dev/demo-CST.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 160,
   "id": "fe99052e-894e-435b-9aed-bc0d98c93bf9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The autoreload extension is already loaded. To reload it, use:\n",
      "  %reload_ext autoreload\n"
     ]
    }
   ],
   "source": [
    "# will autoupdate any of the packages imported:\n",
    "%load_ext autoreload\n",
    "%autoreload 2\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "id": "7d62129d-4284-461f-99a6-38e40452ba6d",
   "metadata": {},
   "outputs": [],
   "source": [
    "import sys\n",
    "sys.path.insert(0, '..')\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pyclifford as pc\n",
    "import time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "13436bca-8a27-47d4-9f5d-807f6fe15345",
   "metadata": {},
   "outputs": [],
   "source": [
    "import seaborn as sns\n",
    "sns.set_theme()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "id": "92340522-cb1a-48f4-a7ae-5054f3ce108c",
   "metadata": {},
   "outputs": [],
   "source": [
    "from numba import njit"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "df936ce5-68be-4953-abd7-447781821cc4",
   "metadata": {},
   "source": [
    "# Global Clifford classical shadow tomography"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "24397c56-2512-465a-8501-1c3facc49cfc",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "id": "5d0bd8a2-ea38-4b77-b5d4-673e83a1c238",
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 300\n",
    "classical_shadows = []\n",
    "num_of_shadow = 1000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "id": "848455c2-1e5a-4b80-89b3-245b97dd139b",
   "metadata": {},
   "outputs": [],
   "source": [
    "circ = pc.Circuit(N)\n",
    "# Create global Clifford random gate\n",
    "circ.gate(*[i for i in range(N)])\n",
    "for povm in circ.povm(num_of_shadow):\n",
    "    state = pc.ghz_state(N)\n",
    "    state.measure(povm)\n",
    "    classical_shadows.append(state)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "id": "59d00321-eae6-467f-9be6-c4479137a9c5",
   "metadata": {},
   "outputs": [],
   "source": [
    "fidelity_predictions = []"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "id": "dfceff8d-9a16-42e9-9e8c-575f366f3590",
   "metadata": {},
   "outputs": [],
   "source": [
    "for shadow in classical_shadows:\n",
    "    fidelity_predictions.append(shadow.expect(pc.ghz_state(N))*(2**N)-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "id": "36ace162-4af5-4335-b1cd-981898a6bdcf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '300 qubit GHZ state')"
      ]
     },
     "execution_count": 165,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "plt.errorbar(x = 0,y = np.mean(np.array(fidelity_predictions)),\\\n",
    "             yerr=np.std(np.array(fidelity_predictions))/np.sqrt(1000),capsize = 5)\n",
    "plt.scatter(x = 0,y = np.mean(np.array(fidelity_predictions)),s= 60,)\n",
    "plt.ylim([0.5,1.1])\n",
    "plt.plot(np.linspace(-0.05,0.05,20),np.ones(20),\"k--\",alpha = 0.6)\n",
    "plt.ylabel(\"Reconstructed Fidelity\",fontsize = 15)\n",
    "plt.title(\"300 qubit GHZ state\",fontsize = 15)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6a8f34f7-fa5b-4f9c-9a81-80f8a5215b65",
   "metadata": {},
   "source": [
    "# Shallow Circuit Shadow Classical"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ff8b004-85a8-4cb8-8d3b-ea0f3a740af6",
   "metadata": {},
   "source": [
    "Let's study the classical shadow tomography with finite depth brick wall circuit. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "36d8d528-c5c7-4a73-a21c-4a85dac7dd46",
   "metadata": {},
   "source": [
    "The reconstruction map has been studied under the formulation of locally scrambled classical shadow tomography."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "596342b9-9a75-40b4-99c2-6d829d093c9d",
   "metadata": {},
   "source": [
    "First, let's get Pauli weights. By definition,"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2e76f6b9-7dca-4e5d-9475-2555ef58c5ac",
   "metadata": {},
   "source": [
    "$\n",
    "W(\\vec{a})=\\underset{U}{\\mathbb{E}}\\underset{b}{\\mathbb{E}}|U^{\\dagger}|b\\rangle \\langle b|U P_{\\vec{a}}|^2\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3b187101-2d5b-46bf-9d83-6475c85949d6",
   "metadata": {},
   "source": [
    "Another we to get the Pauli weights is through experimental calibration:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d67e66b5-3e1f-4dce-b71f-c1cb19a59d76",
   "metadata": {},
   "source": [
    "$W(\\vec{a})=2^N\\dfrac{\\mathbb{E}\\text{Tr}(\\hat{\\sigma}P_{\\vec{a}})}{\\text{Tr}(\\rho P_{\\vec{a}})}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dfb62d27-85c6-456f-8dd2-6e423f994aa7",
   "metadata": {},
   "source": [
    "## Random Pauli measurement"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "4e5d2d0a-3a4c-4d50-a4db-e3d5da3de701",
   "metadata": {},
   "outputs": [],
   "source": [
    "N2 = 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "id": "57e8d289-268f-43d3-a583-e483af3755db",
   "metadata": {},
   "outputs": [],
   "source": [
    "X = pc.pauli([1]+[0 for _ in range(N2-1)])\n",
    "Z = pc.pauli([3]+[0 for _ in range(N2-1)])\n",
    "XX = pc.pauli([1,1]+[0 for _ in range(N2-2)])\n",
    "ZZ = pc.pauli([3,3]+[0 for _ in range(N2-2)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "id": "a2233aaf-e52a-4e16-9fc3-aa71349b9e6f",
   "metadata": {},
   "outputs": [],
   "source": [
    "def brickwall_circuit(N,depth):\n",
    "    circ = pc.circuit.Circuit(N)\n",
    "    for d in range(depth):\n",
    "        for ii in range(int(N//2)):\n",
    "            circ.gate(2*ii,2*ii+1)\n",
    "        for ii in range(int(N//2)):\n",
    "            circ.gate((2*ii+1),(2*ii+2)%N)\n",
    "    return circ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "id": "2b3ed0b0-0f4b-4295-96c3-2f0cdb9009da",
   "metadata": {},
   "outputs": [],
   "source": [
    "def random_pauli_circuit(N):\n",
    "    circ = pc.circuit.Circuit(N)\n",
    "    for ii in range(N):\n",
    "        circ.gate(ii)\n",
    "    return circ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "id": "b2c2480c-abdf-4346-ab9f-ed20e0a8df75",
   "metadata": {},
   "outputs": [],
   "source": [
    "circ = random_pauli_circuit(N2)\n",
    "classical_shadows2 = []\n",
    "num_of_shadow = 10000\n",
    "for povm in circ.povm(num_of_shadow):\n",
    "    state = pc.stabilizer.zero_state(N2)\n",
    "    state.measure(povm)\n",
    "    classical_shadows2.append(state)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "id": "7f0c447a-8e53-4e3b-9ee0-b80509d4eb1f",
   "metadata": {},
   "outputs": [],
   "source": [
    "Wz_lst = []\n",
    "for i in range(num_of_shadow):\n",
    "    Wz_lst.append(np.real(classical_shadows2[i].expect(Z)))\n",
    "Wzz_lst = []\n",
    "for i in range(num_of_shadow):\n",
    "    Wzz_lst.append(np.real(classical_shadows2[i].expect(ZZ)))\n",
    "Wx_lst = []\n",
    "for i in range(num_of_shadow):\n",
    "    Wx_lst.append(np.real(classical_shadows2[i].expect(X)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "id": "ff3e747a-f5d7-4e89-a40d-af88b385f5b4",
   "metadata": {},
   "outputs": [],
   "source": [
    "Wz = (2**N2)*np.mean(np.array(Wz_lst))\n",
    "Wzz = (2**N2)*np.mean(np.array(Wzz_lst))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "id": "1b610763-66a0-47c9-99e8-d3bca2a87945",
   "metadata": {},
   "outputs": [],
   "source": [
    "circ = random_pauli_circuit(N2)\n",
    "classical_shadows_test = []\n",
    "num_of_shadow = 1000\n",
    "for povm in circ.povm(num_of_shadow):\n",
    "    state = pc.stabilizer.ghz_state(N2)\n",
    "    state.measure(povm)\n",
    "    classical_shadows_test.append(state)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a3abac2c-9c04-4da9-8c18-3832d233a75d",
   "metadata": {},
   "source": [
    "Predict ZZ operator:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "id": "4af83f54-80de-4305-9221-3fe4574800e0",
   "metadata": {},
   "outputs": [],
   "source": [
    "predict_zzs = []\n",
    "for i in range(len(classical_shadows_test)):\n",
    "    predict_zzs.append(np.real(classical_shadows_test[i].expect(ZZ)))\n",
    "predict_zs = []\n",
    "for i in range(len(classical_shadows_test)):\n",
    "    predict_zs.append(np.real(classical_shadows_test[i].expect(Z)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "id": "7b01ab67-ae54-439a-8610-d9fff2bab0fe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.954954954954955"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(np.array(predict_zzs))*(2**N2)/Wzz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "id": "aef9f6f6-c6a2-4f1a-9dda-e7954ce0867a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '10 qubit GHZ state (randomized Pauli measurement)')"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.errorbar(x = 1,y = np.mean(np.array(predict_zs))*(2**N2)/Wz,\\\n",
    "             yerr=np.std(np.array(predict_zs))*(2**N2)/(Wz*np.sqrt(1000)),capsize = 5)\n",
    "plt.scatter(x = 1,y = np.mean(np.array(predict_zs))*(2**N2)/Wz,s= 60,)\n",
    "plt.errorbar(x = 2,y = np.mean(np.array(predict_zzs))*(2**N2)/Wzz,\\\n",
    "             yerr=np.std(np.array(predict_zzs))*(2**N2)/(Wzz*np.sqrt(1000)),capsize = 5)\n",
    "plt.scatter(x = 2,y = np.mean(np.array(predict_zzs))*(2**N2)/Wzz,s= 60,)\n",
    "\n",
    "# plt.ylim([0.8,1.1])\n",
    "# plt.plot(np.linspace(-0.05,0.05,20),np.ones(20),\"k--\",alpha = 0.6)\n",
    "plt.xlabel(r\"$k$\",fontsize = 15)\n",
    "plt.ylabel(r\"$\\langle Z^{\\otimes k}\\rangle$\",fontsize = 15)\n",
    "plt.title(\"10 qubit GHZ state (randomized Pauli measurement)\",fontsize = 15)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "66e4b994-a6b0-4fee-8ed3-b7488339aa2a",
   "metadata": {},
   "source": [
    "## Shallow layer brickwall circuit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "id": "f17c08be-618e-48c2-87eb-122701786776",
   "metadata": {},
   "outputs": [],
   "source": [
    "circ = brickwall_circuit(N2,3)\n",
    "classical_shadows3 = []\n",
    "num_of_shadow = 50000\n",
    "for povm in circ.povm(num_of_shadow):\n",
    "    state = pc.stabilizer.zero_state(N2)\n",
    "    state.measure(povm)\n",
    "    classical_shadows3.append(state)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "id": "c73af6db-a9e1-4369-9fe4-258d313b3a36",
   "metadata": {},
   "outputs": [],
   "source": [
    "Wz_lst = []\n",
    "for i in range(num_of_shadow):\n",
    "    Wz_lst.append(np.real(classical_shadows3[i].expect(Z)))\n",
    "Wzz_lst = []\n",
    "for i in range(num_of_shadow):\n",
    "    Wzz_lst.append(np.real(classical_shadows3[i].expect(ZZ)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 144,
   "id": "2a082dec-ab98-408d-a8d8-db311443277a",
   "metadata": {},
   "outputs": [],
   "source": [
    "circ = brickwall_circuit(N2,3)\n",
    "classical_shadows_test = []\n",
    "num_of_shadow = 10000\n",
    "for povm in circ.povm(num_of_shadow):\n",
    "    state = pc.stabilizer.ghz_state(N2)\n",
    "    state.measure(povm)\n",
    "    classical_shadows_test.append(state)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "id": "ab731231-10f6-432f-98a2-3b34eb54ea8b",
   "metadata": {},
   "outputs": [],
   "source": [
    "predict_zzs = []\n",
    "for i in range(len(classical_shadows_test)):\n",
    "    predict_zzs.append(np.real(classical_shadows_test[i].expect(ZZ)))\n",
    "predict_zs = []\n",
    "for i in range(len(classical_shadows_test)):\n",
    "    predict_zs.append(np.real(classical_shadows_test[i].expect(Z)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "id": "f964117c-2ae5-4775-b122-760f6205e476",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '10 qubit GHZ state (3 layer brickwall circuit)')"
      ]
     },
     "execution_count": 148,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Gg4MDBgwYAKVSid27d8PGxkb43VefhwkODsa8efNgb2+PmzdvNumKOXVqbmz7K1euqLyQiImJwYwZMwDUfvyWlJQ0eORQ1+NToVDg4cOHLXJusSkYHBq2YMECpKen49NPP8XVq1cRFhaGCxcu4N13361zfWdnZ0ilUqxatQq//fYbfvrpJ+zYsUNlnYEDB+LEiRM4ceIEcnNzsWHDhlpTSRKJBCtWrMClS5dw5swZbNy4EVOnTkW3bt1q7bN9+/Z48OABrl27Vue5F0NDQ+zcuRNZWVmYP38+jh07hlu3biEjIwOffvop3nvvPXTv3r3On92QhIQExMXFYdmyZejcuTPkcrnKv2efiKt16tQJP/74I4KCgpCZmYkbN24gLi4Of/rTn/D6668L/eTk5KCoqAidO3dGSUkJkpKSkJeXh5SUFOFVe/VVRO3bt0dBQQFu3bqFyspKLF68GPv27UNMTAxu3ryJI0eOYOvWrfU+SM3NzWFubo6MjAzhtjFjxqBHjx748MMP8dtvv+GXX37BsmXLMGjQoHpPoj+rS5cu+O233/Drr7/i1q1b+Oqrr/Dll1+q1A08PUkeFxeHx48fY/To0cLtixcvxtGjRxEeHo7c3Fz88MMPCAoKQtu2bes9CV6ff/7zn/jmm29w9epVrF27FiUlJcK5lMZMnToV7du3x0cffYSsrCycP38eW7ZswbBhw2pN13p6esLS0hKBgYHCE/7o0aORmJgIiUQCKysrGBsbY/DgwYiPj1d5I2mXLl1w7tw5ZGZmIjc3F7t378a3337bpPc6iKm5LvPnz8e9e/ewadMmXL16FT/99BN27dqFt99+G8DTx+/Ro0dx/vx5ZGdnY82aNQ1ekfjs/blaZmYmqqqqYGdnJ7q/lsDg0DArKyvs3r0bSUlJmD59OlJTU/H555/Xe0LS2NhYuGxy9uzZ2Lhxo/BKpZqnpyfGjh0LPz8/zJkzBx06dKh1aaZUKsWECRPw/vvvw9fXF8OHD8eGDRvq3KeTkxO6desGV1dX/PDDD3WuU30idtiwYfjkk0/g4uKCBQsW4NSpU/jggw/wzTffNDjXXZfqk41bt27FqFGjav2r6xp1AwMD7NmzB8DTk7aurq7IyclBRESEcCWWt7c3Dhw4AC8vL0yePBnvvvuucPlqWFgYfH19YWlpiUuXLgF4+uRbVVUFZ2dnXLlyBW5ubvjggw8QEREBZ2dn7Ny5E76+vli6dGm9vYwZM0blUuZ27dohIiIC7du3x4IFC+Dj4wNra2vs3btXrekFPz8/9O3bF15eXpg5cyaSk5OxdetWABDqBp4+sbZt2xYuLi4qgeDo6Ijt27fjyJEjcHFxQVBQEKZPn96kiyx8fX2xd+9eTJs2DdevX0dERIRw0rYxpqamiIiIQElJCWbPno0lS5Zg+PDhWL9+fa11DQ0NsWnTJqSnpwvnZoYPHw5DQ0MMHz5ceOU9YsQIKBQKleBYu3YtOnbsiLlz58LNzQ2XLl3Cxo0bUVRUJPqktJia62Jubo69e/cKb5r19/fHrFmzhPvPBx98AGtrayxcuBDvvfdeg+dPANX7c7WzZ8+ib9++ol+stRQJvwFQ96Wnp2P+/PlISUkR/eRMrSM7OxuzZs3C8ePHNf4GsWfdv38fb731FmJiYhq9Go1eHNOmTYOHh0e95+o0jUccRC2gT58+GD9+vMpVTJp0//59JCUlwd/fH/3792do6JGzZ8+ivLwc06ZN01oNDA6iFhIQEIBDhw7VeW6mpVVUVCAgIAA3btxo0htE6fn1ySefYMuWLU26equlcKqKiIhE4REHERGJwuAgIiJRGBxERCSK3nzkyP37pVAoxJ/O6dq1A4qKShpf8QXCnvWDvvWsb/0CzevZwECCzp3r/jQHvQkOhULZpOCo3lbfsGf9oG8961u/gGZ65lQVERGJwuAgIiJRGBxERCSKzgRHSUkJXFxchO98ftaxY8cwbdo0uLq6wtfXV+Ujx4mIqHXpRHBcuHABbm5uyM3NrTVWUlKC9evXIzw8HAkJCbCysmrSd/cSEVHL0IngiI2Nxbp16+r8LuqKigqsW7dO+MY8KysrlS8oIiKi1qUTl+M29CFtnTt3xoQJEwA8/ZrR8PBwuLu7t1ZpRERUg04EhzoePXqEJUuWwNraWuUrMtXVtWuHxleqh1TascnbPq/Ys37Qt55f5H6Lf4zB7ydiVW57VMd6Zm/9GV0c5zRrX89FcBQWFsLLywsODg7w9/dv0s8oKipp0hthpNKOkMvr+vW/uNizftC3nl/4fvs6o2NfZ2HxjyNbYGRkCKNJq1RWqwLU+j0YGEjqfcGt88FRVVUFHx8fTJ48Gb6+vtouh4hI7+lscHh7e8PPzw/5+fm4cuUKqqqqkJSUBADo378/v7yGiEhLdCo4UlNThf/v3bsXAGBra4vMzExtlURERDXoxOW4RET0/GBwEBGRKAwOIiIShcFBRESiMDiIiEgUBgcREYnC4CAiIlEYHEREJAqDg4iIRGFwEBGRKAwOIiIShcFBRESiMDiIiEgUBgcREYnC4CAieoEolQpUZKehSp6L8tuZKI1bj4rsNCiVihbbh059HwcRETWdUqlAWfJuVOVdBiofAwAU93JRfuJLtLmejnYTlkIiaf7xAo84iIheEJU5p1VC478Dj1F1+zIqr55pkf0wOIiIXhBPLiXXDo1qlY/x5GJSi+yHwUFE9IJQlhY3a1xdDA4ioheEpH2XZo2ri8FBRPSCMLadCBia1D1oaAJjO6cW2Q+Dg4joBWHY2wFtutnUDg9DE7TpbgPDXsNbZj8t8lOIiEjrJBIDtJu4FJVXz6D8x32AohIGXXrA2M4Jhr2Gt8iluACDg4johSKRGMCo9whUZPwAIyNDGE1a1eL70JmpqpKSEri4uOD27du1xjIyMjBjxgw4OTkhICAAlZWVWqiQiIgAHQmOCxcuwM3NDbm5uXWOr1q1CkFBQUhKSoJSqURsbGzrFkhERAKdCI7Y2FisW7cOMpms1lheXh7Ky8sxcOBAAMCMGTOQmJjYyhUSEVE1nTjHsXnz5nrHCgsLIZVKhWWpVIqCggLR++jatUOTanu6z45N3vZ5xZ71g771rE/93jF6+vSuiZ51IjgaolAoIJFIhGWlUqmyrK6iohIoFErR20mlHSGXPxK93fOMPesHfetZ3/qtqKiEkZFhk3s2MJDU+4JbJ6aqGmJhYQG5XC4s37t3r84pLSIiah06HxzdunWDiYkJzp07BwCIj4+Ho6OjlqsiItJfOhsc3t7euHTpEgAgJCQEW7ZswaRJk/DHH3/Aw8NDy9UREekvnTrHkZqaKvx/7969wv+tra1x8OBBbZREREQ16OwRBxER6SYGBxERicLgICIiURgcREQkCoODiIhEYXAQEZEoDA4iIhKFwUFERKLo1BsAiYioaR6nH8KTX+JVbqsCUB6+UOU2Y/tpMBnyTrP2xeAgInoBmAx5p1YgaOoTgTlVRUREojA4iIhIFAYHERGJwuAgIiJRGBxERCQKg4OIiERhcBARkSgMDiIiEoXBQUREojA4iIhIFAYHERGJwuAgIiJRGBxERCSKTgTHkSNH4OzsjIkTJyIqKqrW+OXLlzFz5ky4urpi0aJFePjwoRaqJCIiQAeCo6CgAKGhoYiOjsbhw4cRExODnJwclXU2b94MPz8/JCQk4PXXX0dERISWqiUiIq0HR1paGhwcHGBmZgZTU1M4OTkhMTFRZR2FQoHS0lIAQFlZGdq2bauNUomICDoQHIWFhZBKpcKyTCZDQUGByjpr1qxBYGAgRo0ahbS0NMydO7e1yyQiov9P698AqFAoIJFIhGWlUqmyXF5ejoCAAOzbtw92dnb417/+hb/+9a8IDw8XtZ+uXTs0uUaptGOTt31esWf9oG8961u/gGZ61npwWFhYID09XViWy+WQyWTCclZWFkxMTGBnZwcAmDNnDsLCwkTvp6ioBAqFUvR2mvrqRV3GnvWDvvWsb/0CzevZwEBS7wturU9VjRw5EqdOnUJxcTHKysqQnJwMR0dHYdzS0hL5+fm4du0aACAlJQW2trbaKpeISO9p/YjD3NwcK1asgIeHByoqKjBr1izY2dnB29sbfn5+sLW1xZYtW7B8+XIolUp07doVwcHB2i6biEhvSZRKpfj5m+cQp6rUx571g771rG/9Ai/wVBURET1fGBxERCQKg4OIiERhcBARkSgMDiIiEoXBQUREojA4iIhIFAYHERGJwuAgIiJRGBxERCQKg4OIiERpVnAoFAqcPn0aUVFRuHnzZkvVREREOkz0p+P+8ccfOHHiBFJSUnD8+HE8efIEPXr0wN/+9jf06tUL48aNw7hx44TvzyAioheL2sHx22+/ISQkBGfOnMGf/vQnjBkzBtu2bcPIkSNhbGyMgoICpKamIiUlBV988QU6deqEsWPHYuPGjZqsn4iIWpnawVFcXAxra2ssWbIEAwcOrDVubm4ONzc3uLm5obS0FMePH0dqampL1kpERDpA7eCwsbHBiBEj1Fq3ffv2cHZ2hrOzc5MLIyIi3aT2yfHVq1fXuu3u3bstWgwREek+tYNj2LBh+OSTT4TlixcvwtPTUyNFERGR7lJ7qsrT0xPLli1DcnIyKisr8Y9//AO7d+/WZG1ERKSDGg2OBQsWwMbGBjY2NvD09ERgYCA6d+6M/fv3o1OnTq1RIxER6ZBGp6qCgoLQt29f/Oc//8GOHTtw9+5dlJaWYufOnYiNjW2NGomISIc0esTxxhtv4I033sD06dOF227cuIErV64gIyNDk7UREZEOEv3OcQCwtLSEpaUlJk+e3NL1EBGRjlM7OK5fv47PPvsMeXl5MDMzw7vvvovhw4drsjYiItJBal2OW1hYiJUrV2LixImIjo6Gj48PwsLCkJ6e3iJFHDlyBM7Ozpg4cSKioqJqjV+7dg3u7u5wdXWFl5cXHjx40CL7JSIi8dQKjvDwcLz//vsIDQ1FSkoKAgIC4O3tjdDQ0GYXUFBQgNDQUERHR+Pw4cOIiYlBTk6OMK5UKrF48WJ4e3sjISEBffv2RXh4eLP3S0RETaNWcGRnZ2P06NHYsGEDlixZgpkzZ2LMmDEoLS1tdgFpaWlwcHCAmZkZTE1N4eTkhMTERGH88uXLMDU1haOjIwDAx8cH8+fPb/Z+iYioadQ6x9GtWzecP38en3zyCVavXo3Y2FjY29vD0LBJ59ZVFBYWQiqVCssymQwXL14Ulm/evImXXnoJ/v7+yMjIQM+ePbF27VrR++natUOTa5RKOzZ52+cVe9YP+tazvvULaKZntZ75vby8EBAQgMWLF2P8+PEYOnQodu/eDS8vr2YXoFAoIJFIhGWlUqmyXFlZibNnz2L//v2wtbXFzp07sXXrVmzdulXUfoqKSqBQKEXXJ5V2hFz+SPR2zzP2rB/0rWd96xdoXs8GBpJ6X3CrNVXVq1cvLFu2DF999RUWLFiArVu3Yvz48S1yOa6FhQXkcrmwLJfLIZPJhGWpVApLS0vY2toCAFxcXFSOSIiIqHWpPdc0YsQItT9WXYyRI0di165dKC4uRrt27ZCcnIxNmzYJ44MGDUJxcTEyMzNhbW2N1NRU2NjYtHgdRESknuafpGgmc3NzrFixAh4eHqioqMCsWbNgZ2cHb29v+Pn5wdbWFn//+98RGBiIsrIyWFhYYPv27doum4hIb0mUSqVaE/9ffPEFxo8fj1dffVXTNWkEz3Gojz3rB33rWd/6BbR8jgMAtm/fjtmzZ7fYm/6IiOj5pHZwVPPy8kJycnK941lZWVi6dGmziiIiIt0lKjhCQkJgZ2eH5cuX46uvvqpznbKyMqSkpLRIcUREpHtEBUenTp0QERGBiRMnIjg4mCepiYj0kOipKmNjY+zcuRPu7u744osvsHLlSlRUVGiiNiIi0kFNvhzX398f5ubmCAkJQVFREXbv3o0OHZr+sR5ERPR8EH3E8SwvLy/s2LED586dw/z581FYWNhSdRERkY5qVnAATz8CZO/evcjLy8PcuXORnZ3dEnUREZGOUjs4jh07hl69etU55uDggP3796OiogLr1q1rseKIiEj3qB0c3bt3R/v27esdt7a2RkxMDF577bWWqIuIiHSU2sFx8eJFfP311yguLq53nVdeeQUHDx5EZGQkMjMzsXv37hYpkoiIdIfaV1VVVlZiz549CAoKwoABAzBu3DiMGzdOOMKoqqrC2bNnkZKSgtTUVBQUFGDIkCGaqpuIiLRE7eCwt7dHcnIysrOzkZqaiqSkJISEhKBnz57o1asXTp8+jcrKSowaNQrLli3D6NGj0alTJ03WTkREWiD6fRx9+vRBnz59sGjRIhQWFiI1NRVXr17Fjh07MGLECBgbG2uiTiIi0hHN+j4OmUyGuXPntlQtRET0HGj2+ziIiEi/MDiIiEgUBgcREYnC4CAiIlEYHEREJAqDg4iIRGFwEBGRKAwOIiISRSeC48iRI3B2dsbEiRMRFRVV73o//PADxo4d24qVERFRTc1653hLKCgoQGhoKOLi4mBsbIy5c+di+PDh6N27t8p69+7dw7Zt27RUJRERVdP6EUdaWhocHBxgZmYGU1NTODk5ITExsdZ6gYGBWLp0qRYqJCKiZ2k9OAoLCyGVSoVlmUyGgoIClXUiIyPRr18/DBgwoLXLIyKiGrQ+VaVQKCCRSIRlpVKpspyVlYXk5GTs27cP+fn5Td5P164dmrytVNqxyds+r9izftC3nvWtX0AzPWs9OCwsLJCeni4sy+VyyGQyYTkxMRFyuRwzZ85ERUUFCgsLMW/ePERHR4vaT1FRCRQKpej6pNKOkMsfid7uecae9YO+9axv/QLN69nAQFLvC26tT1WNHDkSp06dQnFxMcrKypCcnAxHR0dh3M/PD0lJSYiPj0d4eDhkMpno0CAiopaj9eAwNzfHihUr4OHhgenTp8PFxQV2dnbw9vbGpUuXtF0eERHVIFEqleLnb55DnKpSH3vWD/rWs771C7zAU1VERPR8YXAQEZEoDA4iIhKFwUFERKIwOIiISBQGBxERicLgICIiURgcREQkCoODiIhEYXAQEZEoDA4iIhKFwUFERKIwOIiISBQGBxERicLgICIiURgcREQkCoODiIhEYXAQEZEohtouQJccPnENCT/lNrqe65uvYfpbPTVfEBGRDmJwPGP6Wz1VAmFb1C8wMm6DD2YP0GJVRES6hVNVREQkCoODiIhEYXAQEZEoDA4iIhJFJ4LjyJEjcHZ2xsSJExEVFVVr/NixY5g2bRpcXV3h6+uLBw8eaKFKIiICdCA4CgoKEBoaiujoaBw+fBgxMTHIyckRxktKSrB+/XqEh4cjISEBVlZW2LVrlxYrJiLSb1oPjrS0NDg4OMDMzAympqZwcnJCYmKiMF5RUYF169bB3NwcAGBlZYW7d+9qq1wiIr2n9fdxFBYWQiqVCssymQwXL14Uljt37owJEyYAAMrLyxEeHg53d3fR++natYPobYyM2wAApNKOord93rFn/aBvPetbv4BmetZ6cCgUCkgkEmFZqVSqLFd79OgRlixZAmtra7zzzjui91NUVAKFQilqm4onVTAybgO5/JHo/T3PpNKO7FkP6FvP+tYv0LyeDQwk9b7g1vpUlYWFBeRyubAsl8shk8lU1iksLMS8efNgZWWFzZs3t3aJRET0DK0Hx8iRI3Hq1CkUFxejrKwMycnJcHR0FMarqqrg4+ODyZMnIyAgoM6jESIiaj1an6oyNzfHihUr4OHhgYqKCsyaNQt2dnbw9vaGn58f8vPzceXKFVRVVSEpKQkA0L9/fx55EBFpidaDAwCmTp2KqVOnqty2d+9eAICtrS0yMzO1URYREdVB61NVRET0fGFwEBGRKAwOIiIShcFRB4VSiVOX85Gb/wgZ14uxYd/POHU5HwqluPeBEBG9iHTi5LguUSiV+HvcJVzJLcbjCgUA4Eb+I0QmZiI9sxBLZtjCgJcEE5Ee4xFHDWeuFKiERrXHFQpcyS3G2SsFWqqMiEg3MDhqSP75Vq3QqPa4QoHkn2+1ckVERLqFwVHD/YflDY4XNzJORPSiY3DU0PlPbRsc79LIOBHRi47BUcPEoT1gYlT3r8XEyAATh/Zo5YqIiHQLg6OG4f3M0e+1LrXCw8TIAP1e64Jh/cy1VBkRkW7g5bg1GEgkWDLDFmevFODLxN9QWaVAD1kHTBzaA8P6mfNSXCLSewyOOhhIJHCwscDxX+/AyLgNPpg9QNslERHpDE5VERGRKAwOIiIShcFBRESiMDiIiEgUBgcREYnC4CAiIlEYHEREJAqDg4iIRGFwEBGRKAwOIiISRSeC48iRI3B2dsbEiRMRFRVVazwjIwMzZsyAk5MTAgICUFlZqYUqiYgI0IHgKCgoQGhoKKKjo3H48GHExMQgJydHZZ1Vq1YhKCgISUlJUCqViI2N1VK1RESk9eBIS0uDg4MDzMzMYGpqCicnJyQmJgrjeXl5KC8vx8CBAwEAM2bMUBknIqLWpfXgKCwshFQqFZZlMhkKCgrqHZdKpSrjRETUurT+seoKhQKSZ77jQqlUqiw3Nq6url07iN7GyLgNAEAq7Sh62+cde9YP+tazvvULaKZnrQeHhYUF0tPThWW5XA6ZTKYyLpfLheV79+6pjKurqKgECoVS1DYVT6pgZNwGcvkj0ft7nkmlHdmzHtC3nvWtX6B5PRsYSOp9wa314Bg5ciR27dqF4uJitGvXDsnJydi0aZMw3q1bN5iYmODcuXMYPHgw4uPj4ejoqJFaDp+4hoSfcmvd7rk1VWXZ9c3XMP2tnhqpgYhI12k9OMzNzbFixQp4eHigoqICs2bNgp2dHby9veHn5wdbW1uEhIQgMDAQJSUlsLGxgYeHh0Zqmf5Wz1qBoI+vUoiIGiJRKpXi5m+eU02ZqgL0MzjYs37Qt571rV9Ac1NVWr+qioiIni8MDiIiEoXBQUREojA4iIhIFAYHERGJwuAgIiJRtP4+jtZiYCD+Y0paYtvnFXvWD/rWs771CzS954a205v3cRARUcvgVBUREYnC4CAiIlEYHEREJAqDg4iIRGFwEBGRKAwOIiIShcFBRESiMDiIiEgUBgcREYnC4Pj/SkpK4OLigtu3b9cay8jIwIwZM+Dk5ISAgABUVlZqocKW11DPx44dw7Rp0+Dq6gpfX188ePBACxW2vIZ6rvbDDz9g7NixrViVZjXU87Vr1+Du7g5XV1d4eXnpxd/58uXLmDlzJlxdXbFo0SI8fPhQCxW2rN27d2PKlCmYMmUKtm/fXmu8pZ/DGBwALly4ADc3N+Tm5tY5vmrVKgQFBSEpKQlKpRKxsbGtW6AGNNRzSUkJ1q9fj/DwcCQkJMDKygq7du1q/SJbWGN/ZwC4d+8etm3b1npFaVhDPSuVSixevBje3t5ISEhA3759ER4e3vpFtrDG/s6bN2+Gn58fEhIS8PrrryMiIqJ1C2xhaWlpOHnyJA4dOoTDhw/j8uXL+P7771XWaennMAYHgNjYWKxbtw4ymazWWF5eHsrLyzFw4EAAwIwZM5CYmNjKFba8hnquqKjAunXrYG5uDgCwsrLC3bt3W7vEFtdQz9UCAwOxdOnSVqxKsxrq+fLlyzA1NYWjoyMAwMfHB/Pnz2/tEltcY39nhUKB0tJSAEBZWRnatm3bmuW1OKlUijVr1sDY2BhGRkbo1asX7ty5I4xr4jlMbz4dtyGbN2+ud6ywsBBSqVRYlkqlKCgoaI2yNKqhnjt37owJEyYAAMrLyxEeHg53d/fWKk1jGuoZACIjI9GvXz8MGDCglSrSvIZ6vnnzJl566SX4+/sjIyMDPXv2xNq1a1uxOs1o7O+8Zs0aeHp6Ijg4GO3atXvuZxD69Okj/D83NxffffcdDhw4INymiecwHnE0QqFQQCL578cLK5VKleUX2aNHj/CXv/wF1tbWeOedd7RdjkZlZWUhOTkZvr6+2i6l1VRWVuLs2bNwc3PDoUOH0KNHD2zdulXbZWlUeXk5AgICsG/fPpw8eRLz5s3DX//6V22X1SKys7Ph6emJ1atX47XXXhNu18RzGIOjERYWFpDL5cLyvXv3GpzqeFEUFhZi3rx5sLKyavQV3IsgMTERcrkcM2fOxF/+8heh/xeZVCqFpaUlbG1tAQAuLi64ePGilqvSrKysLJiYmMDOzg4AMGfOHJw9e1bLVTXfuXPnsHDhQqxcubLWizxNPIcxOBrRrVs3mJiY4Ny5cwCA+Ph4YU74RVVVVQUfHx9MnjwZAQEBenGE5efnh6SkJMTHxyM8PBwymQzR0dHaLkujBg0ahOLiYmRmZgIAUlNTYWNjo+WqNMvS0hL5+fm4du0aACAlJUUIzufV3bt3sWTJEoSEhGDKlCm1xjXxHMZzHPXw9vaGn58fbG1tERISgsDAQJSUlMDGxgYeHh7aLk8jqnvOz8/HlStXUFVVhaSkJABA//79X8gjj2f/zvri2Z7//ve/IzAwEGVlZbCwsKjzUs4XwbM9b9myBcuXL4dSqUTXrl0RHBys7fKaJSIiAo8fP1aZZpw7dy5SU1M19hzGbwAkIiJROFVFRESiMDiIiEgUBgcREYnC4CAiIlEYHEREJAqDg4iIRGFwEGnB0aNH0b9/f1RUVGi7FCLRGBxEWpCZmYnevXvDyMhI26UQicbgINKCjIwM9O3bV9tlEDUJg4NICzIzM1WC49SpUxg+fDg2b96MqqoqLVZG1Dh+VhVRKysuLoZcLheCY//+/dixYwf8/f0xZ84cLVdH1DgGB1Ery8jIgEQiQa9evbB27VokJydjz549cHBw0HZpRGphcBC1sszMTHTq1An/8z//g6KiIsTGxsLS0lLbZRGpjcFB1Mqqv/8iPT0d+/btY2jQc4cfq07UyqZOnQonJyfcvHkTZ8+eRVxcHLp06aLtsojUxquqiFrRkydPcO3aNVhZWWHjxo3o0qULli1bhsrKSm2XRqQ2BgdRK8rJyUFlZSXeeOMNtG3bFrt27UJ2dja2bNmi7dKI1MbgIGpFmZmZMDU1xauvvgrg6fdBh4aG4sCBA4iLi9NydUTq4TkOIiIShUccREQkCoODiIhEYXAQEZEoDA4iIhKFwUFERKIwOIiISBQGBxERicLgICIiURgcREQkyv8D2jvqQjb9KsIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.errorbar(x = 1,y = np.mean(np.array(predict_zs))*(2**N2)/Wz,\\\n",
    "             yerr=np.std(np.array(predict_zs))*(2**N2)/(Wz*np.sqrt(5000)),capsize = 5)\n",
    "plt.scatter(x = 1,y = np.mean(np.array(predict_zs))*(2**N2)/Wz,s= 60,)\n",
    "plt.errorbar(x = 2,y = np.mean(np.array(predict_zzs))*(2**N2)/Wzz,\\\n",
    "             yerr=np.std(np.array(predict_zzs))*(2**N2)/(Wzz*np.sqrt(5000)),capsize = 5)\n",
    "plt.scatter(x = 2,y = np.mean(np.array(predict_zzs))*(2**N2)/Wzz,s= 60,)\n",
    "\n",
    "# plt.ylim([0.8,1.1])\n",
    "# plt.plot(np.linspace(-0.05,0.05,20),np.ones(20),\"k--\",alpha = 0.6)\n",
    "plt.xlabel(r\"$k$\",fontsize = 15)\n",
    "plt.ylabel(r\"$\\langle Z^{\\otimes k}\\rangle$\",fontsize = 15)\n",
    "plt.title(\"10 qubit GHZ state (3 layer brickwall circuit)\",fontsize = 15)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b3e73f8a-d32e-49db-8112-47b5e04d6fef",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "aabb18a6-c200-44b1-8ec2-6c83ab1bcca9",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9188ba9f-43de-4614-9f4d-0f4b03f565d7",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2d458ca5-7f9a-43fb-ae1e-55ca05e5b62e",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4741a379-922f-4d57-9fe0-46d09b9bf840",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "45c5b8ae-0852-4d01-9454-98ee6d5474f4",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "d40f5729-bff3-403c-a7b3-2ff4cf70a4a7",
   "metadata": {
    "jp-MarkdownHeadingCollapsed": true,
    "tags": []
   },
   "source": [
    "# Test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "id": "70627d3f-491e-4176-9b47-b59c4ca6529c",
   "metadata": {},
   "outputs": [],
   "source": [
    "Wx_lst = []\n",
    "for _ in range(10000):\n",
    "    state = pc.stabilizer.random_bit_state(N2)\n",
    "    circ.backward(state)\n",
    "    Wx_lst.append(abs(state.expect(X))**2)\n",
    "Wx = np.mean(np.array(Wx_lst))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "id": "1087a220-9474-42bb-a1e5-e2f9a3297744",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.9797377830750893"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1/Wx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "d46f608b-b339-4883-8d2e-63714f5e35a6",
   "metadata": {},
   "outputs": [],
   "source": [
    "Wz_lst = []\n",
    "for _ in range(10000):\n",
    "    state = pc.stabilizer.random_bit_state(N2)\n",
    "    circ.backward(state)\n",
    "    Wz_lst.append(abs(state.expect(Z))**2)\n",
    "Wz = np.mean(np.array(Wz_lst))\n",
    "Wxx_lst = []\n",
    "for _ in range(10000):\n",
    "    state = pc.stabilizer.random_bit_state(N2)\n",
    "    circ.backward(state)\n",
    "    Wxx_lst.append(abs(state.expect(XX))**2)\n",
    "Wxx = np.mean(np.array(Wxx_lst))\n",
    "Wzz_lst = []\n",
    "for _ in range(10000):\n",
    "    state = pc.stabilizer.random_bit_state(N2)\n",
    "    circ.backward(state)\n",
    "    Wzz_lst.append(abs(state.expect(ZZ))**2)\n",
    "Wzz = np.mean(np.array(Wzz_lst))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "id": "98d8b546-dddc-46ee-9f64-aaca8a930533",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.19963201471941"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1/Wxx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "fb9b4197-059d-4243-a6c3-4992231c5293",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.347245409015025"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1/Wzz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5341b68c-95bd-4de0-8cdf-037e0a2bbe77",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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