dev/demo-MIPT.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "2d6d342c-ca5c-42dc-9d31-a6819cc4f9e8",
   "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": 21,
   "id": "3d72be68-2d1d-4648-8d12-0b4a123970c9",
   "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "9f5c590d-0f48-40ae-afcd-6bfb3b19564e",
   "metadata": {},
   "outputs": [],
   "source": [
    "from numba import njit\n",
    "import seaborn as sns\n",
    "sns.set_theme()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "4e7184cb-8248-4ea5-b4f0-a39a154e5dc5",
   "metadata": {},
   "outputs": [],
   "source": [
    "def one_layer_brick_wall(circ,even=True):\n",
    "    for ii in range(int(circ.N//2)):\n",
    "        if even:\n",
    "            circ.gate(2*ii,2*ii+1)\n",
    "        else:\n",
    "            circ.gate((2*ii+1),(2*ii+2)%circ.N)\n",
    "    return circ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "f288bc50-165e-4c13-8968-9919412e5491",
   "metadata": {},
   "outputs": [],
   "source": [
    "@njit\n",
    "def generate_measurement_position(N,p):\n",
    "    positions = []\n",
    "    for i in range(N):\n",
    "        if np.random.rand()<p:\n",
    "            positions.append(i)\n",
    "    return positions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "f0a2c457-4172-4f4e-b405-badced0c7a48",
   "metadata": {},
   "outputs": [],
   "source": [
    "def one_layer_measurement(circ,p):\n",
    "    '''\n",
    "    Input:\n",
    "    p (float): measurement probability\n",
    "    '''\n",
    "    pos = generate_measurement_position(circ.N,p)\n",
    "    circ.measure(*pos)\n",
    "    return circ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "6ca2cdb6-f21c-4a41-8890-8962c067b107",
   "metadata": {},
   "outputs": [],
   "source": [
    "def create_circuit(N,depth,p):\n",
    "    circ = pc.circuit.Circuit(N)\n",
    "    for i in range(depth):\n",
    "        circ = one_layer_brick_wall(circ,even=True)\n",
    "        circ = one_layer_measurement(circ,p)\n",
    "        circ = one_layer_brick_wall(circ,even=False)\n",
    "        circ = one_layer_measurement(circ,p)\n",
    "    return circ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "10124563-4177-495f-8985-e327edfa26ff",
   "metadata": {},
   "outputs": [],
   "source": [
    "import time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "794d8949-f5fc-4a25-8998-6d8baf860580",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3\n",
      "time:  0.5204460620880127\n"
     ]
    }
   ],
   "source": [
    "ctime = time.time()\n",
    "circ = create_circuit(300,10,0.1)\n",
    "state = pc.zero_state(300)\n",
    "state = circ.forward(state)\n",
    "entropy = state.entropy([i for i in range(150)])\n",
    "print(entropy)\n",
    "print(\"time: \",time.time()-ctime)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d2edf5c6-5d6b-4579-9515-fe5390888068",
   "metadata": {},
   "source": [
    "# Average half-system entropy"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "32fef692-b6ec-4063-8288-a4d852eb0870",
   "metadata": {},
   "source": [
    "When $p<p_c=0.16$, the system is in the entropy volume-law phase."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "5917065f-3319-4688-9b6d-10e940bfa903",
   "metadata": {},
   "outputs": [],
   "source": [
    "ps = np.array([0.1,0.13,0.15,0.17,0.2,0.23])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "9c6f330a-afa1-448c-aa96-2b69165dd7cd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "measurement prob:  0.1\n",
      "measurement prob:  0.13\n",
      "measurement prob:  0.15\n",
      "measurement prob:  0.17\n",
      "measurement prob:  0.2\n",
      "measurement prob:  0.23\n"
     ]
    }
   ],
   "source": [
    "entropy_mean = np.zeros_like(ps)\n",
    "entropy_std = np.zeros_like(ps)\n",
    "for k in range(ps.shape[0]):\n",
    "    print(\"measurement prob: \",ps[k])\n",
    "    entropy_lst = []\n",
    "    for _ in range(1000):\n",
    "        circ = create_circuit(300,10,ps[k])\n",
    "        state = pc.zero_state(300)\n",
    "        state = circ.forward(state)\n",
    "        entropy = state.entropy([i for i in range(150)])\n",
    "        entropy_lst.append(entropy)\n",
    "    entropy_mean[k] = np.mean(np.array(entropy_lst))\n",
    "    entropy_std[k] = np.std(np.array(entropy_lst))/np.sqrt(1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "a619cdc3-b75f-4e49-aa8f-af0046ae839c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '10 layers of brickwall circuit')"
      ]
     },
     "execution_count": 24,
     "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 = ps,y = entropy_mean,\\\n",
    "             yerr=entropy_std,capsize = 5)\n",
    "plt.xlabel(r'$P$',fontsize = 14)\n",
    "plt.ylabel(\"Half-system entropy\",fontsize = 14)\n",
    "plt.title(\"10 layers of brickwall circuit\",fontsize = 14)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "01c7dff5-475a-4424-acd7-111b078317a3",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f0186ab4-0e0f-4771-9ab8-643924116dff",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "979cf2e3-b64b-4443-a350-e20d691be0d5",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
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