algorithms/grover_algorithm.ipynb

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    "<img src=\"https://raw.githubusercontent.com/Qiskit/qiskit-tutorials/master/images/qiskit-heading.png\" alt=\"Note: In order for images to show up in this jupyter notebook you need to select File => Trusted Notebook\" width=\"500 px\" align=\"left\">"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Grover Search for Combinatorial Problems\n",
    "\n",
    "\n",
    "The latest version of this notebook is available on https://github.com/qiskit/qiskit-tutorial.\n",
    "\n",
    "***\n",
    "### Contributors\n",
    "Giacomo Nannicini and Rudy Raymond (based on [this paper](https://arxiv.org/abs/1708.03684)) \n",
    "\n",
    "### Qiskit Package Versions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "qiskit==1.2.0\n",
      "qiskit-aer==0.15.0\n",
      "qiskit-algorithms==0.3.0\n",
      "qiskit-ibm-runtime==0.31.0\n",
      "qiskit-machine-learning==0.7.2\n",
      "qiskit-nature==0.7.2\n",
      "qiskit-nature-pyscf==0.4.0\n",
      "qiskit-qasm3-import==0.5.0\n",
      "qiskit-transpiler-service==0.4.5\n"
     ]
    }
   ],
   "source": [
    "! pip freeze | grep qiskit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction\n",
    "\n",
    "Grover search is one of the most popular algorithms used for searching a solution among many possible candidates using Quantum Computers. If there are $N$ possible solutions among which there is exactly one solution (that can be verified by some function evaluation), then Grover search can be used to find the solution with $O(\\sqrt{N})$ function evaluations. This is in contrast to classical computers that require $\\Omega(N)$ function evaluations: the Grover search is a quantum algorithm that provably can be used search the correct solutions quadratically faster than its classical counterparts.  \n",
    "\n",
    "Here, we are going to illustrate the use of Grover search to solve a combinatorial problem called [Exactly-1 3-SAT problem](https://en.wikipedia.org/wiki/Boolean_satisfiability_problem#Exactly-1_3-satisfiability). The Exactly-1 3-SAT problem is a NP-complete problem, namely, it is one of the most difficult problems that are interconnected (meaning that if we solve any one of them, we essentially can solve all of them). Unfortunately, there are many natural problems that are NP-complete, such as, the Traveling Salesman Problem (TSP), the Maximum Cut (MaxCut) and so on. Up to now, there is no classical and quantum algorithm that can efficiently solve such NP-hard problems. \n",
    "\n",
    "We begin with an example of the Exactly-1 3-SAT problem. Then, we show how to design an evaluation function which is also known as the oracle (or, blackbox) which is essential to Grover search. Finally, we show the circuit of Grover search using the oracle and present their results on simulator and real-device backends."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exactly-1 3-SAT problem\n",
    "\n",
    "The Exactly-1 3-SAT problem is best explained with the following concrete problem. Let us consider a Boolean function $f$ with three Boolean variables $x_1, x_2, x_3$ as below.\n",
    "\n",
    "$$\n",
    "f(x_1, x_2, x_3) = (x_1 \\vee x_2 \\vee \\neg x_3) \\wedge (\\neg x_1 \\vee \\neg x_2 \\vee \\neg x_3) \\wedge (\\neg x_1 \\vee x_2 \\vee x_3) \n",
    "$$\n",
    "\n",
    "In the above function, the terms on the right-hand side equation which are inside $()$ are called clauses. Each clause has exactly three literals. Namely, the first clause has $x_1$, $x_2$ and $\\neg x_3$ as its literals. The symbol $\\neg$ is the Boolean NOT that negates (or, flips) the value of its succeeding literal. The symbols $\\vee$ and $\\wedge$ are, respectively, the Boolean OR and AND. The Boolean $f$ is satisfiable if there is an assignment of $x_1, x_2, x_3$ that evaluates to $f(x_1, x_2, x_3) = 1$ (or, $f$ evaluates to True). The Exactly-1 3-SAT problem requires us to find an assignment such that $f = 1$ (or, True) and there is *exactly* one literal that evaluates to True in every clause of $f$. \n",
    "\n",
    "A naive way to find such an assignment is by trying every possible combinations of input values of $f$. Below is the table obtained from trying all possible combinations of $x_1, x_2, x_3$. For ease of explanation, we interchangably use $0$ and False, as well as $1$ and True.  \n",
    "\n",
    "|$x_1$ | $x_2$ | $x_3$ | $f$ | Comment | \n",
    "|------|-------|-------|-----|---------|\n",
    "| 0    |  0    |  0    |  1  | Not a solution because there are three True literals in the second clause     | \n",
    "| 0    |  0    |  1    |  0  | Not a solution because $f$ is False          | \n",
    "| 0    |  1    |  0    |  1  | Not a solution because there are two True literals in the first clause        | \n",
    "| 0    |  1    |  1    |  1  | Not a solution because there are three True literals in the third clause        | \n",
    "| 1    |  0    |  0    |  0  | Not a solution because $f$ is False        | \n",
    "| 1    |  0    |  1    |  1  | **Solution**. BINGO!!       | \n",
    "| 1    |  1    |  0    |  1  | Not a soluton because there are three True literals in the first clause        | \n",
    "| 1    |  1    |  1    |  0  | Not a solution because $f$ is False        | \n",
    "\n",
    "\n",
    "From the table above, we can see that the assignment $x_1x_2x_3 = 101$ is the solution fo the Exactly-1 3-SAT problem to $f$. In general, the Boolean function $f$ can have many clauses and more Boolean variables. \n",
    "\n",
    "## A blackbox function to check the assignment of Exactly-1 3-SAT problem\n",
    "\n",
    "Here, we describe a method to construct a circuit to check the assignment of Exactly-1 3-SAT problem. The circuit can then be used as a blackbox (or, oracle) in Grover search. To design the blackbox, we do not need to know the solution to the problem in advance: it suffices to design a blackbox that checks if the assignment results in $f$ evaluates to True or False. It turns out that we can design such a blackbox efficiently (in fact, any NP-complete problem has the property that although finding the solution is difficult, checking the solution is easy). \n",
    "\n",
    "For each clause of $f$, we design a sub-circuit that outputs True if and only if there is exactly one True literal in the clause. Combining all sub-circuits for all clauses, we can then obtain the blackbox that outputs True if and only if all clauses are satisfied with exactly one True literal each.   \n",
    "\n",
    "For example, let us consider the clause $(x_1 \\vee \\neg x_2 \\vee x_3)$. It is easy to see that $y$ defined as \n",
    "\n",
    "$$\n",
    "y = x_1 \\oplus \\neg x_2 \\oplus x_3 \\oplus ( x_1 \\wedge \\neg x_2 \\wedge x_3), \n",
    "$$\n",
    "\n",
    "is True if and only if exactly one of $x_1$, $\\neg x_2$, and $x_3$ is True. Using two working qubits, $y$ can be computed by the following sub-circuit. Below, $x_1x_2x_3$ is renamed as $q_1q_2q_3$, $q_4$ is used as a working qubit, and $q_5$ is used to store the value of $y$.  "
   ]
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   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister\n",
    "from qiskit.visualization import plot_distribution\n",
    "from qiskit.primitives import StatevectorSampler"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
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       "<pre style=\"word-wrap: normal;white-space: pre;background: #fff0;line-height: 1.1;font-family: &quot;Courier New&quot;,Courier,monospace\">                                             \n",
       "q_0: ────────────────────────────────────────\n",
       "                                             \n",
       "q_1: ───────■──────────────■─────────■───────\n",
       "     ┌───┐  │              │         │  ┌───┐\n",
       "q_2: ┤ X ├──┼────■─────────■─────────■──┤ X ├\n",
       "     └───┘  │    │         │         │  └───┘\n",
       "q_3: ───────┼────┼────■────┼────■────┼───────\n",
       "            │    │    │  ┌─┴─┐  │  ┌─┴─┐     \n",
       "q_4: ───────┼────┼────┼──┤ X ├──■──┤ X ├─────\n",
       "          ┌─┴─┐┌─┴─┐┌─┴─┐└───┘┌─┴─┐└───┘     \n",
       "q_5: ─────┤ X ├┤ X ├┤ X ├─────┤ X ├──────────\n",
       "          └───┘└───┘└───┘     └───┘          </pre>"
      ],
      "text/plain": [
       "                                             \n",
       "q_0: ────────────────────────────────────────\n",
       "                                             \n",
       "q_1: ───────■──────────────■─────────■───────\n",
       "     ┌───┐  │              │         │  ┌───┐\n",
       "q_2: ┤ X ├──┼────■─────────■─────────■──┤ X ├\n",
       "     └───┘  │    │         │         │  └───┘\n",
       "q_3: ───────┼────┼────■────┼────■────┼───────\n",
       "            │    │    │  ┌─┴─┐  │  ┌─┴─┐     \n",
       "q_4: ───────┼────┼────┼──┤ X ├──■──┤ X ├─────\n",
       "          ┌─┴─┐┌─┴─┐┌─┴─┐└───┘┌─┴─┐└───┘     \n",
       "q_5: ─────┤ X ├┤ X ├┤ X ├─────┤ X ├──────────\n",
       "          └───┘└───┘└───┘     └───┘          "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q = QuantumRegister(6, name='q')\n",
    "qc = QuantumCircuit(q)\n",
    "qc.x(q[2])\n",
    "qc.cx(q[1], q[5])\n",
    "qc.cx(q[2], q[5])\n",
    "qc.cx(q[3], q[5])\n",
    "qc.ccx(q[1], q[2], q[4])\n",
    "qc.ccx(q[3], q[4], q[5])\n",
    "qc.ccx(q[1], q[2], q[4])\n",
    "qc.x(q[2])\n",
    "qc.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the sub-circuit above, the three `ccx` gates on the right are used to compute $( q_1 \\wedge \\neg q_2 \\wedge q_3)$ and write the result to $q_5$, while the three `cx` gates on the left are used to compute $q_1 \\oplus \\neg q_2 \\oplus q_3$ and write the result to $q_5$. Notice that the right-most `ccx` gate is used to reset the value of $q_4$ so that it can be reused in the succeeding sub-circuits. \n",
    "\n",
    "From the above sub-circuit, we can define a blackbox function to check the solution of the Exactly-1 3-SAT problem as follows."
   ]
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   "source": [
    "def black_box_u_f(circuit, f_in, f_out, aux, n, exactly_1_3_sat_formula):\n",
    "    \"\"\"Circuit that computes the black-box function from f_in to f_out.\n",
    "\n",
    "    Create a circuit that verifies whether a given exactly-1 3-SAT\n",
    "    formula is satisfied by the input. The exactly-1 version\n",
    "    requires exactly one literal out of every clause to be satisfied.\n",
    "    \"\"\"\n",
    "    num_clauses = len(exactly_1_3_sat_formula)\n",
    "    for (k, clause) in enumerate(exactly_1_3_sat_formula):\n",
    "        # This loop ensures aux[k] is 1 if an odd number of literals\n",
    "        # are true\n",
    "        for literal in clause:\n",
    "            if literal > 0:\n",
    "                circuit.cx(f_in[literal-1], aux[k])\n",
    "            else:\n",
    "                circuit.x(f_in[-literal-1])\n",
    "                circuit.cx(f_in[-literal-1], aux[k])\n",
    "        # Flip aux[k] if all literals are true, using auxiliary qubit\n",
    "        # (ancilla) aux[num_clauses]\n",
    "        circuit.ccx(f_in[0], f_in[1], aux[num_clauses])\n",
    "        circuit.ccx(f_in[2], aux[num_clauses], aux[k])\n",
    "        # Flip back to reverse state of negative literals and ancilla\n",
    "        circuit.ccx(f_in[0], f_in[1], aux[num_clauses])\n",
    "        for literal in clause:\n",
    "            if literal < 0:\n",
    "                circuit.x(f_in[-literal-1])\n",
    "    # The formula is satisfied if and only if all auxiliary qubits\n",
    "    # except aux[num_clauses] are 1\n",
    "    if (num_clauses == 1):\n",
    "        circuit.cx(aux[0], f_out[0])\n",
    "    elif (num_clauses == 2):\n",
    "        circuit.ccx(aux[0], aux[1], f_out[0])\n",
    "    elif (num_clauses == 3):\n",
    "        circuit.ccx(aux[0], aux[1], aux[num_clauses])\n",
    "        circuit.ccx(aux[2], aux[num_clauses], f_out[0])\n",
    "        circuit.ccx(aux[0], aux[1], aux[num_clauses])\n",
    "    else:\n",
    "        raise ValueError('We only allow at most 3 clauses')\n",
    "    # Flip back any auxiliary qubits to make sure state is consistent\n",
    "    # for future executions of this routine; same loop as above.\n",
    "    for (k, clause) in enumerate(exactly_1_3_sat_formula):\n",
    "        for literal in clause:\n",
    "            if literal > 0:\n",
    "                circuit.cx(f_in[literal-1], aux[k])\n",
    "            else:\n",
    "                circuit.x(f_in[-literal-1])\n",
    "                circuit.cx(f_in[-literal-1], aux[k])\n",
    "        circuit.ccx(f_in[0], f_in[1], aux[num_clauses])\n",
    "        circuit.ccx(f_in[2], aux[num_clauses], aux[k])\n",
    "        circuit.ccx(f_in[0], f_in[1], aux[num_clauses])\n",
    "        for literal in clause:\n",
    "            if literal < 0:\n",
    "                circuit.x(f_in[-literal-1])\n",
    "# -- end function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inversion about the average\n",
    "\n",
    "Another important procedure in Grover search is to have an operation that perfom the *inversion-about-the-average* step, namely, it performs the following transformation:\n",
    "\n",
    "$$\n",
    "\\sum_{j=0}^{2^{n}-1} \\alpha_j |j\\rangle \\rightarrow \\sum_{j=0}^{2^{n}-1}\\left(2 \\left( \\sum_{k=0}^{k=2^{n}-1} \\frac{\\alpha_k}{2^n} \\right) - \\alpha_j  \\right) |j\\rangle \n",
    "$$\n",
    "\n",
    "The above transformation can be used to amplify the probability amplitude $\\alpha_s$ when s is the solution and $\\alpha_s$ is negative (and small), while $\\alpha_j$ for $j \\neq s$ is positive. Roughly speaking, the value of $\\alpha_s$ increases by twice the average of the amplitudes, while others are reduced. The inversion-about-the-average can be realized with the sequence of unitary matrices as below:\n",
    "\n",
    "$$\n",
    "H^{\\otimes n} \\left(2|0\\rangle \\langle 0 | - I \\right) H^{\\otimes n}\n",
    "$$\n",
    "\n",
    "The first and last $H$ are just Hadamard gates applied to each qubit. The operation in the middle requires us to design a sub-circuit that flips the probability amplitude of the component of the quantum state corresponding to the all-zero binary string. The sub-circuit can be realized by the following function, which is a multi-qubit controlled-Z which flips the probability amplitude of the component of the quantum state corresponding to the all-one binary string. Applying X gates to all qubits before and after the function realizes the sub-circuit. "
   ]
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   "source": [
    "def n_controlled_Z(circuit, controls, target):\n",
    "    \"\"\"Implement a Z gate with multiple controls\"\"\"\n",
    "    if (len(controls) > 2):\n",
    "        raise ValueError('The controlled Z with more than 2 ' +\n",
    "                         'controls is not implemented')\n",
    "    elif (len(controls) == 1):\n",
    "        circuit.h(target)\n",
    "        circuit.cx(controls[0], target)\n",
    "        circuit.h(target)\n",
    "    elif (len(controls) == 2):\n",
    "        circuit.h(target)\n",
    "        circuit.ccx(controls[0], controls[1], target)\n",
    "        circuit.h(target)\n",
    "# -- end function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, the inversion-about-the-average circuit can be realized by the following function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
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     "start_time": "2018-09-25T17:32:09.196905Z"
    }
   },
   "outputs": [],
   "source": [
    "def inversion_about_average(circuit, f_in, n):\n",
    "    \"\"\"Apply inversion about the average step of Grover's algorithm.\"\"\"\n",
    "    # Hadamards everywhere\n",
    "    for j in range(n):\n",
    "        circuit.h(f_in[j])\n",
    "    # D matrix: flips the sign of the state |000> only\n",
    "    for j in range(n):\n",
    "        circuit.x(f_in[j])\n",
    "    n_controlled_Z(circuit, [f_in[j] for j in range(n-1)], f_in[n-1])\n",
    "    for j in range(n):\n",
    "        circuit.x(f_in[j])\n",
    "    # Hadamards everywhere again\n",
    "    for j in range(n):\n",
    "        circuit.h(f_in[j])\n",
    "# -- end function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is a circuit of the inversion about the average on three qubits."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-25T17:32:11.403205Z",
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    }
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   "outputs": [
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       "<pre style=\"word-wrap: normal;white-space: pre;background: #fff0;line-height: 1.1;font-family: &quot;Courier New&quot;,Courier,monospace\">     ┌───┐┌───┐          ┌───┐┌───┐     \n",
       "q_0: ┤ H ├┤ X ├───────■──┤ X ├┤ H ├─────\n",
       "     ├───┤├───┤       │  ├───┤├───┤     \n",
       "q_1: ┤ H ├┤ X ├───────■──┤ X ├┤ H ├─────\n",
       "     ├───┤├───┤┌───┐┌─┴─┐├───┤├───┤┌───┐\n",
       "q_2: ┤ H ├┤ X ├┤ H ├┤ X ├┤ H ├┤ X ├┤ H ├\n",
       "     └───┘└───┘└───┘└───┘└───┘└───┘└───┘</pre>"
      ],
      "text/plain": [
       "     ┌───┐┌───┐          ┌───┐┌───┐     \n",
       "q_0: ┤ H ├┤ X ├───────■──┤ X ├┤ H ├─────\n",
       "     ├───┤├───┤       │  ├───┤├───┤     \n",
       "q_1: ┤ H ├┤ X ├───────■──┤ X ├┤ H ├─────\n",
       "     ├───┤├───┤┌───┐┌─┴─┐├───┤├───┤┌───┐\n",
       "q_2: ┤ H ├┤ X ├┤ H ├┤ X ├┤ H ├┤ X ├┤ H ├\n",
       "     └───┘└───┘└───┘└───┘└───┘└───┘└───┘"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qr = QuantumRegister(3, name='q')\n",
    "qInvAvg = QuantumCircuit(qr)\n",
    "inversion_about_average(qInvAvg, qr, 3)\n",
    "qInvAvg.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Grover Search: putting all together\n",
    "\n",
    "The complete steps of Grover search is as follow.\n",
    "\n",
    "1. Create the superposition of all possible solutions as the initial state (with working qubits initialized to zero)\n",
    "$$  \\sum_{j=0}^{2^{n}-1} \\frac{1}{2^n} |j\\rangle |0\\rangle$$\n",
    "2. Repeat for $T$ times:\n",
    "\n",
    "    Apply the blackbox function\n",
    "    \n",
    "    Apply the inversion-about-the-average function\n",
    "    \n",
    "3. Measure to obtain the solution\n",
    "\n",
    "The code for the above steps is as below:"
   ]
  },
  {
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   "execution_count": 23,
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   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 700x500 with 1 Axes>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 700x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"\n",
    "Grover search implemented in Qiskit.\n",
    "\n",
    "This module contains the code necessary to run Grover search on 3\n",
    "qubits, both with a simulator and with a real quantum computing\n",
    "device. This code is the companion for the paper\n",
    "\"An introduction to quantum computing, without the physics\",\n",
    "Giacomo Nannicini, https://arxiv.org/abs/1708.03684. \n",
    "\"\"\"\n",
    "\n",
    "def input_state(circuit, f_in, f_out, n):\n",
    "    \"\"\"(n+1)-qubit input state for Grover search.\"\"\"\n",
    "    for j in range(n):\n",
    "        circuit.h(f_in[j])\n",
    "    circuit.x(f_out)\n",
    "    circuit.h(f_out)\n",
    "# -- end function\n",
    "\n",
    "# Make a quantum program for the n-bit Grover search.\n",
    "n = 3\n",
    "\n",
    "# Exactly-1 3-SAT formula to be satisfied, in conjunctive\n",
    "# normal form. We represent literals with integers, positive or\n",
    "# negative, to indicate a Boolean variable or its negation.\n",
    "exactly_1_3_sat_formula = [[1, 2, -3], [-1, -2, -3], [-1, 2, 3]]\n",
    "\n",
    "# Define three quantum registers: 'f_in' is the search space (input\n",
    "# to the function f), 'f_out' is bit used for the output of function\n",
    "# f, aux are the auxiliary bits used by f to perform its\n",
    "# computation.\n",
    "f_in = QuantumRegister(n)\n",
    "f_out = QuantumRegister(1)\n",
    "aux = QuantumRegister(len(exactly_1_3_sat_formula) + 1)\n",
    "\n",
    "# Define classical register for algorithm result\n",
    "ans = ClassicalRegister(n, name='meas')\n",
    "\n",
    "# Define quantum circuit with above registers\n",
    "grover = QuantumCircuit()\n",
    "grover.add_register(f_in)\n",
    "grover.add_register(f_out)\n",
    "grover.add_register(aux)\n",
    "grover.add_register(ans)\n",
    "\n",
    "input_state(grover, f_in, f_out, n)\n",
    "T = 2\n",
    "for t in range(T):\n",
    "    # Apply T full iterations\n",
    "    black_box_u_f(grover, f_in, f_out, aux, n, exactly_1_3_sat_formula)\n",
    "    inversion_about_average(grover, f_in, n)\n",
    "\n",
    "# Measure the output register in the computational basis\n",
    "for j in range(n):\n",
    "    grover.measure(f_in[j], ans[j])\n",
    "\n",
    "# Execute circuit\n",
    "sampler = StatevectorSampler()\n",
    "result = sampler.run([grover]).result()\n",
    "dist = result[0].data.meas.get_counts()\n",
    "\n",
    "plot_distribution(dist)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Running the circuit in real devices\n",
    "\n",
    "We have seen that the simulator can find the solution to the combinatorial problem. We would like to see what happens if we use the real quantum devices that have noise and imperfect gates. \n",
    "\n",
    "However, due to the restriction on the length of strings that can be sent over the network to the real devices (there are more than sixty thousands charactes of QASM of the circuit), at the moment the above circuit cannot be run on real-device backends. We can see the compiled QASM on real-device `ibmqx5` backend as follows."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-25T18:41:39.542516Z",
     "start_time": "2018-09-25T18:41:28.052236Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gates =  OrderedDict([('rz', 3302), ('sx', 2153), ('ecr', 729), ('x', 37), ('measure', 3)])\n",
      "depth =  3274\n"
     ]
    }
   ],
   "source": [
    "from qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager\n",
    "from qiskit_ibm_runtime.fake_provider import FakeBrisbane\n",
    "\n",
    "backend = FakeBrisbane()#service.backend('ibm_brisbane')\n",
    "target = backend.target\n",
    "pm = generate_preset_pass_manager(target=target, optimization_level=3)\n",
    "\n",
    "grover_compiled = pm.run(grover)\n",
    "\n",
    "print('gates = ', grover_compiled.count_ops())\n",
    "print('depth = ', grover_compiled.depth())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The number of gates is in the order of thousands which is above the limits of decoherence time of the current near-term quantum computers. It is a challenge to design a quantum circuit for Grover search to solve large optimization problems. \n",
    "\n",
    "### Future Work\n",
    "\n",
    "In addition to using too many gates, the circuit in this notebook uses auxiliary qubits. It is left as future work to improve the efficiency of the circuit as to make it possible to run it in the real devices. Below is the original circuit. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-25T18:42:03.016728Z",
     "start_time": "2018-09-25T18:41:52.288291Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<pre style=\"word-wrap: normal;white-space: pre;background: #fff0;line-height: 1.1;font-family: &quot;Courier New&quot;,Courier,monospace\">        ┌───┐                                   ┌───┐                         »\n",
       " q10_0: ┤ H ├───────■──────────────■─────────■──┤ X ├───────■──────────────■──»\n",
       "        ├───┤       │              │         │  ├───┤       │              │  »\n",
       " q10_1: ┤ H ├───────┼────■─────────■─────────■──┤ X ├───────┼────■─────────■──»\n",
       "        ├───┤┌───┐  │    │         │         │  ├───┤┌───┐  │    │         │  »\n",
       " q10_2: ┤ H ├┤ X ├──┼────┼────■────┼────■────┼──┤ X ├┤ X ├──┼────┼────■────┼──»\n",
       "        ├───┤├───┤  │    │    │    │    │    │  └───┘└───┘  │    │    │    │  »\n",
       "   q11: ┤ X ├┤ H ├──┼────┼────┼────┼────┼────┼──────────────┼────┼────┼────┼──»\n",
       "        └───┘└───┘┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │              │    │    │    │  »\n",
       " q12_0: ──────────┤ X ├┤ X ├┤ X ├──┼──┤ X ├──┼──────────────┼────┼────┼────┼──»\n",
       "                  └───┘└───┘└───┘  │  └─┬─┘  │            ┌─┴─┐┌─┴─┐┌─┴─┐  │  »\n",
       " q12_1: ───────────────────────────┼────┼────┼────────────┤ X ├┤ X ├┤ X ├──┼──»\n",
       "                                   │    │    │            └───┘└───┘└───┘  │  »\n",
       " q12_2: ───────────────────────────┼────┼────┼─────────────────────────────┼──»\n",
       "                                 ┌─┴─┐  │  ┌─┴─┐                         ┌─┴─┐»\n",
       " q12_3: ─────────────────────────┤ X ├──■──┤ X ├─────────────────────────┤ X ├»\n",
       "                                 └───┘     └───┘                         └───┘»\n",
       "meas: 3/══════════════════════════════════════════════════════════════════════»\n",
       "                                                                              »\n",
       "«                  ┌───┐┌───┐                              ┌───┐               »\n",
       "« q10_0: ───────■──┤ X ├┤ X ├──■──────────────■─────────■──┤ X ├────────────■──»\n",
       "«               │  ├───┤└───┘  │              │         │  └───┘            │  »\n",
       "« q10_1: ───────■──┤ X ├───────┼────■─────────■─────────■───────────────────┼──»\n",
       "«               │  ├───┤       │    │         │         │  ┌───┐            │  »\n",
       "« q10_2: ──■────┼──┤ X ├───────┼────┼────■────┼────■────┼──┤ X ├────────────┼──»\n",
       "«          │    │  └───┘       │    │    │    │    │    │  └───┘┌───┐       │  »\n",
       "«   q11: ──┼────┼──────────────┼────┼────┼────┼────┼────┼───────┤ X ├───────┼──»\n",
       "«          │    │              │    │    │    │    │    │       └─┬─┘     ┌─┴─┐»\n",
       "« q12_0: ──┼────┼──────────────┼────┼────┼────┼────┼────┼────■────┼────■──┤ X ├»\n",
       "«        ┌─┴─┐  │              │    │    │    │    │    │    │    │    │  └───┘»\n",
       "« q12_1: ┤ X ├──┼──────────────┼────┼────┼────┼────┼────┼────■────┼────■───────»\n",
       "«        └─┬─┘  │            ┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │    │    │    │       »\n",
       "« q12_2: ──┼────┼────────────┤ X ├┤ X ├┤ X ├──┼──┤ X ├──┼────┼────■────┼───────»\n",
       "«          │  ┌─┴─┐          └───┘└───┘└───┘┌─┴─┐└─┬─┘┌─┴─┐┌─┴─┐  │  ┌─┴─┐     »\n",
       "« q12_3: ──■──┤ X ├─────────────────────────┤ X ├──■──┤ X ├┤ X ├──■──┤ X ├─────»\n",
       "«             └───┘                         └───┘     └───┘└───┘     └───┘     »\n",
       "«meas: 3/══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                              »\n",
       "«                                 ┌───┐                                   ┌───┐»\n",
       "« q10_0: ────────────■─────────■──┤ X ├───────■──────────────■─────────■──┤ X ├»\n",
       "«                    │         │  ├───┤       │              │         │  ├───┤»\n",
       "« q10_1: ──■─────────■─────────■──┤ X ├───────┼────■─────────■─────────■──┤ X ├»\n",
       "«          │         │         │  ├───┤┌───┐  │    │         │         │  ├───┤»\n",
       "« q10_2: ──┼────■────┼────■────┼──┤ X ├┤ X ├──┼────┼────■────┼────■────┼──┤ X ├»\n",
       "«          │    │    │    │    │  └───┘└───┘  │    │    │    │    │    │  └───┘»\n",
       "«   q11: ──┼────┼────┼────┼────┼──────────────┼────┼────┼────┼────┼────┼───────»\n",
       "«        ┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │              │    │    │    │    │    │       »\n",
       "« q12_0: ┤ X ├┤ X ├──┼──┤ X ├──┼──────────────┼────┼────┼────┼────┼────┼───────»\n",
       "«        └───┘└───┘  │  └─┬─┘  │            ┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │       »\n",
       "« q12_1: ────────────┼────┼────┼────────────┤ X ├┤ X ├┤ X ├──┼──┤ X ├──┼───────»\n",
       "«                    │    │    │            └───┘└───┘└───┘  │  └─┬─┘  │       »\n",
       "« q12_2: ────────────┼────┼────┼─────────────────────────────┼────┼────┼───────»\n",
       "«                  ┌─┴─┐  │  ┌─┴─┐                         ┌─┴─┐  │  ┌─┴─┐     »\n",
       "« q12_3: ──────────┤ X ├──■──┤ X ├─────────────────────────┤ X ├──■──┤ X ├─────»\n",
       "«                  └───┘     └───┘                         └───┘     └───┘     »\n",
       "«meas: 3/══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                              »\n",
       "«        ┌───┐                              ┌───┐┌───┐┌───┐     ┌───┐┌───┐     »\n",
       "« q10_0: ┤ X ├──■──────────────■─────────■──┤ X ├┤ H ├┤ X ├──■──┤ X ├┤ H ├─────»\n",
       "«        └───┘  │              │         │  ├───┤├───┤└───┘  │  ├───┤├───┤     »\n",
       "« q10_1: ───────┼────■─────────■─────────■──┤ H ├┤ X ├───────■──┤ X ├┤ H ├─────»\n",
       "«               │    │         │         │  ├───┤├───┤┌───┐┌─┴─┐├───┤├───┤┌───┐»\n",
       "« q10_2: ───────┼────┼────■────┼────■────┼──┤ H ├┤ X ├┤ H ├┤ X ├┤ H ├┤ X ├┤ H ├»\n",
       "«               │    │    │    │    │    │  └───┘└───┘└───┘└───┘└───┘└───┘└───┘»\n",
       "«   q11: ───────┼────┼────┼────┼────┼────┼─────────────────────────────────────»\n",
       "«               │    │    │    │    │    │                                     »\n",
       "« q12_0: ───────┼────┼────┼────┼────┼────┼─────────────────────────────────────»\n",
       "«               │    │    │    │    │    │                                     »\n",
       "« q12_1: ───────┼────┼────┼────┼────┼────┼─────────────────────────────────────»\n",
       "«             ┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │                                     »\n",
       "« q12_2: ─────┤ X ├┤ X ├┤ X ├──┼──┤ X ├──┼─────────────────────────────────────»\n",
       "«             └───┘└───┘└───┘┌─┴─┐└─┬─┘┌─┴─┐                                   »\n",
       "« q12_3: ────────────────────┤ X ├──■──┤ X ├───────────────────────────────────»\n",
       "«                            └───┘     └───┘                                   »\n",
       "«meas: 3/══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                              »\n",
       "«                                           ┌───┐                              »\n",
       "« q10_0: ──■───────────────────■─────────■──┤ X ├───────■──────────────■───────»\n",
       "«          │                   │         │  ├───┤       │              │       »\n",
       "« q10_1: ──┼────■──────────────■─────────■──┤ X ├───────┼────■─────────■───────»\n",
       "«          │    │  ┌───┐       │         │  ├───┤┌───┐  │    │         │       »\n",
       "« q10_2: ──┼────┼──┤ X ├──■────┼────■────┼──┤ X ├┤ X ├──┼────┼────■────┼────■──»\n",
       "«          │    │  └───┘  │    │    │    │  └───┘└───┘  │    │    │    │    │  »\n",
       "«   q11: ──┼────┼─────────┼────┼────┼────┼──────────────┼────┼────┼────┼────┼──»\n",
       "«        ┌─┴─┐┌─┴─┐     ┌─┴─┐  │  ┌─┴─┐  │              │    │    │    │    │  »\n",
       "« q12_0: ┤ X ├┤ X ├─────┤ X ├──┼──┤ X ├──┼──────────────┼────┼────┼────┼────┼──»\n",
       "«        └───┘└───┘     └───┘  │  └─┬─┘  │            ┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐»\n",
       "« q12_1: ──────────────────────┼────┼────┼────────────┤ X ├┤ X ├┤ X ├──┼──┤ X ├»\n",
       "«                              │    │    │            └───┘└───┘└───┘  │  └─┬─┘»\n",
       "« q12_2: ──────────────────────┼────┼────┼─────────────────────────────┼────┼──»\n",
       "«                            ┌─┴─┐  │  ┌─┴─┐                         ┌─┴─┐  │  »\n",
       "« q12_3: ────────────────────┤ X ├──■──┤ X ├─────────────────────────┤ X ├──■──»\n",
       "«                            └───┘     └───┘                         └───┘     »\n",
       "«meas: 3/══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                              »\n",
       "«             ┌───┐┌───┐                              ┌───┐                    »\n",
       "« q10_0: ──■──┤ X ├┤ X ├──■──────────────■─────────■──┤ X ├────────────■───────»\n",
       "«          │  ├───┤└───┘  │              │         │  └───┘            │       »\n",
       "« q10_1: ──■──┤ X ├───────┼────■─────────■─────────■───────────────────┼────■──»\n",
       "«          │  ├───┤       │    │         │         │  ┌───┐            │    │  »\n",
       "« q10_2: ──┼──┤ X ├───────┼────┼────■────┼────■────┼──┤ X ├────────────┼────┼──»\n",
       "«          │  └───┘       │    │    │    │    │    │  └───┘┌───┐       │    │  »\n",
       "«   q11: ──┼──────────────┼────┼────┼────┼────┼────┼───────┤ X ├───────┼────┼──»\n",
       "«          │              │    │    │    │    │    │       └─┬─┘     ┌─┴─┐┌─┴─┐»\n",
       "« q12_0: ──┼──────────────┼────┼────┼────┼────┼────┼────■────┼────■──┤ X ├┤ X ├»\n",
       "«          │              │    │    │    │    │    │    │    │    │  └───┘└───┘»\n",
       "« q12_1: ──┼──────────────┼────┼────┼────┼────┼────┼────■────┼────■────────────»\n",
       "«          │            ┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │    │    │    │            »\n",
       "« q12_2: ──┼────────────┤ X ├┤ X ├┤ X ├──┼──┤ X ├──┼────┼────■────┼────────────»\n",
       "«        ┌─┴─┐          └───┘└───┘└───┘┌─┴─┐└─┬─┘┌─┴─┐┌─┴─┐  │  ┌─┴─┐          »\n",
       "« q12_3: ┤ X ├─────────────────────────┤ X ├──■──┤ X ├┤ X ├──■──┤ X ├──────────»\n",
       "«        └───┘                         └───┘     └───┘└───┘     └───┘          »\n",
       "«meas: 3/══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                              »\n",
       "«                            ┌───┐                                   ┌───┐┌───┐»\n",
       "« q10_0: ───────■─────────■──┤ X ├───────■──────────────■─────────■──┤ X ├┤ X ├»\n",
       "«               │         │  ├───┤       │              │         │  ├───┤└───┘»\n",
       "« q10_1: ───────■─────────■──┤ X ├───────┼────■─────────■─────────■──┤ X ├─────»\n",
       "«               │         │  ├───┤┌───┐  │    │         │         │  ├───┤     »\n",
       "« q10_2: ──■────┼────■────┼──┤ X ├┤ X ├──┼────┼────■────┼────■────┼──┤ X ├─────»\n",
       "«          │    │    │    │  └───┘└───┘  │    │    │    │    │    │  └───┘     »\n",
       "«   q11: ──┼────┼────┼────┼──────────────┼────┼────┼────┼────┼────┼────────────»\n",
       "«        ┌─┴─┐  │  ┌─┴─┐  │              │    │    │    │    │    │            »\n",
       "« q12_0: ┤ X ├──┼──┤ X ├──┼──────────────┼────┼────┼────┼────┼────┼────────────»\n",
       "«        └───┘  │  └─┬─┘  │            ┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │            »\n",
       "« q12_1: ───────┼────┼────┼────────────┤ X ├┤ X ├┤ X ├──┼──┤ X ├──┼────────────»\n",
       "«               │    │    │            └───┘└───┘└───┘  │  └─┬─┘  │            »\n",
       "« q12_2: ───────┼────┼────┼─────────────────────────────┼────┼────┼────────────»\n",
       "«             ┌─┴─┐  │  ┌─┴─┐                         ┌─┴─┐  │  ┌─┴─┐          »\n",
       "« q12_3: ─────┤ X ├──■──┤ X ├─────────────────────────┤ X ├──■──┤ X ├──────────»\n",
       "«             └───┘     └───┘                         └───┘     └───┘          »\n",
       "«meas: 3/══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                              »\n",
       "«                                      ┌───┐┌───┐┌───┐     ┌───┐┌───┐     ┌─┐»\n",
       "« q10_0: ──■──────────────■─────────■──┤ X ├┤ H ├┤ X ├──■──┤ X ├┤ H ├─────┤M├»\n",
       "«          │              │         │  ├───┤├───┤└───┘  │  ├───┤├───┤     └╥┘»\n",
       "« q10_1: ──┼────■─────────■─────────■──┤ H ├┤ X ├───────■──┤ X ├┤ H ├──────╫─»\n",
       "«          │    │         │         │  ├───┤├───┤┌───┐┌─┴─┐├───┤├───┤┌───┐ ║ »\n",
       "« q10_2: ──┼────┼────■────┼────■────┼──┤ H ├┤ X ├┤ H ├┤ X ├┤ H ├┤ X ├┤ H ├─╫─»\n",
       "«          │    │    │    │    │    │  └───┘└───┘└───┘└───┘└───┘└───┘└───┘ ║ »\n",
       "«   q11: ──┼────┼────┼────┼────┼────┼──────────────────────────────────────╫─»\n",
       "«          │    │    │    │    │    │                                      ║ »\n",
       "« q12_0: ──┼────┼────┼────┼────┼────┼──────────────────────────────────────╫─»\n",
       "«          │    │    │    │    │    │                                      ║ »\n",
       "« q12_1: ──┼────┼────┼────┼────┼────┼──────────────────────────────────────╫─»\n",
       "«        ┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │                                      ║ »\n",
       "« q12_2: ┤ X ├┤ X ├┤ X ├──┼──┤ X ├──┼──────────────────────────────────────╫─»\n",
       "«        └───┘└───┘└───┘┌─┴─┐└─┬─┘┌─┴─┐                                    ║ »\n",
       "« q12_3: ───────────────┤ X ├──■──┤ X ├────────────────────────────────────╫─»\n",
       "«                       └───┘     └───┘                                    ║ »\n",
       "«meas: 3/══════════════════════════════════════════════════════════════════╩═»\n",
       "«                                                                          0 »\n",
       "«              \n",
       "« q10_0: ──────\n",
       "«        ┌─┐   \n",
       "« q10_1: ┤M├───\n",
       "«        └╥┘┌─┐\n",
       "« q10_2: ─╫─┤M├\n",
       "«         ║ └╥┘\n",
       "«   q11: ─╫──╫─\n",
       "«         ║  ║ \n",
       "« q12_0: ─╫──╫─\n",
       "«         ║  ║ \n",
       "« q12_1: ─╫──╫─\n",
       "«         ║  ║ \n",
       "« q12_2: ─╫──╫─\n",
       "«         ║  ║ \n",
       "« q12_3: ─╫──╫─\n",
       "«         ║  ║ \n",
       "«meas: 3/═╩══╩═\n",
       "«         1  2 </pre>"
      ],
      "text/plain": [
       "        ┌───┐                                   ┌───┐                         »\n",
       " q10_0: ┤ H ├───────■──────────────■─────────■──┤ X ├───────■──────────────■──»\n",
       "        ├───┤       │              │         │  ├───┤       │              │  »\n",
       " q10_1: ┤ H ├───────┼────■─────────■─────────■──┤ X ├───────┼────■─────────■──»\n",
       "        ├───┤┌───┐  │    │         │         │  ├───┤┌───┐  │    │         │  »\n",
       " q10_2: ┤ H ├┤ X ├──┼────┼────■────┼────■────┼──┤ X ├┤ X ├──┼────┼────■────┼──»\n",
       "        ├───┤├───┤  │    │    │    │    │    │  └───┘└───┘  │    │    │    │  »\n",
       "   q11: ┤ X ├┤ H ├──┼────┼────┼────┼────┼────┼──────────────┼────┼────┼────┼──»\n",
       "        └───┘└───┘┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │              │    │    │    │  »\n",
       " q12_0: ──────────┤ X ├┤ X ├┤ X ├──┼──┤ X ├──┼──────────────┼────┼────┼────┼──»\n",
       "                  └───┘└───┘└───┘  │  └─┬─┘  │            ┌─┴─┐┌─┴─┐┌─┴─┐  │  »\n",
       " q12_1: ───────────────────────────┼────┼────┼────────────┤ X ├┤ X ├┤ X ├──┼──»\n",
       "                                   │    │    │            └───┘└───┘└───┘  │  »\n",
       " q12_2: ───────────────────────────┼────┼────┼─────────────────────────────┼──»\n",
       "                                 ┌─┴─┐  │  ┌─┴─┐                         ┌─┴─┐»\n",
       " q12_3: ─────────────────────────┤ X ├──■──┤ X ├─────────────────────────┤ X ├»\n",
       "                                 └───┘     └───┘                         └───┘»\n",
       "meas: 3/══════════════════════════════════════════════════════════════════════»\n",
       "                                                                              »\n",
       "«                  ┌───┐┌───┐                              ┌───┐               »\n",
       "« q10_0: ───────■──┤ X ├┤ X ├──■──────────────■─────────■──┤ X ├────────────■──»\n",
       "«               │  ├───┤└───┘  │              │         │  └───┘            │  »\n",
       "« q10_1: ───────■──┤ X ├───────┼────■─────────■─────────■───────────────────┼──»\n",
       "«               │  ├───┤       │    │         │         │  ┌───┐            │  »\n",
       "« q10_2: ──■────┼──┤ X ├───────┼────┼────■────┼────■────┼──┤ X ├────────────┼──»\n",
       "«          │    │  └───┘       │    │    │    │    │    │  └───┘┌───┐       │  »\n",
       "«   q11: ──┼────┼──────────────┼────┼────┼────┼────┼────┼───────┤ X ├───────┼──»\n",
       "«          │    │              │    │    │    │    │    │       └─┬─┘     ┌─┴─┐»\n",
       "« q12_0: ──┼────┼──────────────┼────┼────┼────┼────┼────┼────■────┼────■──┤ X ├»\n",
       "«        ┌─┴─┐  │              │    │    │    │    │    │    │    │    │  └───┘»\n",
       "« q12_1: ┤ X ├──┼──────────────┼────┼────┼────┼────┼────┼────■────┼────■───────»\n",
       "«        └─┬─┘  │            ┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │    │    │    │       »\n",
       "« q12_2: ──┼────┼────────────┤ X ├┤ X ├┤ X ├──┼──┤ X ├──┼────┼────■────┼───────»\n",
       "«          │  ┌─┴─┐          └───┘└───┘└───┘┌─┴─┐└─┬─┘┌─┴─┐┌─┴─┐  │  ┌─┴─┐     »\n",
       "« q12_3: ──■──┤ X ├─────────────────────────┤ X ├──■──┤ X ├┤ X ├──■──┤ X ├─────»\n",
       "«             └───┘                         └───┘     └───┘└───┘     └───┘     »\n",
       "«meas: 3/══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                              »\n",
       "«                                 ┌───┐                                   ┌───┐»\n",
       "« q10_0: ────────────■─────────■──┤ X ├───────■──────────────■─────────■──┤ X ├»\n",
       "«                    │         │  ├───┤       │              │         │  ├───┤»\n",
       "« q10_1: ──■─────────■─────────■──┤ X ├───────┼────■─────────■─────────■──┤ X ├»\n",
       "«          │         │         │  ├───┤┌───┐  │    │         │         │  ├───┤»\n",
       "« q10_2: ──┼────■────┼────■────┼──┤ X ├┤ X ├──┼────┼────■────┼────■────┼──┤ X ├»\n",
       "«          │    │    │    │    │  └───┘└───┘  │    │    │    │    │    │  └───┘»\n",
       "«   q11: ──┼────┼────┼────┼────┼──────────────┼────┼────┼────┼────┼────┼───────»\n",
       "«        ┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │              │    │    │    │    │    │       »\n",
       "« q12_0: ┤ X ├┤ X ├──┼──┤ X ├──┼──────────────┼────┼────┼────┼────┼────┼───────»\n",
       "«        └───┘└───┘  │  └─┬─┘  │            ┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │       »\n",
       "« q12_1: ────────────┼────┼────┼────────────┤ X ├┤ X ├┤ X ├──┼──┤ X ├──┼───────»\n",
       "«                    │    │    │            └───┘└───┘└───┘  │  └─┬─┘  │       »\n",
       "« q12_2: ────────────┼────┼────┼─────────────────────────────┼────┼────┼───────»\n",
       "«                  ┌─┴─┐  │  ┌─┴─┐                         ┌─┴─┐  │  ┌─┴─┐     »\n",
       "« q12_3: ──────────┤ X ├──■──┤ X ├─────────────────────────┤ X ├──■──┤ X ├─────»\n",
       "«                  └───┘     └───┘                         └───┘     └───┘     »\n",
       "«meas: 3/══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                              »\n",
       "«        ┌───┐                              ┌───┐┌───┐┌───┐     ┌───┐┌───┐     »\n",
       "« q10_0: ┤ X ├──■──────────────■─────────■──┤ X ├┤ H ├┤ X ├──■──┤ X ├┤ H ├─────»\n",
       "«        └───┘  │              │         │  ├───┤├───┤└───┘  │  ├───┤├───┤     »\n",
       "« q10_1: ───────┼────■─────────■─────────■──┤ H ├┤ X ├───────■──┤ X ├┤ H ├─────»\n",
       "«               │    │         │         │  ├───┤├───┤┌───┐┌─┴─┐├───┤├───┤┌───┐»\n",
       "« q10_2: ───────┼────┼────■────┼────■────┼──┤ H ├┤ X ├┤ H ├┤ X ├┤ H ├┤ X ├┤ H ├»\n",
       "«               │    │    │    │    │    │  └───┘└───┘└───┘└───┘└───┘└───┘└───┘»\n",
       "«   q11: ───────┼────┼────┼────┼────┼────┼─────────────────────────────────────»\n",
       "«               │    │    │    │    │    │                                     »\n",
       "« q12_0: ───────┼────┼────┼────┼────┼────┼─────────────────────────────────────»\n",
       "«               │    │    │    │    │    │                                     »\n",
       "« q12_1: ───────┼────┼────┼────┼────┼────┼─────────────────────────────────────»\n",
       "«             ┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │                                     »\n",
       "« q12_2: ─────┤ X ├┤ X ├┤ X ├──┼──┤ X ├──┼─────────────────────────────────────»\n",
       "«             └───┘└───┘└───┘┌─┴─┐└─┬─┘┌─┴─┐                                   »\n",
       "« q12_3: ────────────────────┤ X ├──■──┤ X ├───────────────────────────────────»\n",
       "«                            └───┘     └───┘                                   »\n",
       "«meas: 3/══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                              »\n",
       "«                                           ┌───┐                              »\n",
       "« q10_0: ──■───────────────────■─────────■──┤ X ├───────■──────────────■───────»\n",
       "«          │                   │         │  ├───┤       │              │       »\n",
       "« q10_1: ──┼────■──────────────■─────────■──┤ X ├───────┼────■─────────■───────»\n",
       "«          │    │  ┌───┐       │         │  ├───┤┌───┐  │    │         │       »\n",
       "« q10_2: ──┼────┼──┤ X ├──■────┼────■────┼──┤ X ├┤ X ├──┼────┼────■────┼────■──»\n",
       "«          │    │  └───┘  │    │    │    │  └───┘└───┘  │    │    │    │    │  »\n",
       "«   q11: ──┼────┼─────────┼────┼────┼────┼──────────────┼────┼────┼────┼────┼──»\n",
       "«        ┌─┴─┐┌─┴─┐     ┌─┴─┐  │  ┌─┴─┐  │              │    │    │    │    │  »\n",
       "« q12_0: ┤ X ├┤ X ├─────┤ X ├──┼──┤ X ├──┼──────────────┼────┼────┼────┼────┼──»\n",
       "«        └───┘└───┘     └───┘  │  └─┬─┘  │            ┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐»\n",
       "« q12_1: ──────────────────────┼────┼────┼────────────┤ X ├┤ X ├┤ X ├──┼──┤ X ├»\n",
       "«                              │    │    │            └───┘└───┘└───┘  │  └─┬─┘»\n",
       "« q12_2: ──────────────────────┼────┼────┼─────────────────────────────┼────┼──»\n",
       "«                            ┌─┴─┐  │  ┌─┴─┐                         ┌─┴─┐  │  »\n",
       "« q12_3: ────────────────────┤ X ├──■──┤ X ├─────────────────────────┤ X ├──■──»\n",
       "«                            └───┘     └───┘                         └───┘     »\n",
       "«meas: 3/══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                              »\n",
       "«             ┌───┐┌───┐                              ┌───┐                    »\n",
       "« q10_0: ──■──┤ X ├┤ X ├──■──────────────■─────────■──┤ X ├────────────■───────»\n",
       "«          │  ├───┤└───┘  │              │         │  └───┘            │       »\n",
       "« q10_1: ──■──┤ X ├───────┼────■─────────■─────────■───────────────────┼────■──»\n",
       "«          │  ├───┤       │    │         │         │  ┌───┐            │    │  »\n",
       "« q10_2: ──┼──┤ X ├───────┼────┼────■────┼────■────┼──┤ X ├────────────┼────┼──»\n",
       "«          │  └───┘       │    │    │    │    │    │  └───┘┌───┐       │    │  »\n",
       "«   q11: ──┼──────────────┼────┼────┼────┼────┼────┼───────┤ X ├───────┼────┼──»\n",
       "«          │              │    │    │    │    │    │       └─┬─┘     ┌─┴─┐┌─┴─┐»\n",
       "« q12_0: ──┼──────────────┼────┼────┼────┼────┼────┼────■────┼────■──┤ X ├┤ X ├»\n",
       "«          │              │    │    │    │    │    │    │    │    │  └───┘└───┘»\n",
       "« q12_1: ──┼──────────────┼────┼────┼────┼────┼────┼────■────┼────■────────────»\n",
       "«          │            ┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │    │    │    │            »\n",
       "« q12_2: ──┼────────────┤ X ├┤ X ├┤ X ├──┼──┤ X ├──┼────┼────■────┼────────────»\n",
       "«        ┌─┴─┐          └───┘└───┘└───┘┌─┴─┐└─┬─┘┌─┴─┐┌─┴─┐  │  ┌─┴─┐          »\n",
       "« q12_3: ┤ X ├─────────────────────────┤ X ├──■──┤ X ├┤ X ├──■──┤ X ├──────────»\n",
       "«        └───┘                         └───┘     └───┘└───┘     └───┘          »\n",
       "«meas: 3/══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                              »\n",
       "«                            ┌───┐                                   ┌───┐┌───┐»\n",
       "« q10_0: ───────■─────────■──┤ X ├───────■──────────────■─────────■──┤ X ├┤ X ├»\n",
       "«               │         │  ├───┤       │              │         │  ├───┤└───┘»\n",
       "« q10_1: ───────■─────────■──┤ X ├───────┼────■─────────■─────────■──┤ X ├─────»\n",
       "«               │         │  ├───┤┌───┐  │    │         │         │  ├───┤     »\n",
       "« q10_2: ──■────┼────■────┼──┤ X ├┤ X ├──┼────┼────■────┼────■────┼──┤ X ├─────»\n",
       "«          │    │    │    │  └───┘└───┘  │    │    │    │    │    │  └───┘     »\n",
       "«   q11: ──┼────┼────┼────┼──────────────┼────┼────┼────┼────┼────┼────────────»\n",
       "«        ┌─┴─┐  │  ┌─┴─┐  │              │    │    │    │    │    │            »\n",
       "« q12_0: ┤ X ├──┼──┤ X ├──┼──────────────┼────┼────┼────┼────┼────┼────────────»\n",
       "«        └───┘  │  └─┬─┘  │            ┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │            »\n",
       "« q12_1: ───────┼────┼────┼────────────┤ X ├┤ X ├┤ X ├──┼──┤ X ├──┼────────────»\n",
       "«               │    │    │            └───┘└───┘└───┘  │  └─┬─┘  │            »\n",
       "« q12_2: ───────┼────┼────┼─────────────────────────────┼────┼────┼────────────»\n",
       "«             ┌─┴─┐  │  ┌─┴─┐                         ┌─┴─┐  │  ┌─┴─┐          »\n",
       "« q12_3: ─────┤ X ├──■──┤ X ├─────────────────────────┤ X ├──■──┤ X ├──────────»\n",
       "«             └───┘     └───┘                         └───┘     └───┘          »\n",
       "«meas: 3/══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                              »\n",
       "«                                      ┌───┐┌───┐┌───┐     ┌───┐┌───┐     ┌─┐»\n",
       "« q10_0: ──■──────────────■─────────■──┤ X ├┤ H ├┤ X ├──■──┤ X ├┤ H ├─────┤M├»\n",
       "«          │              │         │  ├───┤├───┤└───┘  │  ├───┤├───┤     └╥┘»\n",
       "« q10_1: ──┼────■─────────■─────────■──┤ H ├┤ X ├───────■──┤ X ├┤ H ├──────╫─»\n",
       "«          │    │         │         │  ├───┤├───┤┌───┐┌─┴─┐├───┤├───┤┌───┐ ║ »\n",
       "« q10_2: ──┼────┼────■────┼────■────┼──┤ H ├┤ X ├┤ H ├┤ X ├┤ H ├┤ X ├┤ H ├─╫─»\n",
       "«          │    │    │    │    │    │  └───┘└───┘└───┘└───┘└───┘└───┘└───┘ ║ »\n",
       "«   q11: ──┼────┼────┼────┼────┼────┼──────────────────────────────────────╫─»\n",
       "«          │    │    │    │    │    │                                      ║ »\n",
       "« q12_0: ──┼────┼────┼────┼────┼────┼──────────────────────────────────────╫─»\n",
       "«          │    │    │    │    │    │                                      ║ »\n",
       "« q12_1: ──┼────┼────┼────┼────┼────┼──────────────────────────────────────╫─»\n",
       "«        ┌─┴─┐┌─┴─┐┌─┴─┐  │  ┌─┴─┐  │                                      ║ »\n",
       "« q12_2: ┤ X ├┤ X ├┤ X ├──┼──┤ X ├──┼──────────────────────────────────────╫─»\n",
       "«        └───┘└───┘└───┘┌─┴─┐└─┬─┘┌─┴─┐                                    ║ »\n",
       "« q12_3: ───────────────┤ X ├──■──┤ X ├────────────────────────────────────╫─»\n",
       "«                       └───┘     └───┘                                    ║ »\n",
       "«meas: 3/══════════════════════════════════════════════════════════════════╩═»\n",
       "«                                                                          0 »\n",
       "«              \n",
       "« q10_0: ──────\n",
       "«        ┌─┐   \n",
       "« q10_1: ┤M├───\n",
       "«        └╥┘┌─┐\n",
       "« q10_2: ─╫─┤M├\n",
       "«         ║ └╥┘\n",
       "«   q11: ─╫──╫─\n",
       "«         ║  ║ \n",
       "« q12_0: ─╫──╫─\n",
       "«         ║  ║ \n",
       "« q12_1: ─╫──╫─\n",
       "«         ║  ║ \n",
       "« q12_2: ─╫──╫─\n",
       "«         ║  ║ \n",
       "« q12_3: ─╫──╫─\n",
       "«         ║  ║ \n",
       "«meas: 3/═╩══╩═\n",
       "«         1  2 "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grover.draw()"
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   "source": [
    "## References\n",
    "\n",
    "[1] \"[A fast quantum mechanical algorithm for database search](https://arxiv.org/abs/quant-ph/9605043)\", L. K. Grover, Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (STOC 1996)\n",
    "\n",
    "[2] \"[Tight bounds on quantum searching](https://arxiv.org/abs/quant-ph/9605034)\", Boyer et al., Fortsch.Phys.46:493-506,1998"
   ]
  }
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