))\n",
    "U = Ty.dot(Tx.dot(Coin))\n",
    "\n",
    "shape(U)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "id": "6bca961a",
   "metadata": {},
   "outputs": [],
   "source": [
    "initialPositionx = zeros(P)\n",
    "initialPositionx[N] = 1\n",
    "initialPositiony = zeros(P)\n",
    "initialPositiony[N] = 1\n",
    "\n",
    "psi0 = kron(initialPositionx,kron(initialPositiony,(ket0+ket1*1j)/sqrt(2.)))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "id": "05817960",
   "metadata": {},
   "outputs": [],
   "source": [
    "#State after N random quantum steps.\n",
    "psiN = linalg.matrix_power(U, N).dot(psi0) \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "id": "1ea7ee2e",
   "metadata": {},
   "outputs": [],
   "source": [
    "xy_probability = csr_matrix((P, P), dtype = float).toarray()\n",
    "for x in range(P):\n",
    "    positionx = zeros(P)\n",
    "    positionx[x] = 1\n",
    "    for y in range(P):\n",
    "        positiony = zeros(P)\n",
    "        positiony[y] = 1\n",
    "        for p in range(2): \n",
    "            polarization = zeros(2)\n",
    "            polarization[p] = 1\n",
    "            posn = kron(positionx,positiony) \n",
    "            M_hat_k = M_hat_k =kron(outer(posn,posn), eye(2)) \n",
    "            proj = M_hat_k.dot(psiN)\n",
    "            xy_probability[x][y] = proj.dot(proj.conjugate()).real"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "90c76bc4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.  , 0.  , 0.25, 0.  , 0.  ],\n",
       "       [0.  , 0.  , 0.  , 0.  , 0.  ],\n",
       "       [0.  , 0.  , 0.5 , 0.  , 0.  ],\n",
       "       [0.  , 0.  , 0.  , 0.  , 0.  ],\n",
       "       [0.  , 0.  , 0.25, 0.  , 0.  ]])"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xy_probability"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "id": "786f98d2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYAAAAEWCAYAAABv+EDhAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjMuNCwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8QVMy6AAAACXBIWXMAAAsTAAALEwEAmpwYAAAch0lEQVR4nO3debgdVZ3u8e9rEhKGSJgEQmKCDCJOUdMMF5AICIFG6W58FBBacEjbVxqwURvEAURtpG0EL6jQYqOMgtJIowwRCGEKhCEgCMpMkCFhCCEyh9/9Y61Dip09nGEndU7W+3me85xdu6Zf1a6qt1bVPnUUEZiZWXneVHcBZmZWDweAmVmhHABmZoVyAJiZFcoBYGZWKAeAmVmhBkUASHpQ0k759Vcl/bSL014k6W359WmSvt3Faf9E0te7Nb0+zPefJT2Rl22t5T1/G5wkHSnpjPx6oqSQNLyL0/+kpMva9J8i6ZE+TG+GpM92qbb9JV3TjWl1W1/XSy+nuZ2kPw10Om0DQNJISadKekjSc5JulbRrpf8USa/lA9EiSY9IOlfS3/S3oIj4bkR03Ch6u/FExGoRcX9/66nMb6kNLCI+HxFHD3TafaxjBHAcsHNetqca+m8q6TeS5kt6WtKlkt7ej/l0faPtw7zHSPqxpMclPS/pD5I+tZzm3fUDZ4f5XSrpK5XuDfL8m7233vKoqZWIODMidq7UFZI2rrOmUkXE1RHR5/26UacWwHBgLrA9sDrwdeBcSRMrwzwaEasBo4GtgLuBqyXtONDiBmJ57cA1WBcYBdzZov8Y4ELg7XnYG4HfLJfKukDSSsDvgQnA1qTt7svAsZIOqrO2ZWQmaf/q8UHSPtT43j0R8fjyLMz6bsgddyKiTz/A7cCe+fUU4JEmw5wI3NRmGvsBDwFPAUcADwI75X5HAmfk16OAM/JwC4DZpIPad4DFwIvAIuDEPHwAXwDuAR6ovLdxfn0a8BNgOvAccBUwIfebmIcdXqlzBvBZ4B15Xovz/BZUpvftyvCfA+4FniYdhMdW+gXw+VzbM8BJgFqsn5HA8cCj+ef4/N6mwF/ztBYBV/Ti81ozD79Wi/67AX/M6+MvwJeAVYEXgNfyfBYBY0knDIcB9+XP5FxgzYb1Ny3X/BhwaGU+WwA3AQuBJ4DjWtTzGWAesGrD+5/I445u/FwbPwtgDeAiYH5e1xcB4xo+16OBa/NyXwasnfs9XFm/i0ghdCR5m2y2reTpfRu4Lo/zv8BawJm55tnAxBbLux1p235T7v4R8E95HVXf+2l+fQLppGwhcDOwXWVar9fZpMY9SfvZu5rUcBVL9ult83i75e6dgDn59f7ANfn1zDzcX/Myf4J8PAAOzZ/hY8ABbbbNlp9D7n8e8DjwbJ7fOyv91iLtYwtJJzlH99TWy/V0HunY8hzwB9K+dXiuey6phd0z/AHAXXnY+4F/qvTrWeZ/y7WeDqxM2h6fIe1bX6bJcbIyjXeSjklP58/9q+2OA82Ovfmz/RLp+Pws8EtgVKfjQ5/uAUhaN6+oVmefPc4H3i9p1SbT2Bz4MSkExpI+yHEtpvMp0hng+Dzc54EXIuII4GrgwEiXQQ6sjPN3wJbA5i2m+UnSxrI2MIe0k7YVEXfleV+f5zemyXLtAPw78HFgfVLAndMw2O7A3wDvzcPt0mKWR5BaU5PysFsAX4uIP5M2FoAxEbFDp9pJZ4+PR8OloopTSRv0aOBdpFD5K7AruXWXfx4FDiKt3+1Jn11PkFV9CNgE2Bk4rOfeDmmHPCEi3gxsRAqPZj4MXJxrqPo1sAppvXTyJuC/Sa2It5LC7MSGYfYh7dhvAVYi7TyQ1hek9btaRFzfi/kB7EXapjcgLd/1uYY1SQePb7YY70bSjv7eyvynk04kqu/NzK9nk7aLNYGzgPMkjWpXmKQDgO+RTrLuaDLIVaQDSs+87mdJC+SDuf8bRETPenpvXk+/zN3rkfbZDUhhfpKkNdqU1+pzALiYtC29BbiFN+6rJ5FOytYHPp1/qjqtp4+QDtZrALcCl5K2mw2AbwEnV4adR9p335xr/YGk91f6r5fnM4F0AvRN0jawEWkfb3n5UtJoUov3EtI+tTFwee7d9DjQalqkY8pUYEPgPaTAbq9TQlQSZkQu9OTG9Gsy7Gaks4MNmvT7BnBOpXtV4GWatwA+TTqrek+Ls4fPNrwXwA5N3qu2AKrzXo10Vj+eNi2AxrOfSv/TWHLWeSpwbMO0XyGf+eVpb1vpfy5wWIt1fR/5DCx37wI82OzMrsNnNo50Vr93m2EeJp1xvrnh/aU+W9KBbMdK9/p5GYdX6tqs0v9Y4NRYcsZ4FJUzvBb1/B44pkW/x4F9Gj/Xxs+iyXiTgGcaPtevVbr/L3BJq/VL71oAR1T6/ycpxHq6P0I+i25R3wzgYNJB5JH83jGV914jt1SbjPsM6SD8hjorNX6JdBY6rs38dwRuz68vIbV6Z+Xuq4B/aLYPNPkMppDCtrru5gFbtVnupp9Dk2HH5PmtDgzL2111W/suDftnh/U0veHzWQQMy92j87zGtJjWBcDBlWV+mcrZNilAp1a6p9GiBQDsDdzaol+748AUlm4B7Nuw7/2k3b4W0csWgKQ3kdLyZeDADoNDStEgNW0bjSU1sQCIdKbX6uz0dFIynyPpUUnH5pug7cztbf+IWERqdo3tME5vjCWd9Ven/RRpXfSoXsN9nhQSHaeVX/epRknrkJrUP4qIs9sMuifpMtBDkq6StHWbYScA/yNpgaQFpEBYTLos16O6/qt1f4bUerxb0mxJu7eYx5OkYGlcnuGkVtv8NvX1DLuKpJPzlxcWksJnjKRhlcF6+1n01hOV1y806W43/ZmkM+3tgJ4vGlxTeW9uRDwEIOlQSXdJejZ/BquT1ksrXwZOioh2N/SvBzbNLfxJwC+A8ZLWJp11zmwzbqOnIuLVSnenddv0c5A0TNIxku7Ln+GDeZi1gXVYcn+yR3V/6c16avx8noyIxZVuKrXsKmlW/lLFAtL+Up3W/Ih4sdL9hmNcY20NxpMO9M309TjQ5226YwBIEunsdl3SdcJXOo0D/D1wSyzdjId0XXB8ZfqrkC7vLCUiXomIoyJic+D/kJph/9jTu8W8W73fozrv1UhnWI+SrmVCuszQo/qti07TfZR0gOyZ9qqk5fpLh/E6Tot0GePR3o6cm9yXARdGxHfaDRsRsyNiD1Iz+wKWXJpptrxzgV0jYkzlZ1REVJdxfOX163VHxD0RsXeez/eAXzW7REhqAezapN+epLO+G3P387T+rA4l3QTfMtIlp57LFWoyv0bNlvuvbebVDTNJB/oPki5tQrouvg2Vyz+StiNda/44sEakS5HP0n65dga+JmnPVgNExPOk6+QHA3dExMuklve/AvdFxJP9XrL+2wfYg3QPYnVSiwbSss4HXmXpbS0N0L/11JSkkaTLj98H1s3T+l3DtBq3mTcc46q1NTGXdKmomQEdB3qjNy2AH5Nugn4kIl5oNZCSDSR9k9SE/GqLQX8F7C5p2/yNj2+1qkPShyS9O5+5LSQdAHpS+gngbb2ov9FulXkfDdwQEXMjYj7pYL1vPvv4NG/8YJ4AxuXxmjkLOEDSpLzRfDdP+8F+1Hg2aaddJ5+FfYN0w6ojSW8mtZqujYjDOgy7ktJ3u1fPwb6QN67ftSStXhnlJ8B3JE3I468jaY+GyX49n4G/k3S99Jd52H0lrRMRr7GkZbiYpZ1Ouql2Xv5K5ghJuwA/JF1iezYPNwfYJ39WU3njt2ZGk87iFkhak9bX35uZT7rkUt225gAflPTWvD4O78P0euM60iWOfckBEBHP5Fr2ZckZ+GjSgW8+MFzSN0jXpdu5k3Rd+CRJH20z3FWk1n3P9f4ZDd3N9Hcf7I3RwEukVvQqpP0JgHymfj5wZN7WNueN19n7s55aWYl0j2Y+8KrS1+B3bj8K5wKHS1pD0jjgX9oMexGwnqRDlL52P1rSlrlfv48DvdXp7wAmkK4PTwIe15Lv+3+yMthYST3fmJgNvBuYEhFN/2AkIu4kfVPnLFJSPkPa4ZtZjxQYC0mXG65iyQo4AfiYpGck/bDTglacRTogPA18gHRTuMfnSE3mp0g3W6+r9LuCtDM9LmmpM6KIuJz0Ndlf5+XaiHRjsD++TfrGzO2kbyjckt/rjb8n3Wg+oPJ5LZLU6ixkP+DB3Mz+POmAQ0TcTdoA78+XfMaS1vmFwGWSngNmkW64V11FuoF5OfD9ynYwFbgzbysnAHs1NJvJ832JdNY3F7iBdCC/hPQNiKMqgx5Muna7gPQZXlDpdzzpmxhP5hovabHsS8lnw98Brs3LvVVETCcF2e2kM+WLeju9PszzZtKBpnqT9mpSi6knAC4l3Rj9M+lywIt0vuRJRNxGaj3/lyp/x9PgKtKBc2aL7maOBH6e19PHO9XRR78gLeNfSPcwZjX0P5B0ieNx0v2f/67069d6aiYiniN9+eFc0rFqH9I+0M5Reb4PkFrip3eY/odJ2/LjpG8Jfij3HshxoFeUbxiYDYjS34Y8AIxouAY80OmOIO3MfwH2D2+wZl0zKB4FYdZKvjS1J+lG2YD/8tHMlnALwLpiWbUAzGzZcQCYmRXKl4DMzAo1tB5cNMitpJEximZfbbeBemn80FqvI+c2+xMY64bneObJiFin7jpWBA6ALhrFqmxZ70NQV1j3HtqbRwANHht/sfFbi9Ytv49ftfvLWusDXwIyMyuUA8DMrFAOADOzQjkAzMwK5QAwMyuUA8DMrFAOADOzQjkAzMwK5QAwMyuUA8DMrFAOADOzQjkAzMwK5QAwMyuUA8DMrFAOADOzQjkAzMwK5QAwMyuUA6ADSVMl/UnSvZIOq7seM7NucQC0IWkYcBKwK7A5sLekzeutysysOxwA7W0B3BsR90fEy8A5wB4112Rm1hUOgPY2AOZWuh/J771O0jRJN0m66RVeWq7FmZkNhAOgPTV5L97QEXFKREyOiMkjGLmcyjIzGzgHQHuPAOMr3eOAR2uqxcysqxwA7c0GNpG0oaSVgL2AC2uuycysK4bXXcBgFhGvSjoQuBQYBvwsIu6suSwzs65wAHQQEb8Dfld3HWZm3eZLQGZmhXIAmJkVygFgZlYoB4CZWaEcAGZmhXIAmJkVygFgZlYoB4CZWaEcAGZmhXIAmJkVygFgZlYoB4CZWaEcAGZmhXIAmJkVygFgZlYoB4CZWaH8D2FsSNj4i7PqLsFsheMWgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAbUj6maR5ku6ouxYzs25zALR3GjC17iLMzJYFB0AbETETeLruOszMlgUHgJlZoYbXXcBQJ2kaMA1gFKvUXI2ZWe+5BTBAEXFKREyOiMkjGFl3OWZmveYAMDMrlAOgDUlnA9cDb5f0iKTP1F2TmVm3+B5AGxGxd901mJktK24BmJkVygFgZlYoB4CZWaEcAGZmhXIAmJkVygFgZlYoB4CZWaEcAGZmhXIAmJkVygFgZlYoB4CZWaEcAGZmhXIAmJkVygFgZlYoB4CZWaEcAGZmhfI/hLEh4dJH59RdQp/sMnZS3SWYdeQWgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgC0IWm8pCsl3SXpTkkH112TmVm3+H8Ct/cqcGhE3CJpNHCzpOkR8ce6CzMzGyi3ANqIiMci4pb8+jngLmCDeqsyM+sOtwB6SdJE4H3ADQ3vTwOmAYxileVfmJlZP7kF0AuSVgN+DRwSEQur/SLilIiYHBGTRzCyngLNzPrBAdCBpBGkg/+ZEXF+3fWYmXWLA6ANSQJOBe6KiOPqrsfMrJscAO1tA+wH7CBpTv7Zre6izMy6wTeB24iIawDVXYeZ2bLgFoCZWaEcAGZmhXIAmJkVygFgZlYoB4CZWaEcAGZmhXIAmJkVygFgZlYoB4CZWaEcAGZmhXIAmJkVygFgZlYoB4CZWaEcAGZmhXIAmJkVygFgZlYo/0MYGxJ2GTup7hLMVjhuAZiZFcoBYGZWKAeAmVmhHABmZoVyAJiZFcoBYGZWKAeAmVmhHABmZoVyAJiZFcoBYGZWKAeAmVmhHABmZoVyAJiZFcoBYGZWKAeAmVmhHABmZoVyAJiZFcoB0IakUZJulHSbpDslHVV3TWZm3eJ/CdneS8AOEbFI0gjgGkkXR8SsugszMxsoB0AbERHAotw5Iv9EfRWZmXWPLwF1IGmYpDnAPGB6RNxQc0lmZl3hAOggIhZHxCRgHLCFpHdV+0uaJukmSTe9wku11Ghm1h8OgF6KiAXADGBqw/unRMTkiJg8gpF1lGZm1i8OgDYkrSNpTH69MrATcHetRZmZdYlvAre3PvBzScNIYXluRFxUc01mZl3hAGgjIm4H3ld3HWZmy4IvAZmZFcoBYGZWKAeAmVmhHABmZoVyAJiZFcoBYGZWKAeAmVmhHABmZoVyAJiZFcoBYGZWKAeAmVmhHABmZoVyAJiZFcoBYGZWKAeAmVmhHABmZoXyP4SxIeHeH2xVdwl9svEXZ9VdgllHbgGYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHQAeShkm6VdJFdddiZtZNDoDODgbuqrsIM7NucwC0IWkc8LfAT+uuxcys2xwA7R0PfAV4rdUAkqZJuknSTa/w0nIrzMxsoBwALUjaHZgXETe3Gy4iTomIyRExeQQjl1N1ZmYD5wBobRvgo5IeBM4BdpB0Rr0lmZl1jwOghYg4PCLGRcREYC/giojYt+ayzMy6xgFgZlao4XUXMBRExAxgRs1lmJl1lVsAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRQRddewwpA0H3hoGUx6beDJZTDdZWUo1TuUaoWhVe+yqnVCRKyzDKZbHAfAECDppoiYXHcdvTWU6h1KtcLQqnco1VoqXwIyMyuUA8DMrFAOgKHhlLoL6KOhVO9QqhWGVr1DqdYi+R6AmVmh3AIwMyuUA8DMrFAOgEFM0lRJf5J0r6TD6q6nHUk/kzRP0h1119IbksZLulLSXZLulHRw3TW1ImmUpBsl3ZZrParumjqRNEzSrZIuqrsWa80BMEhJGgacBOwKbA7sLWnzeqtq6zRgat1F9MGrwKER8Q5gK+ALg3j9vgTsEBHvBSYBUyVtVW9JHR0M3FV3EdaeA2Dw2gK4NyLuj4iXgXOAPWquqaWImAk8XXcdvRURj0XELfn1c6SD1Qb1VtVcJIty54j8M2i/vSFpHPC3wE/rrsXacwAMXhsAcyvdjzBID1BDnaSJwPuAG2oupaV8SWUOMA+YHhGDtlbgeOArwGs112EdOAAGLzV5b9Ce9Q1VklYDfg0cEhEL666nlYhYHBGTgHHAFpLeVXNJTUnaHZgXETfXXYt15gAYvB4Bxle6xwGP1lTLCknSCNLB/8yIOL/uenojIhYAMxi891u2AT4q6UHSZcsdJJ1Rb0nWigNg8JoNbCJpQ0krAXsBF9Zc0wpDkoBTgbsi4ri662lH0jqSxuTXKwM7AXfXWlQLEXF4RIyLiImkbfaKiNi35rKsBQfAIBURrwIHApeSblCeGxF31ltVa5LOBq4H3i7pEUmfqbumDrYB9iOdoc7JP7vVXVQL6wNXSrqddGIwPSL89UobMD8KwsysUG4BmJkVygFgZlYoB4CZWaEcAGZmhXIAmJkVygGwgstPvXxA0pq5e43cPaHJsIvz1yHvkHSepFX6OK+xkn6VX0+qfq1S0ke79URTSStLuio/MK/VMF/t57QP6cdyT2n21EtJkyX9sD91DNQAlv8cSZt0ux4bnPw10AJI+gqwcURMk3Qy8GBE/HuT4RZFxGr59ZnAzf39IylJ+wOTI+LAAZTeatpfAIZHxAlthnl9Wfow3WHAfaS6n+zDeFOAL0XE7n2Z37I0gOXfFtg3Ij63bCqzwcQtgDL8ANhK0iGkHfw/ezHO1cDGktaUdIGk2yXNkvQeAEnbV/6A6lZJoyVNzK2HlYBvAZ/I/T8haX9JJ+ZxJ0i6PE/zcklvze+fJumHkq6TdL+kj7Wo7ZPAb/I460uaWWm5bCfpGGDl/N6ZebgLJN2cn6c/rWdCkhZJ+pakG4AjgLGkP7q6MvffWdL1km7JraKegJwq6W5J1wD/0KzIastA0pFK/zNhRl62g1qMMzXP6zZJl+f3Vs3jzs7reo/8/v6Szpd0iaR7JB2b32+2/Psq/U+BOZJO7mk9NSz/1vlz30nS8HYbh60gIsI/BfwAu5AeJvfhNsMsyr+Hkw6w/wz8P+Cb+f0dgDn59f8C2+TXq+VxJgJ35Pf2B06sTPv17jzup/LrTwMX5NenAeeRTkw2Jz0Ou7HGlYDHK92HAkfk18OA0dVlqQy3Zv69MnAHsFbuDuDjleEeBNbOr9cGZgKr5u5/A74BjCI9qXUT0kP7zgUualLrlJ73gSOB64CRebpPASMahl8nT3fDhpq/SzorBxgD/BlYNa/T+4HVc00PAeMblx94R17nI3L3j4B/bLb8+b3pwAfq3mb9s+x/3AIox67AY0C7p0iurPTI4ZuAh0nPytkWOB0gIq4A1pK0OnAtcFw+kx0T6dEVvbU1cFZ+fXqeR48LIuK1iPgjsG6TcdcGFlS6ZwMHSDoSeHekZ/s3c5Ck24BZpIfs9VznXkx6IFwzW5GC6Nq8Xj4FTAA2Ax6IiHsiIoDePuzstxHxUqTLS/NYevm2AmZGxAMAEdHz/xV2Bg7LNcwgHezfmvtdHhHPRsSLwB9zfY12BD4AzM7T2BF4W+7XbPnnkVpCtoJzM68AkiYBHyYdYK6RdE5EPNZk0BciPXK4Om7Tx1JHxDGSfgvsBsyStBPwYj9LrN6Ieqk6+2Y1kg6APYXMlPRB0j8gOV3Sf0TELxqWYQrpAWpbR8TzkmZUpvFiRCxuUZdIz93Zu2F6k+jfo7mry7aYpfc/tZiugD0j4k8NdWzZi2n2jP/ziDi8Sb9myz+KtJ5tBecWwAouH8B/THre/cPAfwDf78MkZpKuufccSJ+MiIWSNoqIP0TE90gths0axnsOGN1imteRnhRJnvY1vS0mIp4BhkkalWuaQHr+/H+RWizvz4O+ovS4Z0iXSJ7JB//NSEHYSrXuWcA2kjbO81pF0qakJ3FuKGmjPNzeS0+mX64Htpe0YZ7fmvn9S4F/6QljSe/rxbSqy3858DFJb+mZrpp8C6xiU2DQPnjQuscBsOL7HPBwREzP3T8CNpO0fS/HPxKYrPQkymNIl0EADsk3XW8jnS1e3DDelcDmPTeBG/odRLpsczvpiZx9/Yfsl7HkstEUYI6kW4E9gZ5vBp0C3J5vgl4CDM/zO5p0YG/lFOBiSVdGxHzSdfaz87izgM3y5ZZpwG/zTeCH+lh/U3l+04Dz83r9Ze51NOnfQN4u6Y7c3cnry58vp30NuCwvx3TSE0aXImldUkuwWQvRVjD+GqgNOfkM+F8jYr+6a1nRSPoisDAiTq27Flv23AKwIScibiV9VbPlH4JZvy0Afl53EbZ8uAVgZlYotwDMzArlADAzK5QDwMysUA4AM7NCOQDMzAr1/wGmjgCl2P1Z6wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Plot2D(xy_probability,2) "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7247a146",
   "metadata": {},
   "source": [
    "<b>Protocol</b>: Hadamard walk 2D\n",
    "\n",
    "#### Coin operator\n",
    "\n",
    " $C \\ = \\ \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 && 1 \\\\ -1 && 1 \\end{pmatrix} $\n",
    " \n",
    "#### Translation operators \n",
    " $Tx \\ = \\ \\displaystyle\\sum_{x} \\ \\lvert x \\ + \\ 1\\rangle\\langle x\\lvert  \\ \\otimes Iy \\otimes \\lvert R\\rangle\\langle R\\lvert  \\ \\otimes \\ + \\displaystyle\\sum_{x} \\ \\lvert x \\ - \\ 1\\rangle\\langle x\\lvert \\otimes Iy \\otimes\\ \\lvert L \\rangle\\langle L\\lvert  \\  $\n",
    " \n",
    "  $Ty \\ = Ix \\otimes \\ \\displaystyle\\sum_{y} \\ \\lvert y \\ + \\ 1\\rangle\\langle y\\lvert  \\ \\otimes \\ \\lvert R\\rangle\\langle R\\lvert  \\ + Ix \\otimes \\ \\displaystyle\\sum_{y} \\ \\lvert y \\ - \\ 1\\rangle\\langle y\\lvert \\otimes \\ \\lvert L \\rangle\\langle L\\lvert  \\ $\n",
    "#### Protocol \n",
    "$U \\ = \\ Ty Tx (C \\otimes Ix \\otimes Iy) $ \n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "748cfb6f",
   "metadata": {},
   "source": [
    "#### Tx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "8ea3e683",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(50, 50)"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LPs = roll(iden, 1, axis=0)\n",
    "LP = kron(iden,kron(LPs, C00))\n",
    "\n",
    "LMs = roll(iden, -1, axis=0)\n",
    "LM = kron(iden, kron(LMs, C11))\n",
    "\n",
    "Tx = LP + LM\n",
    "shape(Tx)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b17f172a",
   "metadata": {},
   "source": [
    "#### Ty"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "4b34ad59",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(50, 50)"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LPlus_ = eye(P)\n",
    "LPlus_ = LPlus_.tolist()\n",
    "LPluss = [row[1:] + row[:1] for row in LPlus_]\n",
    "LPlus = kron(LPluss, kron(eye(P), C00))\n",
    "\n",
    "LMinus_ = eye(P)\n",
    "LMinus_ = LMinus_.tolist()\n",
    "LMinuss = [row[-1:] + row[:-1] for row in LMinus_]\n",
    "LMinus = kron(LMinuss, kron(eye(P), C11))\n",
    "\n",
    "#Tx\n",
    "Ty = LPlus + LMinus\n",
    "\n",
    "shape(Ty)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bb1b0431",
   "metadata": {},
   "source": [
    "#### Protocol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "id": "f273ad15",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(50, 50)"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Protocol\n",
    "Coin = kron(eye(P), kron(eye(P),C))\n",
    "U = Ty.dot(Coin.dot(Tx.dot(Coin))) \n",
    "\n",
    "shape(U)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1a06c666",
   "metadata": {},
   "source": [
    "#### Initial State"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "id": "b0c80dcc",
   "metadata": {},
   "outputs": [],
   "source": [
    "initialPositionx = zeros(P)\n",
    "initialPositionx[N] = 1\n",
    "initialPositiony = zeros(P)\n",
    "initialPositiony[N] = 1\n",
    "\n",
    "psi0 = kron(initialPositionx,kron(initialPositiony,(ket0+ket1*1j)/sqrt(2.)))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a64d9bfb",
   "metadata": {},
   "source": [
    "#### Final State"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "id": "529b30df",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(50,)"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#State after N random quantum steps.\n",
    "psiN = linalg.matrix_power(U, N).dot(psi0) \n",
    "\n",
    "    \n",
    "shape(psiN)  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ca04ae8f",
   "metadata": {},
   "source": [
    "#### Measurement"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "id": "6546bc1a",
   "metadata": {},
   "outputs": [],
   "source": [
    "xy_probability = csr_matrix((P, P), dtype = float).toarray()\n",
    "for x in range(P):\n",
    "    positionx = zeros(P)\n",
    "    positionx[x] = 1\n",
    "    for y in range(P):\n",
    "        positiony = zeros(P)\n",
    "        positiony[y] = 1\n",
    "        for p in range(2): \n",
    "            polarization = zeros(2)\n",
    "            polarization[p] = 1\n",
    "            posn = kron(positionx,positiony) \n",
    "            M_hat_k = M_hat_k =kron(outer(posn,posn), eye(2)) \n",
    "            proj = M_hat_k.dot(psiN)\n",
    "            xy_probability[x][y] = proj.dot(proj.conjugate()).real\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "id": "6609b282",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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MyZqjoxkV04Ip9xvdIPM978yNMb83xiwAHgC+faMdGWOeMcaUG2PKc3NzxxVURd7ISGZ5/nKyU7Jtxwkpt8vNgfMHaOlusR1FqbAIptybgNFXiSoCztxsY2PMDmC2iET/gDbOHW07SmNnY0yskrmex+XBYK5dUkGpWBNMue8D5opImYgkAY8AL4/eQETmiIgMf34nkARcDHVYFVm+Oh8JksCGkg22o4Sca6qLBdkLdDSjYtaY5W6MGQS+CPiAGuA5Y8wREXlMRB4b3uxjwNsi8hZXV9Z8fNQ/sKooNDKSuafgHqYmT7UdJyzcLjcHWw9ytuus7ShKhVxQ69yNMa8YY+YZY2YbY747/NjTxpinhz//O2PMYmPMMmPMCmPMrnCGVuF3+MJhznSficrL+wZrZAWQvntXsUjPUFU35PV7SXQksq5kne0oYVM8pZjF0xfrZYBVTNJyV+8RMAF8fh8rC1aSmZRpO05YuV1ujlw8QmNn49gbKxVFtNzVexxsPcj5K+dxl8XOiUs3o6MZFau03NV7eOu8JDuTWVccuyOZEQUZBSzJXaLlrmKOlrt6l6HAEJX1lawuXE16YrrtOBHhcXk41nYMf4ffdhSlQkbLXb3L/vP7udBzIaauJTOWkatd6rt3FUu03NW7+Pw+UhNSWVO0xnaUiMlPz+fOGXfqqhkVU7Tc1TWDgUE2129mTdEa0hLTbMeJKLfLTe2lWk5dOmU7ilIhoeWurtl3bh9tvW1xNZIZsal0E4LoaEbFDC13dY3P7yMtIY3VhattR4m43LRcyvPL8fq96JUzVCzQclcADAQGqGqoYm3xWlISUmzHscLj8lDXUceJ9hO2oyh1y7TcFQBvnH2Djr6OmL6WzFg2lm7EIQ4dzaiYoOWugKsnLmUkZrCycKXtKNZkp2SzPH85Pr9PRzMq6lkr98a2K/oLNEkMDA2wtWEr60vWk+RMsh3HKo/LQ0NnAzVtNbajqGH7ffXseekUJqB9MR7Wyv1SzwDPVevFmiaD1868RudAZ1yukrnehpINJEiCrnmfJC40dfHGy6fpaO1BHDe646e6GWvlnp6UwLf/UENT+xVbEdQwr99LZlImK2ausB3FuqyULO4uuJtKf6X+ZWnZ0GCAqp8dJTktgTWfmGc7TtSxVu5F01IxxvDlFw4R0D+3rOkb6mNb4zY2lGwg0ZloO86k4HF5aO5q5u0Lb9uOEteqX/FzsamLdZ9aQGpGfI8LJ8JauSclOPjafYt47dRFfvVGva0YcW9X8y66B7rjepXM9daXrCfRkaijGYvO11/mTW898+/Jp2xpru04UcnqapmPf6CYtfNz+d4rx/Bf6LYZJW756nxkJWexfOZy21EmjcykTFYWrKSyvpKACdiOE3cGB4ao+j9HSctMYvXDc23HiVpWy11E+P5fLCHRKTzx/EGGdDwTUT2DPWxv2s7G0o0kOBJsx5lUKlwVnOs+x6HWQ7ajxJ29L9fRfu4K6z+9gOQ0HRVOlPV17vlTU/jm/Yuprm/np7tO244TV3Y27aRnsEdHMjewrngdSY4kHc1E2NnaSxyoamDx6gJKFk+3HSeqWS93gAeWFeJenMdTlSc42dJpO07c8Pq9TE+ZTnleue0ok05GUgari1ZT6a9kKDBkO05cGOgbYsvPa5iSncIHPzbHdpyoNynKXUT47kdvJyM5gcefP8jgkM45w+3KwBV2Nu1kU+kmnA6n7TiTksflobWnlQPnD9iOEhde//0pOlp72PDoQpJSdEx4qyZFuQPkZCTznQdu41BTBz/ZrtfUDrftjdvpHerVE5fex5qiNaQ4U3Q0EwFNx9o4vL2JJeuLKJw/zXacmDBpyh3gw7fP5CNLC/iHLSc5cqbDdpyY5vP7mJE6gzvz7rQdZdJKS0xjTdEaNtdvZjAwaDtOzOrvGWTrL44xdUYq9zww23acmDGpyh3gW/cvZlp6Eo8/d5D+QR3PhENXfxe7mndR4arAIZPuR2BS8ZR5aOtt482WN21HiVm7XzhJV3svGz+7iMQkHRGGyqT7zc5KS+L7f3E7x8518qMtJ23HiUnbGrfRH+jXkUwQVhWuIjUhVUczYVL/9kWO7j7LHRUl5M+aajtOTJl05Q6wYWEeD91VxD9tr+Wtxku248Qcn99Hfno+S3KX2I4y6aUmpLK2eC1V9VUMBAZsx4kpvd0DbPtlDdkF6Sy/b5btODFnUpY7wNf+fBH5mSk8/txb9A7oUrRQ6ejrYPeZ3bhL3TqSCZLH5eFS3yX2nt1rO0pM2fncCXo6B9j42UU4E/VnMdQm7SuamZLIDx5cyqnWbp7yHbcdJ2ZsbdjKYGAQT5meuBSslYUryUjM0Ds0hdDpA62ceKOFuz7sIrdkiu04MWnSljvAqrk5fOqeEn66u469dW2248QEn99HYUYhi6cvth0laiQ7k1lXvI6qhioGhnQ0c6t6OvvZ/utj5JZM4a57S23HiVlBlbuIeETkuIjUisiTN/j+J0Xk0PDHayKyNFQBv3LvQoqnpfHE8wfp7tPlaLfiUu8l9pzdg9vlRkRvfDAenjIPnf2dvH72ddtRopoxhld/fZy+nkE2fGYhTuekfn8Z1cZ8ZUXECfwYuBdYBHxCRBZdt1kd8CFjzBLg28AzoQqYnpzAUw8tpbH9Ct//07FQ7TYuVTVUMWSG9FoyE7Bi5gqmJE3R0cwtOlndwqkDrdz957OYXphhO05MC+Z/NpcDtcaY08aYfuBZ4P7RGxhjXjPGtA9/uQcoCmXI5WXZfG5lGb/cU8+ukxdCueu44vV7KZlSwoLsBbajRJ1EZyIbSjawtWErfUN9tuNEpe6OPnb85gR5ZZks21RiO07MC6bcC4HRNzttGn7sZj4H/OlG3xCRz4tItYhUt7a2Bp8SeMI9n9m56Xz5hYNc7tW553hd6LnAvnP7dCRzCzwuD10DXexu3m07StQxxrDtV8cYGgiw8bOLcOj9UMMumHK/0X+FG154XUTWcbXc/+ZG3zfGPGOMKTfGlOfmju/uKimJTn748DLOXe7lO384Oq7nKthSv4WACegqmVuwfOZyspKz9ISmCTj2+lnqD1/kngdmk5WXZjtOXAim3JuA4lFfFwFnrt9IRJYA/wzcb4y5GJp477asOIsvrJ3Nc9VNbKlpCcchYpbX72XW1FnMzdI720xUoiORjaUbr150bbDXdpyo0dnWy87nTlIwN4sl60I6sVXvI5hy3wfMFZEyEUkCHgFeHr2BiJQALwKfNsacCH3Md3xpw1wW5E/hyRcP097dH85DxYzzV87zZsubOpIJAbfLTc9gDzubd9qOEhWMMWz9RQ0Y2PCZhYiOYyJmzHI3xgwCXwR8QA3wnDHmiIg8JiKPDW/2dWA68E8i8paIVIcrcHKCkx8+vJT27n6+8fKRcB0mpmyu34zB6CqZECjPKyc7JRtvnY5mgnFkRzNNx9pZ+eAcMnNSbceJK0FdEd8Y8wrwynWPPT3q878E/jK00W5uccFUvrRhLn+/+QSe2/L58O0zI3XoqOTz+5g7bS6zsvT6HbcqwZHAptJN/Fvtv3Fl4AppiTo/vpmO1ivs/l0tJYuyWbSqwHacuBO1ZxB8Ye1slhRN5asvvc2FLl2adjPnus9x4PwB3KV6BchQcbvc9A71sqNph+0ok1YgYNjy8xocTgfrPr1Ax4EWRG25Jzod/PChpXT1DfI/XjyMMTdcwBP3Rk660VUyoXPnjDvJTc3VVTPv49DWRs7WdrD643PJmJZiO05citpyB5ibN4UnKuZRebSFl95qth1nUqr0V7IweyGlmXoNj1BxOpxUuCrY2bSTrv4u23Emnbaz3ex56TSuJTnMvzvfdpy4FdXlDvC5VbMoL53GN/7tCOc6dHnaaM1dzRy6cEhvyhEGHpeH/kA/25u2244yqQSGAmz52VESk52s/eR8HcdYFPXl7nQITz20lIEhw5MvHtLxzCgjI5kKV4XlJLFnSe4S8tLy8NXptWZG21/ZwPn6TtZ8Yh7pU5Ntx4lrUV/uAK6cdJ68dwHbj7fy232NYz8hTvj8Pm6bfhvFU4rH3liNi0McuF1udp3ZxeX+y7bjTAoXmjrZ94c65pTPYG55nu04cS8myh3g0/eUsmLWdL79h6M0tl2xHce6hssNHL14VP8hNYw8Lg+DgUG2NWyzHcW6ocEAVT+rITk9kQ89Mt92HEUMlbvDIfzgwSWICF9+4RCBQHyPZ66NZEp1JBMut+XcRmFGoa6aAapf8XOxqYt1n5xPSkai7TiKGCp3gOLsNL76Zwt5/fRFfrmn3nYcq7x+L0tzlzIzQ0/wChcRocJVwZ4ze7jUe8l2HGta/Jd501vPgnvyKVs6vgsCqvCJqXIH+PgHilk7P5fv/amGugvdtuNYcbrjNCfaT+jlBiLA4/IwaAbZ0rDFdhQrBgeG2PKzo6RlJrHqYb0o3WQSc+UuIvzdx5aQ5HTwxPMHGYrD8YzP70MQNpVush0l5i3MXkjJlJK4vUPTGy/X0X7uCusfXUBymo5jJpOYK3eAvMwUvnX/bbxZ385Pd522HSfifHU+7phxB3npumIh3EQEt8vN3nN7aeuNr5u4n629xFtVDSxeXUDJoum246jrxGS5A9y/rAD34jyeqjzByZZO23Eipra9llMdp3SVTAS5XW6GzBBV9VW2o0TMQN8QVT+vIXN6Ch/82BzbcdQNxGy5iwjf/ejtZCQn8PjzBxkYCtiOFBFevxeHOHQkE0Hzps2jbGpZXI1mXv/9KS639rD+0YUkpQR1cVkVYTFb7gA5Gcl894HbONTUwU+2n7IdJ+yMMfj8Pj6Q9wFyUnNsx4kbIoLH5aG6pZoLPbF/A/emY20c3t7E0vXFFM6bZjuOuomYLneAe2+fyUeWFvCjLSc5cqbDdpywOt5+HP9lv15uwAK3y03ABKj0V9qOElb9PYNs+UUNWXlp3POA3h9gMov5cgf41v2LmZaexOPPHaRvcMh2nLDx+X04xakjGQtmZ81mTtacmB/N7H7hJN3tfWz4zEISkpy246j3ERflnpWWxN997HaOnevkR1tO2o4TFsYYvHVe7p55N9NS9E9lGzwuD/vP76elOzZv3u4/fIGju89yR0Up+bOm2o6jxhAX5Q6wfkEeD5cX8ZPtpzjQ0G47TsgdvXiUpq4mvbyvRSOvfWV97I1mersH2ParY2QXpLP8vjLbcVQQ4qbcAb563yLyM1N4/PmD9A7E1njG6/eSIAlsKNlgO0rcck11sSB7QUxea2bnb0/Q2znAxs8uwpkYV7URteLqv1JmSiI/eHApp1u7ecp33HackBlZJbOiYAVTk/XPZZvcLjeHWg9xpuuM7Sghc/pAKyf2tnDXh13klkyxHUcFKa7KHWDV3Bw+fU8pP91dx9662Dij8NCFQ5ztPqsnLk0C10YzMbJqpqezn+2/PkZuyRTuuldv1RhN4q7cAZ68dwHF09J44vmDdPcN2o5zy7x1XhIdiawrXmc7StwrnlLM4umLY2I0Y4zh1V8fp69nkA2fXYjTGZd1EbXi8r9WenICTz20lMb2K3zvTzW249ySgAlQWV/JysKVTEnSP5knA4/Lw5GLR2i8HN13BTtZ3cKpA63c/eezmF6QYTuOGqe4LHeA5WXZfG5lGb/a08DOk62240zYW+ff4vyV83p530lk5CQyX330rnnvvtTHjt+cIH9WJss2ldiOoyYgbssd4An3fGbnpvPlFw5xuXfAdpwJ8fq9JDuTWVu81nYUNawgo4AluUui9oQmYwzb/vUYQwMBNnxmEQ6H2I6kJiCuyz0l0ckPH15Gy+Vevv3vR23HGbehwBCb6zezunA16YnptuOoUTwuD8fajuHv8NuOMm41r52l/vBF7vnobLLy0mzHURMU1+UOsKw4i/+8dg7Pv9nElproOrNw//n9XOi5gLtMT1yabCpKKxAk6t69d7b1suv5kxTOy2LJ2iLbcdQtiPtyB/jShrksyJ/Cky8epr2733acoHnrvKQmpLKmcI3tKOo6eel53DHjjqhaNWMChq2/qAED6x9diOg4JqppuQNJCQ5++PBS2rv7+cbLR2zHCcpgYJCqhirWFK0hLVH/dJ6M3C43tZdqOXUpOi43/faOZpqOtbPywTlk5qTajqNuUVDlLiIeETkuIrUi8uQNvr9ARF4XkT4ReSL0McNvccFU/mrDXF4+eIZXDp+1HWdM+87to623TVfJTGIVrgoc4oiKd+8drVd47cVaShZls2hVge04KgTGLHcRcQI/Bu4FFgGfEJFF123WBnwJeCrkCSPoC2tns6RoKl996W1aO/tsx3lfPr+PtIQ0VhWush1F3UROag7leeX4/D6Mmbw3ag8EDFt+XoPD6WDdpxcgouOYWBDMO/flQK0x5rQxph94Frh/9AbGmPPGmH1AdK4nHJbgdPDDh5bS1TfI3/7+8KT9hRwIDFDVUMXa4rWkJKTYjqPeh9vlpq6jjhPtJ2xHualDWxs5W9vBmo/PJWOa/jzFimDKvRAYfapd0/Bj4yYinxeRahGpbm2dnCcOzc2bwhMV86g82sJLbzXbjnNDe87soaOvQ0cyUWBj6Uac4py0q2baznaz56XTlC3NYd7d+bbjqBAKptxv9DfahN7SGmOeMcaUG2PKc3NzJ7KLiPjcqlmUl07j6/92hHMdvbbjvIfP72NK4hRWFq60HUWNITslm+X5yyflaCYwFGDLz46SmOxk7Sd1HBNrgin3JqB41NdFQOxcz/QGnA7hqYeWMjhk+JvfHZpUv5T9Q/1sbdjKupJ1JDmTbMdRQfCUeWjobKCmbXJdx2h/ZQPn6zv50H+YT1qm/izFmmDKfR8wV0TKRCQJeAR4Obyx7HPlpPOVDy/g1ROt/Hbf5LkA1GtnXqNzoFPvuBRFNpRsIEESJtWqmQtNnez7Qx1zymcw564ZtuOoMBiz3I0xg8AXAR9QAzxnjDkiIo+JyGMAIpIvIk3AfwO+KiJNIpIZzuCR8Km7S/ng7Ol8+w9HaWy7YjsOcHUkk5mUyYqZK2xHUUGamjyVewruodJfOSn+ChwaDFD1sxqS0xP50CPzbcdRYRLUOndjzCvGmHnGmNnGmO8OP/a0Mebp4c/PGWOKjDGZxpis4c8vhzN4JDgcwg8eXIKI8OUXDhEI2P3F7BvqY1vjNjaWbiTRmWg1ixofj8tDc1czb19423YUql/xc7Gpi3WfWkBKhv4cxSo9Q3UMRdPS+Np9C3n99EV+8brfapZdTbvoHujWkUwUWleyjkRHovXRTIv/Mm9661mwIp+yJTlWs6jw0nIPwsPlxaybn8v3vceou9BtLYfX72Va8jSW5y+3lkFNTGZSJisLVuLz+wiYgJUMgwNDbPnZUdKnJrHq4XlWMqjI0XIPgojw/Y8tITnByePPvcWQhfFMz2APrza9ysbSjSQ4EiJ+fHXr3GVuWq60cKj1kJXjv/FyHe3nrrDu0wtITtWfoVin5R6kvMwUvvmRxexvuMQ/7zwd8ePvaNpBz2CPnrgUxdYVryPJkWRlNHO29hJvVTWweE0hJYumR/z4KvK03Mfh/mUFeBbn88PKE5xo6YzosX1+H9NTpnNX3l0RPa4KnfTEdFYXrabSX8lQYChixx3oG6Lq5zVkTk/hg38xO2LHVXZpuY+DiPCdj95GRkoCjz93kIGhyMxOuwe62dG0g02lm3A6nBE5pgoPj8tDa08r+8/vj9gxX//9KS5f6GH9owtJStFxTLzQch+nnIxkvvvAbRxu7uAn2yNzne5XG1+lb6gPT5mOZKLdmqI1pCakRuxaM43H2ji8vYml64opnDctIsdUk4OW+wTce/tM7l9WwI+2nOTt5o6wH8/r9zIjdQZ3zLgj7MdS4ZWWmMaaojVsrt/MYGAwrMfq7xlk6y9qyMpL454HZoX1WGry0XKfoG9+ZDHZ6Uk88fxB+gbDNz/t7O9kV/Ouazd+UNHP7XLT1ttGdUt1WI+z64WTdLf3seGzC0lI0nFevNG2mKCstCS+/7HbOXauk3+oOhm242xv3M5AYEBPXIohqwtXk5aQFtbRjP/wBWp2n+UOdyn5ZVPDdhw1eWm534L1C/J4uLyIp189xYGG9rAcw+v3MjN9Jktzl4Zl/yryUhJSWFu8lqr6KgYCob+/TW/3ANt+dYzpheks/7OykO9fRQct91v0tfsWMXNqKo8/f5DegdCOZzr6OnjtzGtUlFbotbZjjNvl5lLfJfae3Rvyfe/87Ql6OwfY8JlFOBP1Vzxe6X/5WzQlJZEfPLiE063d/C/f8ZDue2vDVgYDg7pKJgatKlxFRmJGyE9oOnXgPCf2tlD+Zy5yS6aEdN8qumi5h8DKOTk8uqKUf9ldxxunL4Zsvz6/j6KMIhZPXxyyfarJIcmZxPqS9Wxp2MLAUGhGMz2d/bz66+PklkzhTk9pSPapopeWe4g8ee8CSrLTeOKFg3T33foSt/bedvac3YPb5daRTIxyu9x09nfy+tnXb3lfxhi2//o4fT2DbPjsQpxO/dWOd/oTECJpSQk89dBSmtp7+N6fbv12alUNVQyZIV0lE8NWzFxBZlIm3rpbH82c3NfC6QOt3P3ns5hekBGCdCraabmH0Adc2fzlqjJ+taeBnSdbb2lfPr+P0sxSFmQvCFE6NdkkOhPZULKBrY1b6Rvqm/B+ui/1sePZE+TPymTZppIQJlTRTMs9xB6vmM/s3HS+/MIhLvdObJZ6oecC+87t05FMHPC4PHQPdLO7efeEnm+MYdu/HmNoIMCGzyzC4dCfF3WVlnuIpSQ6+eHDyzjf2ce3/v3ohPZRVV9FwAR0JBMHPjDzA2QlZ0141UzNa2epP3yRFX8xm6y8tBCnU9FMyz0MlhVn8YUPzeaFN5uoOtoy7ud7/V5mTZ3F3Ky5YUinJpNERyIbSzeyvXE7PYM943ru5Ys97Hr+JIXzs7j9Q0XhCaiilpZ7mHxpw1wW5E/hyRcP097dH/Tzzl85z/6W/XhcHh3JxAmPy0PPYA+7mncF/RwTMGz75TEwsP7TCxEdx6jraLmHSVKCg79/eBkdPf18/eUjQT9vc/1mDEZHMnGkPK+c7JTsca2aeXtHM03H2ln54Bwyc1LDmE5FKy33MFpUkMmX1s/l3w+e4Y+Hzgb1HG+dl7nT5jIrSy/RGi+cDiebSjexo2kHVwaujLn9pfNXeO3FWkoWZ7NoVUEEEqpopOUeZl9YO5ulRVP56kuHae18/+Vu57rP8VbrW3qf1DjkcXnoHerl1aZX33e7QMCw9Rc1OBMcrPvUQh3dqZvScg+zBKeDHz68lO7+If7294cxxtx025FLwOpIJv7cmXcnuam5Y14G+NDWRs7WdrD64blkTEuOUDoVjbTcI2DOjCn894r5VB5t4fcHmm+6nc/vY2H2Qkoz9bog8cYhDipcFexs2klXf9cNt2k7282el05TtjSHeXfnRzihijZa7hHyH1eVUV46jW+8fISzHe9d8tbU2cThC4f1XXsc87g89Af62da47T3fCwwF2PKzoyQmO1n7yQU6jlFj0nKPEKdDeOqhpQwOGf7md+8dz1TWVwI6kolnS3KXkJ+eT6W/8j3f2+9r4Hx9Jx/6D/NJy0yykE5FGy33CHLlpPOVDy9gx4lWnt3X+K7veeu83J5zO0VT9GSUeOUQB+5SN7vO7OJy/+Vrj19o6mTfH+uYWz6DOXfNsJhQRRMt9wj71N2lfHD2dL7zh6M0tl1d9tZwuYGathp9165wu9wMBgbZ2rAVgKHBAFX/p4aU9ETWPDLfcjoVTbTcI8zhEH7w4BJEhP/+wkECAaOrZNQ1t+XcRmFG4bVrzVS/4udicxdrP7WAlIxEy+lUNAmq3EXEIyLHRaRWRJ68wfdFRH40/P1DInJn6KPGjqJpaXztvoXsOd3GL1734/V7WZa7jPx0XQER70QEt8vNG2fe4NSJZt701rNgRT5lS3JsR1NRZsxyFxEn8GPgXmAR8AkRWXTdZvcCc4c/Pg/8JMQ5Y87D5cWsm5/L97fs4ET7Cb1PqrrG4/JghmDzz46SPjWJVQ/Psx1JRSF5v5NqAERkBfA/jTHu4a+/AmCM+d6obf5/YLsx5jfDXx8H1hpjbnrOfWnOLPO3H/n2rf9/EOXM8EfyyCdKAX3OZIYSs8lt/kdSe47ZjmNd+4wUHv31ftsxJgURedMYUz7WdglB7KsQGL20owm4O4htCoF3lbuIfJ6r7+xxZReR0n8uiMPHNgM4DSQZXbes3pEshsT2Sqb2HAf0ZyM9qKpSowXzit3oJ+v695jBbIMx5hngGYDy8nLz6K8eD+LwSimlxiuYf1BtAopHfV0EnJnANkoppSIkmHLfB8wVkTIRSQIeAV6+bpuXgUeHV83cA3S837xdKaVUeI05ljHGDIrIFwEf4AT+xRhzREQeG/7+08ArwIeBWuAK8P+EL7JSSqmxBPWvFMaYV7ha4KMfe3rU5wb4L6GNppRSaqL0DFWllIpBWu5KKRWDtNyVUioGabkrpVQMGvPyA2E7sEgncNzKwSefHOCC7RCThL4W79DX4h36WrxjvjFmylgb2Tyn93gw10eIByJSra/FVfpavENfi3foa/EOEakOZjsdyyilVAzScldKqRhks9yfsXjsyUZfi3foa/EOfS3eoa/FO4J6Laz9g6pSSqnw0bGMUkrFIC13pZSKQVbKfawbbscLEfkXETkvIm/bzmKbiBSLyDYRqRGRIyLyV7Yz2SIiKSKyV0QODr8W37SdySYRcYrIARH5g+0stomIX0QOi8hbYy2JjPjMffiG2yeATVy9ycc+4BPGmKMRDTIJiMgaoAv4hTHmNtt5bBKRmcBMY8x+EZkCvAk8EKc/FwKkG2O6RCQR2AX8lTFmj+VoVojIfwPKgUxjzH2289gkIn6g3Bgz5gldNt65LwdqjTGnjTH9wLPA/RZyWGeM2QG02c4xGRhjzhpj9g9/3gnUcPU+vHHHXNU1/GXi8EdcrnwQkSLgz4B/tp0l2tgo95vdTFspAETEBdwBvGE5ijXDo4i3gPPAZmNMvL4W/xv4MhCwnGOyMECliLwpIp9/vw1tlHtQN9NW8UlEMoDfAX9tjLlsO48txpghY8wyrt6PeLmIxN3YTkTuA84bY960nWUSWWmMuRO4F/gvw6PdG7JR7nozbXVDw/Pl3wH/aox50XaeycAYcwnYDnjsJrFiJfCR4Tnzs8B6EfmV3Uh2GWPODP/f88DvuTrmviEb5R7MDbdVnBn+R8SfAjXGmL+3nccmEckVkazhz1OBjcAxq6EsMMZ8xRhTZIxxcbUnthpjPmU5ljUikj682AARSQcqgJuutIt4uRtjBoGRG27XAM8ZY45EOsdkICK/AV4H5otIk4h8znYmi1YCn+bqu7O3hj8+bDuUJTOBbSJyiKtvhjYbY+J+GaAiD9glIgeBvcAfjTHem22slx9QSqkYpGeoKqVUDNJyV0qpGKTlrpRSMUjLXSmlYpCWu1JKxSAtd6WUikFa7kopFYP+L7vc8hrKqgoHAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Plot\n",
    "fig = figure()\n",
    "ax = fig.add_subplot(111)\n",
    "\n",
    "plot(arange(P), xy_probability)\n",
    "#plot(arange(P), xy_probability, 'o')\n",
    "title(\"N=2\")\n",
    "xlim(0, P)\n",
    "\n",
    "show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3e65a534",
   "metadata": {},
   "source": [
    "### Results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "99cec89d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.25, 0.  , 0.25],\n",
       "       [0.  , 0.  , 0.  ],\n",
       "       [0.25, 0.  , 0.25]])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xy_probability #N =1 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "61ac2fc8",
   "metadata": {},
   "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": [
    "Plot2D(xy_probability,1) #-- N = 2 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "35e04131",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.0625, 0.    , 0.125 , 0.    , 0.0625],\n",
       "       [0.    , 0.    , 0.    , 0.    , 0.    ],\n",
       "       [0.125 , 0.    , 0.25  , 0.    , 0.125 ],\n",
       "       [0.    , 0.    , 0.    , 0.    , 0.    ],\n",
       "       [0.0625, 0.    , 0.125 , 0.    , 0.0625]])"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xy_probability #N =2 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "2752e907",
   "metadata": {},
   "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": [
    "Plot2D(xy_probability,2) #-- N = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "58a9dc59",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.125, 0.   , 0.   , 0.   , 0.   , 0.   , 0.   ],\n",
       "       [0.   , 0.   , 0.   , 0.   , 0.   , 0.   , 0.   ],\n",
       "       [0.   , 0.   , 0.375, 0.   , 0.   , 0.   , 0.   ],\n",
       "       [0.   , 0.   , 0.   , 0.   , 0.   , 0.   , 0.   ],\n",
       "       [0.   , 0.   , 0.   , 0.   , 0.375, 0.   , 0.   ],\n",
       "       [0.   , 0.   , 0.   , 0.   , 0.   , 0.   , 0.   ],\n",
       "       [0.   , 0.   , 0.   , 0.   , 0.   , 0.   , 0.125]])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xy_probability #N =3 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "0ffc5f59",
   "metadata": {},
   "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": [
    "Plot2D(xy_probability,3) #-- N = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "id": "b0f29c7c",
   "metadata": {},
   "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": [
    "Plot2D(xy_probability,10) #-- N = 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4f0e8d9d",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}