tutorials/algorithms/09_IQPE.ipynb

{
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   "source": [
    "# Iterative Quantum Phase Estimation Algorithm\n",
    "\n",
    "The goal of this tutorial is to understand how the Iterative Phase Estimation (IPE) algorithm works, why would we use the IPE algorithm instead of the QPE (Quantum Phase Estimation) algorithm and how to build it with Qiskit using the same circuit exploiting reset gate and the `c_if` method that allows to apply gates conditioned by the values stored in a classical register, resulting from previous measurements.\n",
    "\n",
    "**References**\n",
    "\n",
    "- [Section 2 of Lab 6: Iterative Phase Estimation (IPE) Algorithm](https://learn.qiskit.org/course/ch-labs/lab-6-iterative-phase-estimation-algorithm) \n",
    "\n",
    "- [Ch.3.6 Quantum Phase Estimation](https://learn.qiskit.org/course/ch-algorithms/quantum-phase-estimation)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "edf04b07",
   "metadata": {},
   "source": [
    "# Conditioned gates: the ``c_if`` method"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4eb985e3",
   "metadata": {},
   "source": [
    "Before starting the IPE algorithm, we will give a brief tutorial about the Qiskit conditional method, `c_if`, as it goes into building the IPE circuit.\n",
    "\n",
    "`c_if` is a function (actually a method of the gate class) to perform conditioned operations based on the value stored previously in a classical register. With this feature you can apply gates after a measurement in the same circuit conditioned by the measurement outcome.\n",
    "\n",
    "For example, the following code will execute the $X$ gate if the value of the classical register is $0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "8d3ed2e3",
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   "outputs": [
    {
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      "text/plain": [
       "<Figure size 253.926x144.48 with 1 Axes>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from qiskit import QuantumCircuit\n",
    "\n",
    "qc = QuantumCircuit(1, 1)\n",
    "qc.h(0)\n",
    "qc.measure(0,0)\n",
    "qc.x(0).c_if(0, 0)\n",
    "qc.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7bb05bbd",
   "metadata": {},
   "source": [
    "We highlight that the method ``c_if`` expects as the first argument a whole classical register, not a single classical bit (or a list of classical bits), and as the second argument a value in decimal representation (a non-negative integer), not the value of a single bit, 0, or 1 (or a list/string of binary digits).\n",
    "\n",
    "Let's make another example. Consider that we want to perform a bit flip on the third qubit after the measurements in the following circuit, when the results of the measurement of $q_0$ and $q_1$ are both $1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "00b7d3a1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 387.597x264.88 with 1 Axes>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from qiskit import QuantumRegister, ClassicalRegister\n",
    "\n",
    "q = QuantumRegister(3, 'q')\n",
    "c = ClassicalRegister(3, 'c')\n",
    "qc = QuantumCircuit(q, c)\n",
    "\n",
    "qc.h([0, 1, 2])\n",
    "qc.barrier()\n",
    "qc.measure(q, c)\n",
    "\n",
    "qc.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ae92e111",
   "metadata": {},
   "source": [
    "We want to apply the $X$ gate, only if both the results of the measurement of $q_0$ and $q_1$ are $1$. We can do this using the ``c_if`` method, conditioning the application of $X$ depending on the value passed as argument to ``c_if``.\n",
    "\n",
    "We will have to encode the value to pass to the ``c_if`` method such that it will check the values 011 and 111 (in binary representation), since it does not matter what $q_2$ is measured as.\n",
    "\n",
    "The 2 integer values in decimal representation:\n",
    "\n",
    "$$\\begin{alignedat}{2}\n",
    "\\left.\\begin{array}{ccc}2^2 & 2^1 & 2^0\\\\\\hline 0 & 1 & 1\\end{array}\\ \\right\\vert&\\ (0\\times2^2) + (1\\times2^1) + (1\\times2^0) = &&2 + 1 = 3 \\\\[10px]\n",
    "\\left.\\begin{array}{ccc}2^2 & 2^1 & 2^0\\\\\\hline 1 & 1 & 1\\end{array}\\ \\right\\vert&\\ (1\\times2^2) + (1\\times2^1) + (1\\times2^0) = 4 + {}&&2 + 1 = 7\n",
    "\\end{alignedat}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "25fa9853",
   "metadata": {},
   "source": [
    "We can check the solutions using the ``bin()`` method in python (the prefix `0b` indicates the binary format)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "19b565e1",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0b11\n",
      "0b111\n"
     ]
    }
   ],
   "source": [
    "print(bin(3))\n",
    "print(bin(7))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c4c65c39",
   "metadata": {},
   "source": [
    "So we have to apply $X$ to $q_2$ using ``c_if`` two times, one for each value corresponding to 011 and 111."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "1bbd6b37",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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2b94sOgIRyRRLgwx4e3uLjkBEMsXSIANBQUGiIxCRTLE0iIhIMpYGGQgICBAdgYhkiqVBBrKyskRHICKZYmkQEZFkLA0iIpKMpUEGkpOTRUcgIpliaRARkWQsDTIQHh4uOgIRyRTvcmtCSjZQUi5m2449H8677FLbmzdvHnJycoRsW6VStejOwADw6vfnce727dYNJIHvI4/gbQ+vFq0rKjPw23I3haVhQkk5cPGq6BRErSsnJwcajUZ0jGY7d/s2viq/ITpGs3TEzMZweIoMxMTEiI5ARDLF0iADc+bMER2BiGSKpUEGAgMDRUcgIpliaZCBsrIy0RGISKZYGkREJBlLgwx4ebXuKXpE1HmwNMjAnj17REegTsLGxkZ0BGplLA0ysGTJEtERSEYsLCwQFhaGt99+G+np6Th37hzOnj2Lffv2YfHixU0+fyUhIQHHjx9H79692zkxtSVZl4ZWq8WGDRvQv39/WFlZwdfXFxqNBh4eHpg1a5boeJ1WUlKS6AgkAxYWFli0aBGKioqwZ88eLFiwAGq1Gj4+PlCpVJgwYQJWrFiBU6dOISsrC2PHjtWvm5CQgLlz58LNzQ0DBw4U+Cmotcn6ivCoqCikpKRg8eLF8PPzQ2ZmJqZOnYqysjIsWLBAdLxGJa9So+/AEAyduEjSdCI58vX1xaeffgpvb28AwLfffoukpCRkZ2fjX//6F7p06QIPDw8MHz4c06ZNg7+/Pw4ePIhdu3bh9u3biI6ORnV1NcLCwnDkyBEhn0F37x5+/eN8KAf7wmz2S/rptXv3QbtnL7psfR8Ka2sh2YyRe27ZlkZiYiJ27tyJjIwMBAUFAQCCg4Nx5swZpKSkYMiQIYITEnVOI0aMwKFDh2BjY4Pvv/8e0dHRSE9PN1juzJkzSExMxBtvvIGXX34Zq1evxgsvvAAA+sJITU1t7/h6CnNzdIl9Hb/+cR4UQwOgHKyC7vJlaD/8GGarV8iyMAD555bt8FRcXBxCQ0P1hVGvX79+MDc3h4+PD8rLyzF27Fi4u7vD19cXTz/9NAoLCwUl7jw64j2JqHU89thjSE1NhY2NDRITE6FSqRotjPtVVVUhISGhwbDm+fPncejQobaOa5LiMRco//Aiajdsgu7GDfy6dj2UE8ZB6TNIdDSj5JxblqVRXFyM7777DlOmTDGYV1RUBG9vb1haWkKhUGDevHkoKCjAuXPnMHbsWMyYMUNA4s4lLy9PdAQSQKFQ4MMPP4StrS3279+PF154AVVVVZLWTUhIwIsvvojq6mqUl5dj8ODBiI6ObuPE0ignToCirzN+nR0DmJlB+WKk6EiSyDW3LIeniouLAQB9+vRpMP3u3bvQaDQYM2YMAKBHjx4ICQnRzx8xYgTi4+MlbUOhUEhabvJb6XDyVEtatt6p/atxOnVDg2n3qirRd2BIE2s0TqPJwJ+eDm7WOlLMnz/f6PxNmzZJWkYOFq79C4C6n+f9r+VOjrknTZqE4OBgXL16FTNnzoRWq5W0Xv1B7/ohKUtLS6SkpGDNmjX45JNPcPuB24JrNJoWf1az9Wuh9PVp1joKhQIKn0HQnT4D5fMRUJibN3u7GRkZUAQMa/Z6QMsyA+2fW6fTSVpOlnsadnZ2AICCgoIG0+Pj41FaWgo/P79G10tISMDEiRPbOp5JQye8hehtNxv8cXB/QnQsIqPq7268atUqXLt2TdI6DxZGamoq9u7di4yMDNjY2OiPcYiku3wZ2r99BuXvp0C762/QXe0YzzqQa25Z7mm4urrCx8cHcXFx6NWrFxwdHZGcnKw/qNZYaSxfvhyFhYU4evSopG1IbdV308Q9TyMoSI3kVdJyNkd+fr7R+Zs2bTJ5SvPGjRtbM1KLxa7bBqDu53n/a7kTnVutVjc4dmVvb4/Ro0fjl19+wccffyzpPRorjHpbt26FWq3G1KlTsWXLlgbrBQUFISMjo0W5Q7JPNuvZFLqae3XHA8ImwmzGi9CVl6N2/UaYrYuDQin9O7NarcbhFv58mpsZkEfupshyT0OpVCIpKQne3t6Ijo7GjBkzYGdnh5iYGJiZmcHHp+Gu3qpVq/D3v/8dX3zxBbp16yYodeexfPly0RGonfn71z0i8uTJk6ioqDC5vLHCAIDDhw8DAIYMGQJlM37JtTbthx9B0aULlJH/DwBg9srL0P30M7R79grLJIWcc8uyNADA3d0d6enpuHPnDoqKirBy5Urk5ubCy8sLXbt21S+3fPlyHDx4EGlpaejRo4e4wJ1IRESE6AjUzuqvxzh37pzJZU0VBgBcv34dV65cQffu3eHi4tLqeaXQns2BNvULmMW+AUWXukEVRbduMFv4GrR/3QXd5ctCcpki99yyHJ5qSnZ2NoYN+89Bnby8PCxbtgxubm5Qq9X66aKefQwA4YsymjVdjjw9PXHhwgXRMagdnThxAitWrMCxY8eMLjd58mSThVFv48aNsLW1NTgQ3l6Ug1VQHkgxnD7QG8qD4r+xN0XuuTtMaVRWVqKgoACvvPKKfpq3t3eHGL8mkrtjx46ZLAyg7maWCQkJSEtLM3nhXkJCQiulIznpMKVhbW2N2tpa0TGIHnqmTsemzk22xzRInPuH+oiI7sfSIAMPniJJRFSPpUEG5HL7ByKSH5YGGWjphVdE1PmxNIiISDKWBhERScbSIAO8sI+ImsLSIAO7d+8WHYGIZKrDXNwnimPPh2/bS5cu5f2nOjmVStXsdS4VlQIAXPs+2uB1e2y7nu8jjzR7nUtF/wIAuPZ1aPC6rbf7W9Ztjcwt3bYpLA0TwvxFJyBqfS25xUf9LdzXLpzV4HV7etvDq9nrxB75d9awSQ1et5eOmNkYDk8REZFkLA0ysHnzZtERiEimWBpkoP7ZCkRED2JpkIGgoCDREYhIplgaREQkGUuDiIgk4ym3D6EBAwYYnb906VKTyxDRw4l7GmRg2bJloiMQkUyxNIiISDKWBhERScbSICIiyVgaREQkGUuDiIgkY2kQEZFkLI0mPP3001CpVBg0aBDCw8NRUVEhOhK1gYyMDHh7e6Nfv36YOXMmamtrRUcyae7cuXByckKXLh3nMqsrV67gySefhKenJ7y9vfHmm2+KjkQtxNJoQlJSEnJycpCbmwsnJyds3LhRdCRqZVqtFjNnzkRSUhIKCwtRUVGBXbt2iY5l0pQpU5CdnS06RrN06dIF69atw4ULF3D27Fl8/fXX2L9/v+hY1AIsjSbY2toCqPvFUlVVBYVCITgRtbasrCw4ODjAy6vuITlRUVHYs2eP4FSmPfHEE+jTp4/oGM3y6KOPwt+/7olmFhYWGDx4MIqKigSnopZgaRgxadIk2Nvb4/vvv8err74qOg61suLiYjg7O+v/3rdvX1y5ckVgoofDjRs3sG/fPjz11FOio1ALdJxBUQH27t2LmpoaREVFITk5GdOnTxcdiQAcPJyJS1dKG0z7n4/2NPpa5dUPQb/zbfR9dDpd2wRsRMlP15B8SGMwvbHclhbm+P+Tn0E3K8t2y9cYnU6Hz/+ejp+vlTeY3tS/9Ui/gfD38TD6njU1NQgPD8fcuXN5f7MOinsaJlhYWOD555/H3r17RUehf3siYBBu3KxA6dXr+mkPvi69eh13q6rxO5Vnk+/j7OzcYM+iqKgITk5ObZLZsY8dHHvb6bMZy+030F14YQCAQqFA4FAfXL1WbjIzAKi8+xl9v9raWkybNg0qlYp77h0YS6MRt2/fRmlp3TdZrVaLAwcO8Gl2MtLT9hGMDxlpdBkFgCnPqWFladHkMv7+/iguLsb58+cBADt27EBYWFhrRm1g7JPD0dP2EaPLePZzMfltvT059LbDU6P8jS5jZqbE78cGo4uZmdHlZs2ahUceeQRvv/12a0akdsbSaMTt27cxfvx4+Pj4wMfHB7/++isWLVokOhbdZ8jA/vB2f6zJ+U8E+MCtr4PR9zAzM8MHH3yA8PBwuLm5wdraGpGRka2c9D+sLC0Q8ZwaTZ1S0b2bFSaHBpo86WL27NlwcnJCbW0tnJycEBMT0/ph7xM41Acujr2bnP9M4FD0+e9eRt/jm2++wYcffojs7GwMHjwYKpUK77zzTmtHpXbAYxqNcHBwQFZWVqPzdDodz6SSAYVCgUnPjMKPJT+j8s7dBvN62/XE04HGvx3XGz16tH5Poz087vwoAn/nC83JcwbzwkIDYd29q8n3+Mtf/tIW0ZqkVCoRMTYY//PRHtTU3Gsw73HnR/FEwCCT7zFy5Mh2PYZEbYd7Gs2kOXkOn+5Lw68d4CKwzs66W1eEj2n4PHMzZd1QibmML3x76gl/g2/m/oM84N3/MTGBJPivHjYYO3p4g2mWFuaIeE4NpYy/RGm1Wlz6/jv8Y++niIyMxNeHD6Li5g3RsUyquFWOb478LyIjI/FFyi5czM+VzYWn8v2fJVFubi6WLVsGjUaDyspKODs7IyIiAqtXr271bVVX1+Crk+fg7GBvcvyW2scAt74Y6jsAp87lAwCeGuUPh952glMZ16WLGZ4fNxrvfpyC2lotetk+gnFPDje9omABPh64UPgDLhTWXV8xPmSkyWM0Iv3www947rnncP78eSgUCuRmK1FbW4tvjvwvbHQ3sWbNGtmNGuh0Oixbtgxb166ETqfDcTMzaLVanDt1DF6ZXyI1NRVubm5CM3boPY3Tp09j2LBhyMvLw/r165GamorY2Fj89NNPbbK9zDN5+KWqGiEj/drk/allnhs9HP/VwwYujr0RONRHdBxJ+vx3Lzw9KkB/wN7SyAF7uVAoFAgLDUT3blbwdn8MQwb2Fx2pSbdv38bo0aORn1/3ZUKn0/3nm7pOh3Xr1iEuLk5gwsZt2LABK1as0A/l1dbW6l8XFhYiODgY5eXlxt6izSl0HXigUa1W4/z58/jnP/+pv4Jbqth129ooFRGJduZ4OtL2f2Z0GXNzC8QsWg9LS6t2SmXcvZoavLf6ddRUVxldLvi5cAwd1foXRq5dOEvSch12T+OXX37BsWPHMHXq1GYXBhF1bt9mZ5ocerp3rwb/zMtpn0ASFOZ/a7IwACA3O7Md0jStwx7TKC8vh1arbfHFWFJbFag7lrFuayKcHewxY8qYFm2PiNrPrndXSDpbK8jfE6+/Lv13QVt6991qHPibhAVra5r1+6u1ddjS6NmzJ5RKJUpKSlq0fkuGp76/dIXDWkQdwD2tAnWXeBovjvRT3+G6TP5Pn88xPA27MVqFWZv8Hur0w1PdunVDYGAgEhMT+awLImrAe8gwmCqMLl3M4e49uH0CSeDm6QNzc9MnRAwcIvZMuw59IPz06dMIDAyEi4sL3njjDbi4uKCoqAjHjh3DBx980CrbSD9+Fv/4KgsxkRPh7GDfKu9JRG2rvLwcAwcOxE8//QStVtvoMm+++abszqBavnw5li1b1ug8pVIJOzs75OXlwc5O3GnlHXZPAwD8/Pxw/Phx9O/fH/Pnz8ezzz6L1atXw8HB+O0jpKqursGxU9/Cw9WZhUHUgfTs2RNHjx7V3/rezMwMSqVSf3A8OjoaK1euFBmxUYsXL8bcuXMB1J3irFQqYfbva8IcHBxw9OhRoYUBdPA9jbZWevU6du1Lw/NjR7M0iDqgmpoa7Nu3D3v37sWdO3fg5uaGl156Sf/gLbnKz8/H9u3bUVhYiK5du2LChAkICwuDpaUM7n7M0jBOq9VCqezQO2RERK2GpUFERJLxKzQREUnG0iAiIslYGkTUqcyfPx+jRo3Sn4X0oI8//hiBgYEICAjA5s2b2zld00zlXrt2LdRqNdRqNbp3744bN8Tc4p2lQUSdxpkzZ1BZWYljx46hpqam0YepTZs2DV999RVOnDjR7g+0aoqU3LGxscjIyEBycjICAgLQq5fxpyW2FZYGEXUaJ06cwFNP1d0BNiQkBMePH8frr78OrVaL0NBQFBUVwdzcHEDd6bienp4i4+pJyV3vwIEDGD9+vKioLA0i6jxu3rwJGxsbAICtrS1u3ryJn3/+GbNmzcK4cePQt29fAMCKFSvQv39/+PnJ49k4UnMDwN69ezFp0iRRUVkaRNR52Nra6u9FV1FRgR49emD27NnYvXs3Zs6cqV9uyZIluHjxIpKSknD9+nVRcfWk5r59+zauXbuGxx9/XFRUlgYRdR7Dhw/HkSNHAACHDx/GsGHDsHLlSixduhRr164FAFRXVwMALCws0K1bN1lcZS0lNwAcOnQIY8aIfTwDS4OIOo0hQ4bAysoKo0aNgpmZGTIzMzFp0iS8+uqryM3NRV5eHtasWQO1Wo2RI0fi97//PaytrUXHlpQbqBuaCgsLE5qVV4QTEZFk3NMgIiLJWBpERCQZS4OIiCRjaRARkWQsDSIikoylQUREkrE0iIhIMpYGERFJxtIgIiLJWBpERCQZS4OIiCRjaRARkWQsDSIikoylQUREkrE0iIhIMpYGERFJxtIgIiLJ/g/LPUw7nBVLagAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 507.997x264.88 with 1 Axes>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qc.x(2).c_if(c, 3) # for the 011 case\n",
    "qc.x(2).c_if(c, 7) # for the 111 case\n",
    "\n",
    "qc.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a50c886",
   "metadata": {},
   "source": [
    "# IPE\n",
    "\n",
    "The motivation for using the IPE algorithm is that QPE algorithm works fine for short depth circuits but when the circuit starts to grow, it doesn't work properly due to gate noise and decoherence times.\n",
    "\n",
    "The detailed explanation of how the algorithm works can be found in [Iterative Phase Estimation (IPE) Algorithm](https://learn.qiskit.org/course/ch-labs/lab-6-iterative-phase-estimation-algorithm). To understand QPE in depth, you can see also [Ch.3.6 Quantum Phase Estimation](https://learn.qiskit.org/course/ch-algorithms/quantum-phase-estimation).\n",
    "\n",
    "## IPE example with a 1-qubit gate for $U$\n",
    "\n",
    "We want to apply the IPE algorithm to estimate the phase for a 1-qubit operator $U$.  For example, here we use the $S$-gate.\n",
    "\n",
    "Let's apply the IPE algorithm to estimate the phase for $S$-gate.\n",
    "Its matrix is \n",
    "$$ S = \n",
    "\\begin{bmatrix}\n",
    "1 & 0\\\\\n",
    "0 & e^\\frac{i\\pi}{2}\\\\ \n",
    "\\end{bmatrix}$$\n",
    "\n",
    "That is, the $S$-gate adds a phase $\\pi/2$ to the state $|1\\rangle$, leaving unchanged the phase of the state $|0\\rangle$\n",
    "\n",
    "$$ S|1\\rangle = e^\\frac{i\\pi}{2}|1\\rangle $$\n",
    "\n",
    "In the following, we will use the notation and terms used in [Section 2 of lab 6](https://learn.qiskit.org/course/ch-labs/lab-6-iterative-phase-estimation-algorithm#lab-15-0).\n",
    "\n",
    "Let's consider to estimate the phase $\\phi=\\frac{\\pi}{2}$ for the eigenstate $|1\\rangle$, we should find $\\varphi=\\frac{1}{4}$ (where $\\phi = 2 \\pi \\varphi$). Therefore to estimate the phase we need exactly 2 phase bits, i.e. $m=2$, since $1/2^2=1/4$. So $\\varphi=0.\\varphi_1\\varphi_2$.\n",
    "\n",
    "Remember from the theory that for the IPE algorithm, $m$ is also the number of iterations, so we need only $2$ iterations or steps.\n",
    "\n",
    "First, we initialize the circuit. IPE works with only 1 auxiliary qubit, instead of $m$ counting qubits of the QPE algorithm. Therefore, we need 2 qubits, 1 auxiliary qubit and 1 for the eigenstate of $U$-gate, and a classical register of 2 bits, for the phase bits $\\varphi_1$, $\\varphi_2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c82b1da1",
   "metadata": {},
   "outputs": [],
   "source": [
    "nq = 2\n",
    "m = 2\n",
    "q = QuantumRegister(nq, 'q')\n",
    "c = ClassicalRegister(m, 'c')\n",
    "\n",
    "qc_S = QuantumCircuit(q,c)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fead17a6",
   "metadata": {},
   "source": [
    "### First step\n",
    "\n",
    "Now we build the quantum circuit for the first step, that is, the first iteration of the algorithm, to estimate the least significant phase bit $\\varphi_m$, in this case $\\varphi_2$. For the first step we have 3 sub-steps:\n",
    "- initialization\n",
    "- application of the Controlled-$U$ gates\n",
    "- measure of the auxiliary qubit in X-basis\n",
    "\n",
    "#### Initialization\n",
    "\n",
    "The initialization consists of application the Hadamard gate to the auxiliary qubit and the preparation of the eigenstate $|1\\rangle$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "b0eb6f14",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 146.652x204.68 with 1 Axes>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qc_S.h(0)\n",
    "qc_S.x(1)\n",
    "qc_S.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "30757254",
   "metadata": {},
   "source": [
    "#### Application of the Controlled-$U$ gates\n",
    "\n",
    "Then we have to apply $2^t$ times the Controlled-$U$ operators (see also in the docs [Two qubit gates](https://qiskit.org/documentation/tutorials/circuits/3_summary_of_quantum_operations.html#Two-qubit-gates)), that, in this example, is the Controlled-$S$ gate ($CS$ for short).\n",
    "\n",
    "To implement $CS$ in the circuit, since $S$ is a phase gate, we can use the controlled phase gate $\\text{CP}(\\theta)$, with $\\theta=\\pi/2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "4629d696",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKoAAAB7CAYAAADkFBsIAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjYuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8o6BhiAAAACXBIWXMAAAsTAAALEwEAmpwYAAAIK0lEQVR4nO3dW0xU+R3A8e8MC8KWrcWQYsSIIowKkUHxFmK51OqiL9h4t0yiIdGI2kTb6kOD7Ut5IJPGtMHGpgYTH2iUkE2aGFI2wiBC6rWD2rqjQURdBNSyLlRQYfow67AoF+mGc85v+H2SCXBgwm/CN+fMmXDmb/P7/X6Usji72QMo9SE0VCWChqpE0FCVCBqqEkFDVSJoqEoEDVWJoKEqETRUJYKGqkTQUJUIGqoSQUNVImioSgQNVYmgoSoRNFQlgoaqRNBQlQgaqhJBQ1UiaKhKBA1ViaChKhE0VCWChqpE0FCVCBqqEkFDVSJoqEoEDVWJoKEqETRUJYKGqkTQUJUIGqpAgwPQ3wsDr8yexDiWDnVwcBC3201ycjKRkZE4nU48Hg8LFixgz549Zo9nuL4XcOdzqPsjXPwT1P4BblTB8zazJ5t8H5k9wFgKCwupqqqiuLiYjIwMGhsb2bFjB11dXRw+fNjs8QzV8xSu/RVe9w3f/uw+PGuBRZ9C/GJzZjOCZUOtqKjg9OnT1NXVkZ2dDUBubi7Xr1+nqqqKpUuXmjyhcfx+uPk3eN0/0jcDH/79d4iZDR/HGDqaYSx76C8pKSEvLy8Y6VtJSUmEh4eTlpYGQGtrK9nZ2TgcDhYvXszFixfNGHdSdT+C3mcEoxyRHx57jZrIeJYM9dGjR9y6dYstW7a89722tjZSU1OZNm0aAHv37mXbtm34fD5OnjzJ9u3befVq/LMMm80m5vbzwt+M+3j8fj81n10zfdaJ3j6UZUMFmDlz5rDtL1++xOPxBA/7T58+paGhgcLCQgAyMzOZNWsWtbW1xg48ycJsYYy3EqjNZiPMHmbQRMaz5HPU2NhYAHw+Hxs2bAhuLy0tpb29nYyMDCCwd42LiwvuXQHmzZvHgwcPxv0dkpaA7bwHzZ+N/3M/+jQd/+/lPK6JsGSoiYmJpKWlUVJSwowZM4iPj6eyspLz588DBEOdKmITYVp04LXTsZ6nzk43aiLjWfLQb7fbOXfuHKmpqezbt4/du3cTGxvL/v37CQsLC55IzZkzh46ODvr7h06H79+/T0JCglmjTwq7HVI3gM0GjPK0bu5K+H6coWMZyiZpGXSXy4XX66W5uTm4bd26dWzcuJGioiIaGxvZvHkzra2tREREmDjp5PjqS7h3Ef7zcGhb5HSYtxJmLf4m5BAlKtRFixaxatUqysvLg9taWlrYtWsXT548ISIigrKysvde0go1L7vh0l8Cn6/5RWgH+pYln6OOpKenB5/PR1FR0bDtiYmJ1NfXmzSVOaJ+MPT5VIgUBIUaHR3NwMCA2WMok1jyZEqpd2moSgQNVYmgoSoRNFQlgoaqRNBQlQgaqhJBQ1UiaKhKBA01BNXV1ZGQkEBOTg75+fn09Q2/dNXtdnPjxo1R7//48WPWr19PZmYmq1ev5tChQwB4vV5KS0sndfbRaKghyuVyUVdXR2ZmJpWVlcHtg4ODXLp0iSVLlox635qaGgoKCrhw4QINDQ10dnZy8+ZNnE4nTU1NplwdoaGGuPT09OA1aBDYKyYlJQGBPW9MTAw5OTkkJCSQn58PgMfjIT8/n8jISADCw8MJCwtcj5WcnDzm3niyaKghrr6+HofDEfz67t27zJ07F4CsrCxWrFgRfO+EEydO4Pf76e3tJTo6GoDm5ma6urpISUkBAv9WeefOHcMfh4Yaos6cOUNubi7d3d3BPeW7WlpaSExMBODhw4fEx8fj9XpxOp0APH/+nAMHDnDq1CnD5h6NhhqiXC4XtbW1lJWVBQ/bEDh0t7a2AnD79m1SU1MZGBjAbg+kUFNTw9q1a3nz5g0FBQW43e5hl623tLSwcOFCQx8LaKhTjtPpxOfzAUOh9vf309nZSXt7O1evXmXZsmWcO3eOK1eucOTIEXJycmhqagICl7Cnp6cbPreoa6bUkM/dgY8/+eXE7+t2u1mzZs2IZ/5nz55l69atI97P6/VSXV3N0aNHJ/5LvyMNVajvEqpEeuhXImioSgQNVYmgoSoRNFQlgoaqRNBQlQgaqhJBQ1UiaKhKBA1ViaChKhE0VCWCpUPVRXvf999uuNcw9PXztsASlKHO0u84rYv2DvEPwhcX4NE/h2+/fhY+iYP0nwaW+AlVlv1/1IqKCnbu3Dls0V6ATZs2UVVVxeXLl1m+fLmJExrLVwdtV0f5pg2+NwNWusBu6V3P/8+yh/4PXbT32LFjOBwO7Hb7sOvXQ8mrXnh4fYwf8AcW9e3wGTaS4SwZ6kQW7c3Ly6O6upqsrCyjxzTMky8Ch/4x2aD9tiHjmMKyocL4i/ZCYKHet5f8ToTZqypP5PbbX/9u/Hcn8cO1f9wyfdYptbr0txft/bZ3F+2dKr7qfTruH3VwcIDurzsMmsh4lnzqbcSivRY9hxxR39fQ8GfGXLDXbg/jZwfW8KuTch7XRFhyj/qhi/ZOFZGfwOyxHrINoqZDnPHvC2EYS+5RARwOB7W1tcO2uVwuUlJSiIqKMmkq8zh+DG9ew5N/fbPh7TMBP3wcA0s2QVi4WdNNPsu+jjqSkRbtLS4upry8nK6uLqKjo4mKisLj8TB//nwTJ508Lzrgy1vQ9wI+ioAfLoDYxMBS6aFMTKg9PT1Mnz6d48ePc/DgQbPHUQYTE6qa2kL8gKFChYaqRNBQlQgaqhJBQ1UiaKhKBA1ViaChKhE0VCWChqpE0FCVCBqqEkFDVSJoqEoEDVWJoKEqETRUJYKGqkTQUJUIGqoSQUNVImioSgQNVYmgoSoRNFQlgoaqRNBQlQj/A08ZhshjQJjAAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 206.852x144.48 with 1 Axes>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from math import pi\n",
    "\n",
    "cu_circ = QuantumCircuit(2)\n",
    "cu_circ.cp(pi/2, 0, 1)\n",
    "cu_circ.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a09a586",
   "metadata": {},
   "source": [
    "Let's apply $2^t$ times $\\text{CP}(\\pi/2)$. Since for the first step $t=m-1$, and $m=2$, we have $2^t=2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "3e94b96d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 387.452x204.68 with 1 Axes>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "for _ in range(2 ** (m - 1)):\n",
    "    qc_S.cp(pi/2, 0, 1)\n",
    "qc_S.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "db469592",
   "metadata": {},
   "source": [
    "#### Measure in X-basis\n",
    "\n",
    "Finally, we perform the measurement of the auxiliary qubit in X-basis. So we will define a function to perform the ``x_measurement`` and then apply it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "aa1cd4fb",
   "metadata": {},
   "outputs": [],
   "source": [
    "def x_measurement(qc, qubit, cbit):\n",
    "    \"\"\"Measure 'qubit' in the X-basis, and store the result in 'cbit'\"\"\"\n",
    "    qc.h(qubit)\n",
    "    qc.measure(qubit, cbit)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e4417ce",
   "metadata": {},
   "source": [
    "In this way we obtain the phase bit $\\varphi_2$ and store it in the classical bit $c_0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "9d767f96",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 507.852x204.68 with 1 Axes>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_measurement(qc_S, q[0], c[0])\n",
    "qc_S.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "99727bfd",
   "metadata": {},
   "source": [
    "### Subsequent steps (2nd step)\n",
    "\n",
    "Now we build the quantum circuit for the other remaining steps, in this example, only the second one.\n",
    "In these steps we have 4 sub-steps: the 3 sub-steps as in the first step and, in the middle, the additional step of the phase correction\n",
    "- initialization with reset\n",
    "- phase correction\n",
    "- application of the Control-$U$ gates\n",
    "- measure of the auxiliary qubit in X-basis\n",
    "\n",
    "#### Initialization with reset\n",
    "\n",
    "As we want to perform an iterative algorithm in the same circuit, we need to reset the auxiliary qubit $q_0$ after the measument gate and initialize it again as before to recycle the qubit. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "c69ed53a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 628.252x204.68 with 1 Axes>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qc_S.reset(0)\n",
    "qc_S.h(0)\n",
    "qc_S.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "60060d00",
   "metadata": {},
   "source": [
    "#### Phase correction (for step 2)\n",
    "As seen in the theory, in order to extract the phase bit $\\varphi_{1}$, we perform a phase correction of $-\\pi\\varphi_2/2$.\n",
    "Of course, we need to apply the phase correction in the circuit only if the phase bit $\\varphi_2=1$, i.e. we have to apply the phase correction of $-\\pi/2$ only if the classical bit $c_0$ is 1.\n",
    "\n",
    "So, after the reset we apply the phase gate $P(\\theta)$ with phase $\\theta=-\\pi/2$ conditioned by the classical bit $c_0$ ($=\\varphi_2$) using the `c_if` method.\n",
    "So as we saw in the first part of this tutorial, we have to use the `c_if` method with a value of 1, as $1_{10} = 001_{2}$ (the subscripts $_{10}$ and $_2$ indicate the decimal and binary representations)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "4d7b8a7b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 688.452x204.68 with 1 Axes>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qc_S.p(-pi/2, 0).c_if(c, 1)\n",
    "qc_S.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9b096b64",
   "metadata": {},
   "source": [
    "#### Application of the Control-$U$ gates and x-measurement (for step 2)\n",
    "\n",
    "We apply the $CU$ operations as we did in the first step. For the second step we have $t=m-2$, hence $2^t=1$. So we apply $\\text{CP}(\\pi/2)$ once. And then we perform the X-measurement of the qubit $q_0$, storing the result, the phase bit $\\varphi_1$, in the bit $c_1$ of classical register."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "8d0b95ef",
   "metadata": {},
   "outputs": [],
   "source": [
    "## 2^t c-U operations (with t=m-2)\n",
    "for _ in range(2 ** (m - 2)):\n",
    "    qc_S.cp(pi/2, 0, 1)\n",
    "\n",
    "x_measurement(qc_S, q[0], c[1]) "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "02d9b94c",
   "metadata": {},
   "source": [
    "Et voilà, we have our final circuit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "31d410f2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 929.252x204.68 with 1 Axes>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qc_S.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d5b1356c",
   "metadata": {},
   "source": [
    "Let's sample the circuit with Qiskit Aer's `Sampler` primitive, a local simulator without noise that runs locally."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "1c984758",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from qiskit.tools.visualization import plot_histogram\n",
    "from qiskit_aer.primitives import Sampler\n",
    "\n",
    "sampler = Sampler()\n",
    "\n",
    "job = sampler.run(qc_S)\n",
    "result = job.result()\n",
    "\n",
    "dist0 = result.quasi_dists[0]\n",
    "\n",
    "key_new = [str(key/2**m) for key in list(dist0.keys())]\n",
    "dist1 = dict(zip(key_new, dist0.values()))\n",
    "\n",
    "fig, ax = plt.subplots(1,2)\n",
    "plot_histogram(dist0, ax=ax[0])\n",
    "plot_histogram(dist1, ax=ax[1])\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2d830d23",
   "metadata": {},
   "source": [
    "In the picture we have the same histograms but on the left we have on the x-axis the string with phase bits $\\varphi_1$, $\\varphi_2$ and on the right the actual phase $\\varphi$ in decimal representation.\n",
    "\n",
    "As we expected we have found $\\varphi=\\frac{1}{4}=0.25$ with a $100\\%$ probability."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2314dbb2",
   "metadata": {},
   "source": [
    "## IPE example with a 2-qubit gate\n",
    "\n",
    "Now, we want to apply the IPE algorithm to estimate the phase for a 2-qubit gate $U$. For this example, let's consider the controlled version of the $T$ gate, i.e. the gate $U=\\textrm{Controlled-}T$ (that from now we will express more compactly with $CT$). Its matrix is\n",
    "$$ CT = \n",
    "\\begin{bmatrix}\n",
    "1 & 0 & 0 & 0\\\\\n",
    "0 & 1 & 0 & 0\\\\\n",
    "0 & 0 & 1 & 0\\\\\n",
    "0 & 0 & 0 & e^\\frac{i\\pi}{4}\\\\ \n",
    "\\end{bmatrix} $$\n",
    "That is, the $CT$ gate adds a phase $\\pi/4$ to the state $|11\\rangle$, leaving unchanged the phase of the other computational basis states $|00\\rangle$, $|01\\rangle$, $|10\\rangle$.\n",
    "\n",
    "Let's consider to estimate the phase $\\phi=\\pi/4$ for the eigenstate $|11\\rangle$, we should find $\\varphi=1/8$, since $\\phi = 2 \\pi \\varphi$. Therefore to estimate the phase we need exactly 3 classical bits, i.e. $m=3$, since $1/2^3=1/8$. So $\\varphi=0.\\varphi_1\\varphi_2\\varphi_3$.\n",
    "\n",
    "As done with the example for the 1-qubit $U$ operator we will go through the same steps but this time we will have $3$ steps since $m=3$, and we will not repeat all the explanations. So for details see the above example for 1-qubit $U$ gate.\n",
    "\n",
    "First, we initialize the circuit with 3 qubits, 1 for the auxiliary qubit and 2 for the 2-qubit gate, and 3 classical bits to store the phase bits $\\varphi_1$, $\\varphi_2$, $\\varphi_3$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "c13208f6",
   "metadata": {},
   "outputs": [],
   "source": [
    "nq = 3    # number of qubits\n",
    "m = 3    # number of classical bits\n",
    "q = QuantumRegister(nq,'q')\n",
    "c = ClassicalRegister(m,'c')\n",
    "\n",
    "qc = QuantumCircuit(q,c)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8107bb94",
   "metadata": {},
   "source": [
    "### First step\n",
    "\n",
    "Now we build the quantum circuit for the first step, to estimate the least significant phase bit $\\varphi_m=\\varphi_3$.\n",
    "\n",
    "#### Initialization\n",
    "\n",
    "We initialize the auxiliary qubit and the other qubits with the eigenstate $|11\\rangle$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "982d416d",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 146.797x264.88 with 1 Axes>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qc.h(0)\n",
    "qc.x([1, 2])\n",
    "qc.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "739dbd79",
   "metadata": {
    "tags": []
   },
   "source": [
    "#### Application of the Controlled-$U$ gates\n",
    "\n",
    "Then we have to apply multiple times the $CU$ operator, that, in this example, is the Controlled-$CT$ gate ($CCT$ for short).\n",
    "\n",
    "To implement $CCT$ in the circuit, since $T$ is a phase gate, we can use the multi-controlled phase gate $\\text{MCP}(\\theta)$, with $\\theta=\\pi/4$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "280518a7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 206.997x204.68 with 1 Axes>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cu_circ = QuantumCircuit(nq)\n",
    "cu_circ.mcp(pi/4, [0, 1], 2)\n",
    "cu_circ.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cd174450",
   "metadata": {},
   "source": [
    "Let's apply $2^t$ times $\\text{MCP}(\\pi/4)$. Since for the first step $t=m-1$ and $m=3$, we have $2^t=4$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "24c8f9b9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 628.397x264.88 with 1 Axes>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "for _ in range(2 ** (m - 1)):\n",
    "    qc.mcp(pi/4, [0, 1], 2)\n",
    "qc.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "09e40686",
   "metadata": {},
   "source": [
    "#### Measure in X-basis\n",
    "\n",
    "Finally, we perform the measurement of the auxiliary qubit in X-basis.\n",
    "We can use the `x_measurement` function defined above in the example for 1-qubit gate. In this way we have obtained the phase bit $\\varphi_3$ and stored it in the classical bit $c_0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "6fa62739",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 748.797x264.88 with 1 Axes>"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_measurement(qc, q[0], c[0])\n",
    "qc.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3ac0df28",
   "metadata": {},
   "source": [
    "### Subsequent steps (2nd, 3rd)\n",
    "\n",
    "Now we build the quantum circuit for the other remaining steps, the second and the third ones.\n",
    "As said in the first example, in these steps we have the additional sub-step of the phase correction.\n",
    "\n",
    "#### Initialization with reset\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "1520c0da",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 869.197x264.88 with 1 Axes>"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qc.reset(0)\n",
    "qc.h(0)\n",
    "qc.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f7da1b6a",
   "metadata": {},
   "source": [
    "#### Phase correction (for step 2)\n",
    "In order to extract the phase bit $\\varphi_{2}$, we perform a phase correction of $-\\pi\\varphi_3/2$.\n",
    "\n",
    "So, after the reset we apply the phase gate $P(\\theta)$ with phase $\\theta=-\\pi/2$ conditioned by the classical bit $c_0$ ($=\\varphi_3$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "99a64b7b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 929.397x264.88 with 1 Axes>"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qc.p(-pi/2, 0).c_if(c, 1)\n",
    "qc.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68be63a1",
   "metadata": {},
   "source": [
    "#### Application of the Control-$U$ gates and x-measurement (for step 2)\n",
    "We apply the $CU$ operations as we did in the first step. For the second step we have $t=m-2$, hence $2^t=2$. So we apply $\\text{MCP}(\\pi/4)$ $2$ times. And then we perform the X-measurement of the qubit $q_0$, storing the phase bit $\\varphi_2$ in the bit $c_1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "fa0d6cef",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1290.6x264.88 with 1 Axes>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "for _ in range(2 ** (m - 2)):\n",
    "    qc.mcp(pi/4, [0, 1], 2)\n",
    "x_measurement(qc, q[0], c[1])\n",
    "qc.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "25949dc2",
   "metadata": {},
   "source": [
    "#### All substeps of the 3rd step\n",
    "For the 3rd and last step, we perform the reset and initialization of the auxiliary qubit as done in the second step.\n",
    "\n",
    "Then at the 3rd step we have to perform the phase correction of $-2\\pi 0.0\\varphi_{2}\\varphi_{3}= -2\\pi \\left(\\frac{\\varphi_2}{4}+\\frac{\\varphi_3}{8}\\right)=-\\frac{\\varphi_2\\pi}{2}-\\frac{ \\varphi_3\\pi}{4}$, thus we have to apply 2 conditioned  phase corrections, one conditioned by $\\varphi_3$ ($=c_0$) and the other by $\\varphi_2$($=c_1$). To do this we have to apply the following:\n",
    "- gate $P(-\\pi/4)$ conditioned by $c_0=1$, that is, by $c=001$ (``c_if`` with value $1$)\n",
    "- gate $P(-\\pi/2)$ conditioned by $c_1=1$, that is, the gate is applied when $c=010$ (``c_if`` with values $2$)\n",
    "- gate $P(-3\\pi/4)$ conditioned by $c_1=1$ and $c_0=1$ that is, the gate is applied when $c=011$ (``c_if`` with values $3$)\n",
    "\n",
    "Next, the $CU$ operations: we apply $2^t$ times the $\\text{MCP}(\\pi/4)$ gate and since at the 3rd step $t=m-3=0$, we apply the gate only once."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "fee5769f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1591.6x565.88 with 1 Axes>"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# initialization of qubit q0\n",
    "qc.reset(0)\n",
    "qc.h(0)\n",
    "\n",
    "# phase correction\n",
    "qc.p(-pi/4, 0).c_if(c, 1)\n",
    "\n",
    "qc.p(-pi/2, 0).c_if(c, 2)\n",
    "qc.p(-3*pi/2, 0).c_if(c, 3)\n",
    "\n",
    "# c-U operations\n",
    "for _ in range(2 ** (m - 3)):\n",
    "    qc.mcp(pi/4, [0, 1], 2)\n",
    "\n",
    "# X measurement  \n",
    "qc.h(0)\n",
    "qc.measure(0, 2)\n",
    "\n",
    "qc.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f32900a5",
   "metadata": {},
   "source": [
    "Now, we execute the circuit with the simulator without noise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "c6e0a06f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "result = sampler.run(qc).result()\n",
    "dist0 = result.quasi_dists[0]\n",
    "\n",
    "key_new = [str(key/2**m) for key in list(dist0.keys())]\n",
    "dist1 = dict(zip(key_new, dist0.values()))\n",
    "\n",
    "fig, ax = plt.subplots(1,2)\n",
    "plot_histogram(dist0, ax=ax[0])\n",
    "plot_histogram(dist1, ax=ax[1])\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a992b916",
   "metadata": {},
   "source": [
    "We have obtained $100\\%$ probability to find $\\varphi=0.125$, that is, $1/8$, as expected."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "7280b7b7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<h3>Version Information</h3><table><tr><th>Qiskit Software</th><th>Version</th></tr><tr><td><code>qiskit-terra</code></td><td>0.23.0.dev0+1b4fed3</td></tr><tr><td><code>qiskit-aer</code></td><td>0.11.1</td></tr><tr><td><code>qiskit-nature</code></td><td>0.5.0</td></tr><tr><th>System information</th></tr><tr><td>Python version</td><td>3.9.13</td></tr><tr><td>Python compiler</td><td>Clang 12.0.0 </td></tr><tr><td>Python build</td><td>main, Oct 13 2022 16:12:30</td></tr><tr><td>OS</td><td>Darwin</td></tr><tr><td>CPUs</td><td>4</td></tr><tr><td>Memory (Gb)</td><td>32.0</td></tr><tr><td colspan='2'>Fri Dec 09 16:18:07 2022 CET</td></tr></table>"
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      "text/html": [
       "<div style='width: 100%; background-color:#d5d9e0;padding-left: 10px; padding-bottom: 10px; padding-right: 10px; padding-top: 5px'><h3>This code is a part of Qiskit</h3><p>&copy; Copyright IBM 2017, 2022.</p><p>This code is licensed under the Apache License, Version 2.0. You may<br>obtain a copy of this license in the LICENSE.txt file in the root directory<br> of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.<p>Any modifications or derivative works of this code must retain this<br>copyright notice, and modified files need to carry a notice indicating<br>that they have been altered from the originals.</p></div>"
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       "<IPython.core.display.HTML object>"
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   "source": [
    "import qiskit.tools.jupyter\n",
    "%qiskit_version_table\n",
    "%qiskit_copyright"
   ]
  }
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