10_8-Puzzle.ipynb

{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Chapter 10: 8 Puzzle\n",
    "\n",
    "\n",
    "For  $8$-puzzle, there are $9$   different objects: eight cells and one empty cell. Each object has to be represented by four $4$ qubits since $3$ qubits allow only to represent $2^3=8$ different states.\n",
    "The object $1$  is represented by $001$,  $2$  is represented by $010$ and  $3$  is represented by $011$, and we continue the representation as binary numbers with $8$ represented as $1000$. We represent the empty space $x$   by $1111$. The~state is represented by 36 qubits  $x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, \\cdots, x_{35}$.\n",
    "\n",
    "The empty cell can be present in $9$ different positions. The~empty cell  can move either  up, down, left or right.   The new board configuration is determined by the function $p$.  The~input of the function $p$ is the current board configuration and two bits\\mbox{ $m=m_1, m_2$} (qubits $46$ and $47$) indicating whether the blank cell should perform move  right $(m=0=|00 \\rangle)$, left $(m=1=|01 \\rangle)$, up $(m=2=|10\\rangle)$ or  down $(m=3=|11\\rangle)$. There are $36$ qubits to represent the state and $9$ qubits for the trace, together with the auxiliary qubit $49$ qubits are represented by the quantum circuit. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from qiskit import QuantumCircuit, transpile, Aer,assemble,execute\n",
    "from qiskit.visualization import plot_histogram, plot_state_hinton, plot_bloch_multivector\n",
    "from qiskit.quantum_info import Statevector\n",
    "from qiskit.circuit.library import MCXGate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the case  the empty cell is in the center, four movements are possible. For a cell in the edge only three movements are possible, for the corner only two movements are possible"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "def if_rules():\n",
    "    qc = QuantumCircuit(46)\n",
    "    #Marke the trace indicate the rule group through trace\n",
    "    gate = MCXGate(4)\n",
    "    #Empty Space in corner, 2 movements\n",
    "    qc.append(gate, [0, 1, 2, 3, 37])  \n",
    "    #Empty Space in edge, 3 movements\n",
    "    qc.append(gate, [4, 5, 6, 7, 38])\n",
    "    #Empty Space in corner, 2 movements\n",
    "    qc.append(gate, [8, 9, 10, 11, 39])\n",
    "    #Empty Space in edge, 3 movements\n",
    "    qc.append(gate, [12, 13, 14, 15, 40])\n",
    "    #Empty Space in center, 4 movements\n",
    "    qc.append(gate, [16, 17, 18, 19, 41])\n",
    "    #Empty Space in edge, 3 movements\n",
    "    qc.append(gate, [20, 21, 22, 23, 42])\n",
    "    #Empty Space in corner, 2 movements\n",
    "    qc.append(gate, [24, 25, 26, 27, 43])\n",
    "     #Empty Space in edge, 3 movements\n",
    "    qc.append(gate, [28, 29, 30, 31, 44])\n",
    "    #Empty Space in corner, 2 movements\n",
    "    qc.append(gate, [32, 33, 34, 35, 45])\n",
    "    \n",
    "    qc.name=\"IF\"\n",
    "    return qc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the empty space in the center there are four instantiations corresponding to the four movemen"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "def rules1():\n",
    "\n",
    "\n",
    "    qc = QuantumCircuit(48)\n",
    "    #Depth1\n",
    "    #Flag 36\n",
    "    #Path Descriptor 46, 47\n",
    "    #Trace 37-45\n",
    "    \n",
    "    flag_gate = MCXGate(3)\n",
    "\n",
    "    #In the center  qc.append(gate, [16, 17, 18, 19, 41])\n",
    "\n",
    "    #If then rule (c 1) right, for empty at 16, 17, 18, 19  -> 12, 13, 14, 15\n",
    "    \n",
    "    qc.append(flag_gate, [41, 46, 47, 36])\n",
    "\n",
    "    #Move\n",
    "\n",
    "    qc.cswap(36,16,12)\n",
    "    qc.cswap(36,17,13)\n",
    "    qc.cswap(36,18,14)\n",
    "    qc.cswap(36,19,15)\n",
    "\n",
    "    #Clear Flag\n",
    "\n",
    "    qc.append(flag_gate, [41, 46, 47, 36])\n",
    "\n",
    "    #If then rule (c 2) left, for empty at 16, 17, 18, 19  -> 20, 21, 22, 23\n",
    "\n",
    "    qc.x(46)\n",
    "\n",
    "    qc.append(flag_gate, [41, 46, 47, 36])\n",
    "\n",
    "    #Move\n",
    "\n",
    "    qc.cswap(36,16,20)\n",
    "    qc.cswap(36,17,21)\n",
    "    qc.cswap(36,18,22)\n",
    "    qc.cswap(36,19,23)\n",
    "\n",
    "    #Clear Flag\n",
    "\n",
    "    qc.append(flag_gate, [41, 46, 47, 36])\n",
    "    \n",
    "    qc.x(46)\n",
    "   \n",
    "    #If then rule (c 3) up, for empty at 16, 17, 18, 19  -> 28, 29, 30, 31\n",
    "\n",
    "    qc.x(47)\n",
    "\n",
    "    qc.append(flag_gate, [41, 46, 47, 36])\n",
    "\n",
    "    #Move\n",
    "\n",
    "    qc.cswap(36,16,28)\n",
    "    qc.cswap(36,17,29)\n",
    "    qc.cswap(36,18,30)\n",
    "    qc.cswap(36,19,31)\n",
    "\n",
    "    #Clear Flag\n",
    "\n",
    "    qc.append(flag_gate, [41, 46, 47, 36])\n",
    "    \n",
    "    qc.x(47)\n",
    "\n",
    "    #If then rule (c 4) down, for empty at 16, 17, 18, 19  -> 4, 5, 6, 7\n",
    "    \n",
    "    qc.x(46)\n",
    "    qc.x(47)\n",
    "\n",
    "    qc.append(flag_gate, [41, 46, 47, 36])\n",
    "\n",
    "    #Move\n",
    "\n",
    "    qc.cswap(36,16,4)\n",
    "    qc.cswap(36,17,5)\n",
    "    qc.cswap(36,18,6)\n",
    "    qc.cswap(36,19,7)\n",
    "\n",
    "    #Clear Flag\n",
    "\n",
    "    qc.append(flag_gate, [41, 46, 47, 36])\n",
    "    \n",
    "    qc.x(47)\n",
    "    qc.x(46)\n",
    "\n",
    "    qc.name=\"R1\"\n",
    "    return qc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [],
   "source": [
    "def rules1_inv():\n",
    "    qc=rules1()\n",
    "    qc_inv=qc.inverse()\n",
    "    qc_inv.name=\"R1†\"\n",
    "    return qc_inv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [],
   "source": [
    "def rules2():\n",
    "    qc = QuantumCircuit(48)\n",
    "    #Depth1\n",
    "    #Flag 36\n",
    "    #Path Descriptor 46, 47\n",
    "    #Trace 37-45\n",
    "    \n",
    "    flag_gate = MCXGate(3)\n",
    "\n",
    "    #Rules in the middle\n",
    "\n",
    "    #2th type\n",
    "\n",
    "    #qc.append(gate, [12, 13, 14, 15, 40])\n",
    "    \n",
    "    #If then rule (c 1) up, for empty at 12, 13, 14, 15  -> 24, 25, 26, 27\n",
    "\n",
    "    qc.append(flag_gate, [40, 46, 47, 36])\n",
    "\n",
    "    #Move\n",
    "\n",
    "    qc.cswap(36,12,24)\n",
    "    qc.cswap(36,13,25)\n",
    "    qc.cswap(36,14,26)\n",
    "    qc.cswap(36,15,27)\n",
    "\n",
    "    #Clear Flag\n",
    "\n",
    "    qc.append(flag_gate, [40, 46, 47, 36])\n",
    "\n",
    "\n",
    "    #If then rule (c 2) down, for empty at 12, 13, 14, 15  -> 0, 1, 2, 3\n",
    "   \n",
    "    qc.x(46)\n",
    "    qc.append(flag_gate, [40, 46, 47, 36])\n",
    "\n",
    "    #Move\n",
    "\n",
    "    qc.cswap(36,12,0)\n",
    "    qc.cswap(36,13,1)\n",
    "    qc.cswap(36,14,2)\n",
    "    qc.cswap(36,15,3)\n",
    "\n",
    "\n",
    "    #Clear Flag\n",
    "\n",
    "    qc.append(flag_gate, [40, 46, 47, 36])\n",
    "    qc.x(46)\n",
    "\n",
    "    #If then rule (c 3) left, for empty at 12, 13, 14, 15  -> 16, 17, 18, 19\n",
    "\n",
    "\n",
    "    qc.x(47)\n",
    "    qc.append(flag_gate, [40, 46, 47, 36])\n",
    "\n",
    "    #Move\n",
    "\n",
    "    qc.cswap(36,12,16)\n",
    "    qc.cswap(36,13,17)\n",
    "    qc.cswap(36,14,18)\n",
    "    qc.cswap(36,15,19)\n",
    "\n",
    "\n",
    "    #Clear Flag\n",
    "\n",
    "    qc.append(flag_gate, [40, 46, 47, 36])\n",
    "    qc.x(47)\n",
    "\n",
    "    #Again\n",
    "    qc.x(46)\n",
    "    qc.x(47)\n",
    "    qc.append(flag_gate, [40, 46, 47, 36])\n",
    "\n",
    "    #Move\n",
    "\n",
    "    qc.cswap(36,12,16)\n",
    "    qc.cswap(36,13,17)\n",
    "    qc.cswap(36,14,18)\n",
    "    qc.cswap(36,15,19)\n",
    "\n",
    "\n",
    "    #Clear Flag\n",
    "\n",
    "    qc.append(flag_gate, [40, 46, 47, 36])\n",
    "    qc.x(47)\n",
    "    qc.x(46)\n",
    "\n",
    "\n",
    "    qc.name=\"R2\"\n",
    "    return qc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [],
   "source": [
    "def rules2_inv():\n",
    "    qc=rules2()\n",
    "    qc_inv=qc.inverse()\n",
    "    qc_inv.name=\"R2†\"\n",
    "    return qc_inv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [],
   "source": [
    "def if_rules_inv():\n",
    "    qc=if_rules()\n",
    "    qc_inv=qc.inverse()\n",
    "    qc_inv.name=\"IF†\"\n",
    "    return qc_inv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [],
   "source": [
    "def oracle():\n",
    "    qc = QuantumCircuit(49)\n",
    "    gate = MCXGate(4)\n",
    "    #The Position of the Blank\n",
    "    #Four possible goals after moving the space from the center for depth one\n",
    "      \n",
    "    #qc.append(gate, [12, 13, 14, 15, 48])\n",
    "    #qc.append(gate, [4, 5, 6, 7, 48])\n",
    "    #qc.append(gate, [20, 21, 22, 23, 48])\n",
    "    qc.append(gate, [28, 29, 30, 31, 48])\n",
    "\n",
    "    qc.name=\"O\"\n",
    "    return qc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Grover():\n",
    "\n",
    "    qc = QuantumCircuit(48)\n",
    "    #Diffusor\n",
    "    qc.h([46,47])\n",
    "    qc.z([46,47])\n",
    "    qc.cz(46,47)\n",
    "    qc.h([46,47])\n",
    "\n",
    "    qc.name=\"G\"\n",
    "    return qc\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
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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: ┤ X ├──────░─┤0     ├┤0     ├┤0    ├┤0      ├┤0      ├─░─┤ X ├┤0    ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       " q_1: ┤ X ├──────░─┤1     ├┤1     ├┤1    ├┤1      ├┤1      ├─░─┤ X ├┤1    ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       " q_2: ───────────░─┤2     ├┤2     ├┤2    ├┤2      ├┤2      ├─░──────┤2    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       " q_3: ───────────░─┤3     ├┤3     ├┤3    ├┤3      ├┤3      ├─░──────┤3    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       " q_4: ───────────░─┤4     ├┤4     ├┤4    ├┤4      ├┤4      ├─░──────┤4    ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       " q_5: ┤ X ├──────░─┤5     ├┤5     ├┤5    ├┤5      ├┤5      ├─░─┤ X ├┤5    ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       " q_6: ───────────░─┤6     ├┤6     ├┤6    ├┤6      ├┤6      ├─░──────┤6    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       " q_7: ───────────░─┤7     ├┤7     ├┤7    ├┤7      ├┤7      ├─░──────┤7    ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       " q_8: ┤ X ├──────░─┤8     ├┤8     ├┤8    ├┤8      ├┤8      ├─░─┤ X ├┤8    ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       " q_9: ───────────░─┤9     ├┤9     ├┤9    ├┤9      ├┤9      ├─░──────┤9    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_10: ───────────░─┤10    ├┤10    ├┤10   ├┤10     ├┤10     ├─░──────┤10   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_11: ───────────░─┤11    ├┤11    ├┤11   ├┤11     ├┤11     ├─░──────┤11   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_12: ┤ X ├──────░─┤12    ├┤12    ├┤12   ├┤12     ├┤12     ├─░─┤ X ├┤12   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_13: ───────────░─┤13    ├┤13    ├┤13   ├┤13     ├┤13     ├─░──────┤13   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_14: ┤ X ├──────░─┤14    ├┤14    ├┤14   ├┤14     ├┤14     ├─░─┤ X ├┤14   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_15: ───────────░─┤15    ├┤15    ├┤15   ├┤15     ├┤15     ├─░──────┤15   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_16: ┤ X ├──────░─┤16    ├┤16    ├┤16   ├┤16     ├┤16     ├─░─┤ X ├┤16   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_17: ┤ X ├──────░─┤17    ├┤17    ├┤17   ├┤17     ├┤17     ├─░─┤ X ├┤17   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_18: ┤ X ├──────░─┤18    ├┤18    ├┤18   ├┤18     ├┤18     ├─░─┤ X ├┤18   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_19: ┤ X ├──────░─┤19    ├┤19    ├┤19   ├┤19     ├┤19     ├─░─┤ X ├┤19   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_20: ───────────░─┤20    ├┤20    ├┤20   ├┤20     ├┤20     ├─░──────┤20   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_21: ───────────░─┤21    ├┤21    ├┤21   ├┤21     ├┤21     ├─░──────┤21   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_22: ┤ X ├──────░─┤22    ├┤22    ├┤22   ├┤22     ├┤22     ├─░─┤ X ├┤22   ├──────\n",
       "      └───┘      ░ │   IF ││      ││     ││       ││   IF† │ ░ └───┘│     │      \n",
       "q_23: ───────────░─┤23    ├┤23    ├┤23   ├┤23     ├┤23     ├─░──────┤23   ├──────\n",
       "                 ░ │      ││   R1 ││     ││   R1† ││       │ ░      │   G │      \n",
       "q_24: ───────────░─┤24    ├┤24    ├┤24 O ├┤24     ├┤24     ├─░──────┤24   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_25: ┤ X ├──────░─┤25    ├┤25    ├┤25   ├┤25     ├┤25     ├─░─┤ X ├┤25   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_26: ┤ X ├──────░─┤26    ├┤26    ├┤26   ├┤26     ├┤26     ├─░─┤ X ├┤26   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_27: ───────────░─┤27    ├┤27    ├┤27   ├┤27     ├┤27     ├─░──────┤27   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_28: ───────────░─┤28    ├┤28    ├┤28   ├┤28     ├┤28     ├─░──────┤28   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_29: ───────────░─┤29    ├┤29    ├┤29   ├┤29     ├┤29     ├─░──────┤29   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_30: ───────────░─┤30    ├┤30    ├┤30   ├┤30     ├┤30     ├─░──────┤30   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_31: ┤ X ├──────░─┤31    ├┤31    ├┤31   ├┤31     ├┤31     ├─░─┤ X ├┤31   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_32: ┤ X ├──────░─┤32    ├┤32    ├┤32   ├┤32     ├┤32     ├─░─┤ X ├┤32   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_33: ┤ X ├──────░─┤33    ├┤33    ├┤33   ├┤33     ├┤33     ├─░─┤ X ├┤33   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_34: ┤ X ├──────░─┤34    ├┤34    ├┤34   ├┤34     ├┤34     ├─░─┤ X ├┤34   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_35: ───────────░─┤35    ├┤35    ├┤35   ├┤35     ├┤35     ├─░──────┤35   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_36: ───────────░─┤36    ├┤36    ├┤36   ├┤36     ├┤36     ├─░──────┤36   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_37: ───────────░─┤37    ├┤37    ├┤37   ├┤37     ├┤37     ├─░──────┤37   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_38: ───────────░─┤38    ├┤38    ├┤38   ├┤38     ├┤38     ├─░──────┤38   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_39: ───────────░─┤39    ├┤39    ├┤39   ├┤39     ├┤39     ├─░──────┤39   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_40: ───────────░─┤40    ├┤40    ├┤40   ├┤40     ├┤40     ├─░──────┤40   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_41: ───────────░─┤41    ├┤41    ├┤41   ├┤41     ├┤41     ├─░──────┤41   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_42: ───────────░─┤42    ├┤42    ├┤42   ├┤42     ├┤42     ├─░──────┤42   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_43: ───────────░─┤43    ├┤43    ├┤43   ├┤43     ├┤43     ├─░──────┤43   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_44: ───────────░─┤44    ├┤44    ├┤44   ├┤44     ├┤44     ├─░──────┤44   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_45: ───────────░─┤45    ├┤45    ├┤45   ├┤45     ├┤45     ├─░──────┤45   ├──────\n",
       "      ┌───┐      ░ └──────┘│      ││     ││       │└───────┘ ░      │     │┌─┐   \n",
       "q_46: ┤ H ├──────░─────────┤46    ├┤46   ├┤46     ├──────────░──────┤46   ├┤M├───\n",
       "      ├───┤      ░         │      ││     ││       │          ░      │     │└╥┘┌─┐\n",
       "q_47: ┤ H ├──────░─────────┤47    ├┤47   ├┤47     ├──────────░──────┤47   ├─╫─┤M├\n",
       "      ├───┤┌───┐ ░         └──────┘│     │└───────┘          ░ ┌───┐└─────┘ ║ └╥┘\n",
       "q_48: ┤ X ├┤ H ├─░─────────────────┤48   ├───────────────────░─┤ H ├────────╫──╫─\n",
       "      └───┘└───┘ ░                 └─────┘                   ░ └───┘        ║  ║ \n",
       " c: 2/══════════════════════════════════════════════════════════════════════╩══╩═\n",
       "                                                                            0  1 </pre>"
      ],
      "text/plain": [
       "      ┌───┐      ░ ┌──────┐┌──────┐┌─────┐┌───────┐┌───────┐ ░ ┌───┐┌─────┐      \n",
       " q_0: ┤ X ├──────░─┤0     ├┤0     ├┤0    ├┤0      ├┤0      ├─░─┤ X ├┤0    ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       " q_1: ┤ X ├──────░─┤1     ├┤1     ├┤1    ├┤1      ├┤1      ├─░─┤ X ├┤1    ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       " q_2: ───────────░─┤2     ├┤2     ├┤2    ├┤2      ├┤2      ├─░──────┤2    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       " q_3: ───────────░─┤3     ├┤3     ├┤3    ├┤3      ├┤3      ├─░──────┤3    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       " q_4: ───────────░─┤4     ├┤4     ├┤4    ├┤4      ├┤4      ├─░──────┤4    ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       " q_5: ┤ X ├──────░─┤5     ├┤5     ├┤5    ├┤5      ├┤5      ├─░─┤ X ├┤5    ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       " q_6: ───────────░─┤6     ├┤6     ├┤6    ├┤6      ├┤6      ├─░──────┤6    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       " q_7: ───────────░─┤7     ├┤7     ├┤7    ├┤7      ├┤7      ├─░──────┤7    ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       " q_8: ┤ X ├──────░─┤8     ├┤8     ├┤8    ├┤8      ├┤8      ├─░─┤ X ├┤8    ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       " q_9: ───────────░─┤9     ├┤9     ├┤9    ├┤9      ├┤9      ├─░──────┤9    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_10: ───────────░─┤10    ├┤10    ├┤10   ├┤10     ├┤10     ├─░──────┤10   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_11: ───────────░─┤11    ├┤11    ├┤11   ├┤11     ├┤11     ├─░──────┤11   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_12: ┤ X ├──────░─┤12    ├┤12    ├┤12   ├┤12     ├┤12     ├─░─┤ X ├┤12   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_13: ───────────░─┤13    ├┤13    ├┤13   ├┤13     ├┤13     ├─░──────┤13   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_14: ┤ X ├──────░─┤14    ├┤14    ├┤14   ├┤14     ├┤14     ├─░─┤ X ├┤14   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_15: ───────────░─┤15    ├┤15    ├┤15   ├┤15     ├┤15     ├─░──────┤15   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_16: ┤ X ├──────░─┤16    ├┤16    ├┤16   ├┤16     ├┤16     ├─░─┤ X ├┤16   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_17: ┤ X ├──────░─┤17    ├┤17    ├┤17   ├┤17     ├┤17     ├─░─┤ X ├┤17   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_18: ┤ X ├──────░─┤18    ├┤18    ├┤18   ├┤18     ├┤18     ├─░─┤ X ├┤18   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_19: ┤ X ├──────░─┤19    ├┤19    ├┤19   ├┤19     ├┤19     ├─░─┤ X ├┤19   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_20: ───────────░─┤20    ├┤20    ├┤20   ├┤20     ├┤20     ├─░──────┤20   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_21: ───────────░─┤21    ├┤21    ├┤21   ├┤21     ├┤21     ├─░──────┤21   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_22: ┤ X ├──────░─┤22    ├┤22    ├┤22   ├┤22     ├┤22     ├─░─┤ X ├┤22   ├──────\n",
       "      └───┘      ░ │   IF ││      ││     ││       ││   IF† │ ░ └───┘│     │      \n",
       "q_23: ───────────░─┤23    ├┤23    ├┤23   ├┤23     ├┤23     ├─░──────┤23   ├──────\n",
       "                 ░ │      ││   R1 ││     ││   R1† ││       │ ░      │   G │      \n",
       "q_24: ───────────░─┤24    ├┤24    ├┤24 O ├┤24     ├┤24     ├─░──────┤24   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_25: ┤ X ├──────░─┤25    ├┤25    ├┤25   ├┤25     ├┤25     ├─░─┤ X ├┤25   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_26: ┤ X ├──────░─┤26    ├┤26    ├┤26   ├┤26     ├┤26     ├─░─┤ X ├┤26   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_27: ───────────░─┤27    ├┤27    ├┤27   ├┤27     ├┤27     ├─░──────┤27   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_28: ───────────░─┤28    ├┤28    ├┤28   ├┤28     ├┤28     ├─░──────┤28   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_29: ───────────░─┤29    ├┤29    ├┤29   ├┤29     ├┤29     ├─░──────┤29   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_30: ───────────░─┤30    ├┤30    ├┤30   ├┤30     ├┤30     ├─░──────┤30   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_31: ┤ X ├──────░─┤31    ├┤31    ├┤31   ├┤31     ├┤31     ├─░─┤ X ├┤31   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_32: ┤ X ├──────░─┤32    ├┤32    ├┤32   ├┤32     ├┤32     ├─░─┤ X ├┤32   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_33: ┤ X ├──────░─┤33    ├┤33    ├┤33   ├┤33     ├┤33     ├─░─┤ X ├┤33   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_34: ┤ X ├──────░─┤34    ├┤34    ├┤34   ├┤34     ├┤34     ├─░─┤ X ├┤34   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_35: ───────────░─┤35    ├┤35    ├┤35   ├┤35     ├┤35     ├─░──────┤35   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_36: ───────────░─┤36    ├┤36    ├┤36   ├┤36     ├┤36     ├─░──────┤36   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_37: ───────────░─┤37    ├┤37    ├┤37   ├┤37     ├┤37     ├─░──────┤37   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_38: ───────────░─┤38    ├┤38    ├┤38   ├┤38     ├┤38     ├─░──────┤38   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_39: ───────────░─┤39    ├┤39    ├┤39   ├┤39     ├┤39     ├─░──────┤39   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_40: ───────────░─┤40    ├┤40    ├┤40   ├┤40     ├┤40     ├─░──────┤40   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_41: ───────────░─┤41    ├┤41    ├┤41   ├┤41     ├┤41     ├─░──────┤41   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_42: ───────────░─┤42    ├┤42    ├┤42   ├┤42     ├┤42     ├─░──────┤42   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_43: ───────────░─┤43    ├┤43    ├┤43   ├┤43     ├┤43     ├─░──────┤43   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_44: ───────────░─┤44    ├┤44    ├┤44   ├┤44     ├┤44     ├─░──────┤44   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_45: ───────────░─┤45    ├┤45    ├┤45   ├┤45     ├┤45     ├─░──────┤45   ├──────\n",
       "      ┌───┐      ░ └──────┘│      ││     ││       │└───────┘ ░      │     │┌─┐   \n",
       "q_46: ┤ H ├──────░─────────┤46    ├┤46   ├┤46     ├──────────░──────┤46   ├┤M├───\n",
       "      ├───┤      ░         │      ││     ││       │          ░      │     │└╥┘┌─┐\n",
       "q_47: ┤ H ├──────░─────────┤47    ├┤47   ├┤47     ├──────────░──────┤47   ├─╫─┤M├\n",
       "      ├───┤┌───┐ ░         └──────┘│     │└───────┘          ░ ┌───┐└─────┘ ║ └╥┘\n",
       "q_48: ┤ X ├┤ H ├─░─────────────────┤48   ├───────────────────░─┤ H ├────────╫──╫─\n",
       "      └───┘└───┘ ░                 └─────┘                   ░ └───┘        ║  ║ \n",
       " c: 2/══════════════════════════════════════════════════════════════════════╩══╩═\n",
       "                                                                            0  1 "
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qc = QuantumCircuit(49,2)\n",
    "\n",
    "#State Preparation 0-35\n",
    "\n",
    "#N3\n",
    "qc.x(0)\n",
    "qc.x(1)\n",
    "#2\n",
    "#3\n",
    "\n",
    "#N2\n",
    "#4\n",
    "qc.x(5)\n",
    "#6\n",
    "#7\n",
    "\n",
    "#N1\n",
    "qc.x(8)\n",
    "#9\n",
    "#10\n",
    "#11\n",
    "\n",
    "#N5\n",
    "qc.x(12)\n",
    "#13\n",
    "qc.x(14)\n",
    "#15\n",
    "\n",
    "#NX\n",
    "qc.x(16)\n",
    "qc.x(17)\n",
    "qc.x(18)\n",
    "qc.x(19)\n",
    "\n",
    "#N4\n",
    "#20\n",
    "#21\n",
    "qc.x(22)\n",
    "#23\n",
    "\n",
    "#N6\n",
    "#24\n",
    "qc.x(25)\n",
    "qc.x(26)\n",
    "#27\n",
    "\n",
    "#N8\n",
    "#28\n",
    "#29\n",
    "#30\n",
    "qc.x(31)\n",
    "\n",
    "#N7\n",
    "qc.x(32)\n",
    "qc.x(33)\n",
    "qc.x(34)\n",
    "#35\n",
    "\n",
    "\n",
    "\n",
    "#Working Memory, Flag 36\n",
    "\n",
    "#1St Trace 37-45\n",
    "\n",
    "#1St Descriptor in superposition 46, 47\n",
    "\n",
    "qc.h(46)\n",
    "qc.h(47)\n",
    "\n",
    "#Preparation of Aux\n",
    "qc.x(48)\n",
    "qc.h(48)\n",
    "\n",
    "\n",
    "qc.barrier()\n",
    "\n",
    "\n",
    "\n",
    "qc.append(if_rules(),range(46))\n",
    "qc.append(rules1(),range(48))\n",
    "#Oracle\n",
    "qc.append(oracle(),range(49))\n",
    "\n",
    "qc.append(rules1_inv(),range(48))\n",
    "qc.append(if_rules_inv(),range(46))\n",
    "\n",
    "#Clear Input\n",
    "\n",
    "qc.barrier()\n",
    "\n",
    "#State Preparation 0-35\n",
    "\n",
    "#N3\n",
    "qc.x(0)\n",
    "qc.x(1)\n",
    "#2\n",
    "#3\n",
    "\n",
    "#N2\n",
    "#4\n",
    "qc.x(5)\n",
    "#6\n",
    "#7\n",
    "\n",
    "#N1\n",
    "qc.x(8)\n",
    "#9\n",
    "#10\n",
    "#11\n",
    "\n",
    "#N5\n",
    "qc.x(12)\n",
    "#13\n",
    "qc.x(14)\n",
    "#15\n",
    "\n",
    "#NX\n",
    "qc.x(16)\n",
    "qc.x(17)\n",
    "qc.x(18)\n",
    "qc.x(19)\n",
    "\n",
    "#N4\n",
    "#20\n",
    "#21\n",
    "qc.x(22)\n",
    "#23\n",
    "\n",
    "#N6\n",
    "#24\n",
    "qc.x(25)\n",
    "qc.x(26)\n",
    "#27\n",
    "\n",
    "#N8\n",
    "#28\n",
    "#29\n",
    "#30\n",
    "qc.x(31)\n",
    "\n",
    "#N7\n",
    "qc.x(32)\n",
    "qc.x(33)\n",
    "qc.x(34)\n",
    "#35\n",
    "\n",
    "\n",
    "\n",
    "qc.h(48)\n",
    "\n",
    "qc.append(Grover(),range(48))\n",
    "\n",
    "qc.measure(46,0)\n",
    "qc.measure(47,1)\n",
    "\n",
    "\n",
    "qc.draw(fold=140)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The quantum circuit for the 9-puzzle task of the depth search 1. The empty space is in the center and the desired state  is represented by the empty space at top.  Simulating $49$ qubits requires higher memory capacity, we cannot use the statevector simulator or a search depth of two due to memory constraints. The path descriptor has four possible states represented by two qubits and is measured  after  applying the function $rules1$, which corresponds in moving the empty space up."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Total count are: {'01': 1000}\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simulator = Aer.get_backend('aer_simulator')\n",
    "compiled_circuit = transpile(qc, simulator)\n",
    "job = simulator.run(compiled_circuit, shots=1000)\n",
    "result = job.result()\n",
    "counts = result.get_counts(compiled_circuit)\n",
    "print(\"\\nTotal count are:\",counts)\n",
    "plot_histogram(counts)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we change the desired state in the oracle which will marked two state (out of four states...)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [],
   "source": [
    "def oracle():\n",
    "    qc = QuantumCircuit(49)\n",
    "    gate = MCXGate(4)\n",
    "     \n",
    "    #The Position of the Blank, three possible movements\n",
    "     \n",
    "    #qc.append(gate, [24, 25, 26, 27, 48])\n",
    "    #qc.append(gate, [0, 1, 2, 3, 48])\n",
    "    qc.append(gate, [16, 17, 18, 19, 48])\n",
    "    \n",
    "    qc.name=\"O\"\n",
    "    return qc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and apply applying the function $rules2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "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",
       " q_0: ┤ X ├──────░─┤0     ├┤0     ├┤0    ├┤0      ├┤0      ├─░─┤ X ├┤0    ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       " q_1: ┤ X ├──────░─┤1     ├┤1     ├┤1    ├┤1      ├┤1      ├─░─┤ X ├┤1    ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       " q_2: ───────────░─┤2     ├┤2     ├┤2    ├┤2      ├┤2      ├─░──────┤2    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       " q_3: ───────────░─┤3     ├┤3     ├┤3    ├┤3      ├┤3      ├─░──────┤3    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       " q_4: ───────────░─┤4     ├┤4     ├┤4    ├┤4      ├┤4      ├─░──────┤4    ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       " q_5: ┤ X ├──────░─┤5     ├┤5     ├┤5    ├┤5      ├┤5      ├─░─┤ X ├┤5    ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       " q_6: ───────────░─┤6     ├┤6     ├┤6    ├┤6      ├┤6      ├─░──────┤6    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       " q_7: ───────────░─┤7     ├┤7     ├┤7    ├┤7      ├┤7      ├─░──────┤7    ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       " q_8: ┤ X ├──────░─┤8     ├┤8     ├┤8    ├┤8      ├┤8      ├─░─┤ X ├┤8    ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       " q_9: ───────────░─┤9     ├┤9     ├┤9    ├┤9      ├┤9      ├─░──────┤9    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_10: ───────────░─┤10    ├┤10    ├┤10   ├┤10     ├┤10     ├─░──────┤10   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_11: ───────────░─┤11    ├┤11    ├┤11   ├┤11     ├┤11     ├─░──────┤11   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_12: ┤ X ├──────░─┤12    ├┤12    ├┤12   ├┤12     ├┤12     ├─░─┤ X ├┤12   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_13: ┤ X ├──────░─┤13    ├┤13    ├┤13   ├┤13     ├┤13     ├─░─┤ X ├┤13   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_14: ┤ X ├──────░─┤14    ├┤14    ├┤14   ├┤14     ├┤14     ├─░─┤ X ├┤14   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_15: ┤ X ├──────░─┤15    ├┤15    ├┤15   ├┤15     ├┤15     ├─░─┤ X ├┤15   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_16: ┤ X ├──────░─┤16    ├┤16    ├┤16   ├┤16     ├┤16     ├─░─┤ X ├┤16   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_17: ───────────░─┤17    ├┤17    ├┤17   ├┤17     ├┤17     ├─░──────┤17   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_18: ┤ X ├──────░─┤18    ├┤18    ├┤18   ├┤18     ├┤18     ├─░─┤ X ├┤18   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_19: ───────────░─┤19    ├┤19    ├┤19   ├┤19     ├┤19     ├─░──────┤19   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_20: ───────────░─┤20    ├┤20    ├┤20   ├┤20     ├┤20     ├─░──────┤20   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_21: ───────────░─┤21    ├┤21    ├┤21   ├┤21     ├┤21     ├─░──────┤21   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_22: ┤ X ├──────░─┤22    ├┤22    ├┤22   ├┤22     ├┤22     ├─░─┤ X ├┤22   ├──────\n",
       "      └───┘      ░ │   IF ││      ││     ││       ││   IF† │ ░ └───┘│     │      \n",
       "q_23: ───────────░─┤23    ├┤23    ├┤23   ├┤23     ├┤23     ├─░──────┤23   ├──────\n",
       "                 ░ │      ││   R2 ││     ││   R2† ││       │ ░      │   G │      \n",
       "q_24: ───────────░─┤24    ├┤24    ├┤24 O ├┤24     ├┤24     ├─░──────┤24   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_25: ┤ X ├──────░─┤25    ├┤25    ├┤25   ├┤25     ├┤25     ├─░─┤ X ├┤25   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_26: ┤ X ├──────░─┤26    ├┤26    ├┤26   ├┤26     ├┤26     ├─░─┤ X ├┤26   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_27: ───────────░─┤27    ├┤27    ├┤27   ├┤27     ├┤27     ├─░──────┤27   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_28: ───────────░─┤28    ├┤28    ├┤28   ├┤28     ├┤28     ├─░──────┤28   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_29: ───────────░─┤29    ├┤29    ├┤29   ├┤29     ├┤29     ├─░──────┤29   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_30: ───────────░─┤30    ├┤30    ├┤30   ├┤30     ├┤30     ├─░──────┤30   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_31: ┤ X ├──────░─┤31    ├┤31    ├┤31   ├┤31     ├┤31     ├─░─┤ X ├┤31   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_32: ┤ X ├──────░─┤32    ├┤32    ├┤32   ├┤32     ├┤32     ├─░─┤ X ├┤32   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_33: ┤ X ├──────░─┤33    ├┤33    ├┤33   ├┤33     ├┤33     ├─░─┤ X ├┤33   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_34: ┤ X ├──────░─┤34    ├┤34    ├┤34   ├┤34     ├┤34     ├─░─┤ X ├┤34   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_35: ───────────░─┤35    ├┤35    ├┤35   ├┤35     ├┤35     ├─░──────┤35   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_36: ───────────░─┤36    ├┤36    ├┤36   ├┤36     ├┤36     ├─░──────┤36   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_37: ───────────░─┤37    ├┤37    ├┤37   ├┤37     ├┤37     ├─░──────┤37   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_38: ───────────░─┤38    ├┤38    ├┤38   ├┤38     ├┤38     ├─░──────┤38   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_39: ───────────░─┤39    ├┤39    ├┤39   ├┤39     ├┤39     ├─░──────┤39   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_40: ───────────░─┤40    ├┤40    ├┤40   ├┤40     ├┤40     ├─░──────┤40   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_41: ───────────░─┤41    ├┤41    ├┤41   ├┤41     ├┤41     ├─░──────┤41   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_42: ───────────░─┤42    ├┤42    ├┤42   ├┤42     ├┤42     ├─░──────┤42   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_43: ───────────░─┤43    ├┤43    ├┤43   ├┤43     ├┤43     ├─░──────┤43   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_44: ───────────░─┤44    ├┤44    ├┤44   ├┤44     ├┤44     ├─░──────┤44   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_45: ───────────░─┤45    ├┤45    ├┤45   ├┤45     ├┤45     ├─░──────┤45   ├──────\n",
       "      ┌───┐      ░ └──────┘│      ││     ││       │└───────┘ ░      │     │┌─┐   \n",
       "q_46: ┤ H ├──────░─────────┤46    ├┤46   ├┤46     ├──────────░──────┤46   ├┤M├───\n",
       "      ├───┤      ░         │      ││     ││       │          ░      │     │└╥┘┌─┐\n",
       "q_47: ┤ H ├──────░─────────┤47    ├┤47   ├┤47     ├──────────░──────┤47   ├─╫─┤M├\n",
       "      ├───┤┌───┐ ░         └──────┘│     │└───────┘          ░ ┌───┐└─────┘ ║ └╥┘\n",
       "q_48: ┤ X ├┤ H ├─░─────────────────┤48   ├───────────────────░─┤ H ├────────╫──╫─\n",
       "      └───┘└───┘ ░                 └─────┘                   ░ └───┘        ║  ║ \n",
       " c: 2/══════════════════════════════════════════════════════════════════════╩══╩═\n",
       "                                                                            0  1 </pre>"
      ],
      "text/plain": [
       "      ┌───┐      ░ ┌──────┐┌──────┐┌─────┐┌───────┐┌───────┐ ░ ┌───┐┌─────┐      \n",
       " q_0: ┤ X ├──────░─┤0     ├┤0     ├┤0    ├┤0      ├┤0      ├─░─┤ X ├┤0    ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       " q_1: ┤ X ├──────░─┤1     ├┤1     ├┤1    ├┤1      ├┤1      ├─░─┤ X ├┤1    ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       " q_2: ───────────░─┤2     ├┤2     ├┤2    ├┤2      ├┤2      ├─░──────┤2    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       " q_3: ───────────░─┤3     ├┤3     ├┤3    ├┤3      ├┤3      ├─░──────┤3    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       " q_4: ───────────░─┤4     ├┤4     ├┤4    ├┤4      ├┤4      ├─░──────┤4    ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       " q_5: ┤ X ├──────░─┤5     ├┤5     ├┤5    ├┤5      ├┤5      ├─░─┤ X ├┤5    ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       " q_6: ───────────░─┤6     ├┤6     ├┤6    ├┤6      ├┤6      ├─░──────┤6    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       " q_7: ───────────░─┤7     ├┤7     ├┤7    ├┤7      ├┤7      ├─░──────┤7    ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       " q_8: ┤ X ├──────░─┤8     ├┤8     ├┤8    ├┤8      ├┤8      ├─░─┤ X ├┤8    ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       " q_9: ───────────░─┤9     ├┤9     ├┤9    ├┤9      ├┤9      ├─░──────┤9    ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_10: ───────────░─┤10    ├┤10    ├┤10   ├┤10     ├┤10     ├─░──────┤10   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_11: ───────────░─┤11    ├┤11    ├┤11   ├┤11     ├┤11     ├─░──────┤11   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_12: ┤ X ├──────░─┤12    ├┤12    ├┤12   ├┤12     ├┤12     ├─░─┤ X ├┤12   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_13: ┤ X ├──────░─┤13    ├┤13    ├┤13   ├┤13     ├┤13     ├─░─┤ X ├┤13   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_14: ┤ X ├──────░─┤14    ├┤14    ├┤14   ├┤14     ├┤14     ├─░─┤ X ├┤14   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_15: ┤ X ├──────░─┤15    ├┤15    ├┤15   ├┤15     ├┤15     ├─░─┤ X ├┤15   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_16: ┤ X ├──────░─┤16    ├┤16    ├┤16   ├┤16     ├┤16     ├─░─┤ X ├┤16   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_17: ───────────░─┤17    ├┤17    ├┤17   ├┤17     ├┤17     ├─░──────┤17   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_18: ┤ X ├──────░─┤18    ├┤18    ├┤18   ├┤18     ├┤18     ├─░─┤ X ├┤18   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_19: ───────────░─┤19    ├┤19    ├┤19   ├┤19     ├┤19     ├─░──────┤19   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_20: ───────────░─┤20    ├┤20    ├┤20   ├┤20     ├┤20     ├─░──────┤20   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_21: ───────────░─┤21    ├┤21    ├┤21   ├┤21     ├┤21     ├─░──────┤21   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_22: ┤ X ├──────░─┤22    ├┤22    ├┤22   ├┤22     ├┤22     ├─░─┤ X ├┤22   ├──────\n",
       "      └───┘      ░ │   IF ││      ││     ││       ││   IF† │ ░ └───┘│     │      \n",
       "q_23: ───────────░─┤23    ├┤23    ├┤23   ├┤23     ├┤23     ├─░──────┤23   ├──────\n",
       "                 ░ │      ││   R2 ││     ││   R2† ││       │ ░      │   G │      \n",
       "q_24: ───────────░─┤24    ├┤24    ├┤24 O ├┤24     ├┤24     ├─░──────┤24   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_25: ┤ X ├──────░─┤25    ├┤25    ├┤25   ├┤25     ├┤25     ├─░─┤ X ├┤25   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_26: ┤ X ├──────░─┤26    ├┤26    ├┤26   ├┤26     ├┤26     ├─░─┤ X ├┤26   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_27: ───────────░─┤27    ├┤27    ├┤27   ├┤27     ├┤27     ├─░──────┤27   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_28: ───────────░─┤28    ├┤28    ├┤28   ├┤28     ├┤28     ├─░──────┤28   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_29: ───────────░─┤29    ├┤29    ├┤29   ├┤29     ├┤29     ├─░──────┤29   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_30: ───────────░─┤30    ├┤30    ├┤30   ├┤30     ├┤30     ├─░──────┤30   ├──────\n",
       "      ┌───┐      ░ │      ││      ││     ││       ││       │ ░ ┌───┐│     │      \n",
       "q_31: ┤ X ├──────░─┤31    ├┤31    ├┤31   ├┤31     ├┤31     ├─░─┤ X ├┤31   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_32: ┤ X ├──────░─┤32    ├┤32    ├┤32   ├┤32     ├┤32     ├─░─┤ X ├┤32   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_33: ┤ X ├──────░─┤33    ├┤33    ├┤33   ├┤33     ├┤33     ├─░─┤ X ├┤33   ├──────\n",
       "      ├───┤      ░ │      ││      ││     ││       ││       │ ░ ├───┤│     │      \n",
       "q_34: ┤ X ├──────░─┤34    ├┤34    ├┤34   ├┤34     ├┤34     ├─░─┤ X ├┤34   ├──────\n",
       "      └───┘      ░ │      ││      ││     ││       ││       │ ░ └───┘│     │      \n",
       "q_35: ───────────░─┤35    ├┤35    ├┤35   ├┤35     ├┤35     ├─░──────┤35   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_36: ───────────░─┤36    ├┤36    ├┤36   ├┤36     ├┤36     ├─░──────┤36   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_37: ───────────░─┤37    ├┤37    ├┤37   ├┤37     ├┤37     ├─░──────┤37   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_38: ───────────░─┤38    ├┤38    ├┤38   ├┤38     ├┤38     ├─░──────┤38   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_39: ───────────░─┤39    ├┤39    ├┤39   ├┤39     ├┤39     ├─░──────┤39   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_40: ───────────░─┤40    ├┤40    ├┤40   ├┤40     ├┤40     ├─░──────┤40   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_41: ───────────░─┤41    ├┤41    ├┤41   ├┤41     ├┤41     ├─░──────┤41   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_42: ───────────░─┤42    ├┤42    ├┤42   ├┤42     ├┤42     ├─░──────┤42   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_43: ───────────░─┤43    ├┤43    ├┤43   ├┤43     ├┤43     ├─░──────┤43   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_44: ───────────░─┤44    ├┤44    ├┤44   ├┤44     ├┤44     ├─░──────┤44   ├──────\n",
       "                 ░ │      ││      ││     ││       ││       │ ░      │     │      \n",
       "q_45: ───────────░─┤45    ├┤45    ├┤45   ├┤45     ├┤45     ├─░──────┤45   ├──────\n",
       "      ┌───┐      ░ └──────┘│      ││     ││       │└───────┘ ░      │     │┌─┐   \n",
       "q_46: ┤ H ├──────░─────────┤46    ├┤46   ├┤46     ├──────────░──────┤46   ├┤M├───\n",
       "      ├───┤      ░         │      ││     ││       │          ░      │     │└╥┘┌─┐\n",
       "q_47: ┤ H ├──────░─────────┤47    ├┤47   ├┤47     ├──────────░──────┤47   ├─╫─┤M├\n",
       "      ├───┤┌───┐ ░         └──────┘│     │└───────┘          ░ ┌───┐└─────┘ ║ └╥┘\n",
       "q_48: ┤ X ├┤ H ├─░─────────────────┤48   ├───────────────────░─┤ H ├────────╫──╫─\n",
       "      └───┘└───┘ ░                 └─────┘                   ░ └───┘        ║  ║ \n",
       " c: 2/══════════════════════════════════════════════════════════════════════╩══╩═\n",
       "                                                                            0  1 "
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qc = QuantumCircuit(49,2)\n",
    "\n",
    "#State Preparation 0-35\n",
    "\n",
    "#N3\n",
    "qc.x(0)\n",
    "qc.x(1)\n",
    "#2\n",
    "#3\n",
    "\n",
    "#N2\n",
    "#4\n",
    "qc.x(5)\n",
    "#6\n",
    "#7\n",
    "\n",
    "#N1\n",
    "qc.x(8)\n",
    "#9\n",
    "#10\n",
    "#11\n",
    "\n",
    "#N5\n",
    "qc.x(12)\n",
    "qc.x(13)\n",
    "qc.x(14)\n",
    "qc.x(15)\n",
    "\n",
    "#NX\n",
    "qc.x(16)\n",
    "#qc.x(17)\n",
    "qc.x(18)\n",
    "#qc.x(19)\n",
    "\n",
    "#N4\n",
    "#20\n",
    "#21\n",
    "qc.x(22)\n",
    "#23\n",
    "\n",
    "#N6\n",
    "#24\n",
    "qc.x(25)\n",
    "qc.x(26)\n",
    "#27\n",
    "\n",
    "#N8\n",
    "#28\n",
    "#29\n",
    "#30\n",
    "qc.x(31)\n",
    "\n",
    "#N7\n",
    "qc.x(32)\n",
    "qc.x(33)\n",
    "qc.x(34)\n",
    "#35\n",
    "\n",
    "\n",
    "\n",
    "#Working Memory, Flag 36\n",
    "\n",
    "#1St Trace 37-45\n",
    "\n",
    "#1St Descriptor in superposition 46, 47\n",
    "\n",
    "qc.h(46)\n",
    "qc.h(47)\n",
    "\n",
    "#Preparation of Aux\n",
    "qc.x(48)\n",
    "qc.h(48)\n",
    "\n",
    "\n",
    "qc.barrier()\n",
    "\n",
    "\n",
    "\n",
    "qc.append(if_rules(),range(46))\n",
    "qc.append(rules2(),range(48))\n",
    "#Oracle\n",
    "qc.append(oracle(),range(49))\n",
    "qc.append(rules2_inv(),range(48))\n",
    "qc.append(if_rules_inv(),range(46))\n",
    "\n",
    "\n",
    "#Clear Input\n",
    "\n",
    "qc.barrier()\n",
    "\n",
    "#State Preparation 0-35 clear\n",
    "\n",
    "#N3\n",
    "qc.x(0)\n",
    "qc.x(1)\n",
    "#2\n",
    "#3\n",
    "\n",
    "#N2\n",
    "#4\n",
    "qc.x(5)\n",
    "#6\n",
    "#7\n",
    "\n",
    "#N1\n",
    "qc.x(8)\n",
    "#9\n",
    "#10\n",
    "#11\n",
    "\n",
    "#N5\n",
    "qc.x(12)\n",
    "qc.x(13)\n",
    "qc.x(14)\n",
    "qc.x(15)\n",
    "\n",
    "#NX\n",
    "qc.x(16)\n",
    "#qc.x(17)\n",
    "qc.x(18)\n",
    "#qc.x(19)\n",
    "\n",
    "#N4\n",
    "#20\n",
    "#21\n",
    "qc.x(22)\n",
    "#23\n",
    "\n",
    "#N6\n",
    "#24\n",
    "qc.x(25)\n",
    "qc.x(26)\n",
    "#27\n",
    "\n",
    "#N8\n",
    "#28\n",
    "#29\n",
    "#30\n",
    "qc.x(31)\n",
    "\n",
    "#N7\n",
    "qc.x(32)\n",
    "qc.x(33)\n",
    "qc.x(34)\n",
    "#35\n",
    "\n",
    "\n",
    "qc.h(48)\n",
    "\n",
    "qc.append(Grover(),range(48))\n",
    "\n",
    "qc.measure(46,0)\n",
    "qc.measure(47,1)\n",
    "\n",
    "\n",
    "qc.draw(fold=140)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Total count are: {'01': 257, '00': 253, '11': 240, '10': 250}\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simulator = Aer.get_backend('aer_simulator')\n",
    "compiled_circuit = transpile(qc, simulator)\n",
    "job = simulator.run(compiled_circuit, shots=1000)\n",
    "result = job.result()\n",
    "counts = result.get_counts(compiled_circuit)\n",
    "print(\"\\nTotal count are:\",counts)\n",
    "plot_histogram(counts)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The path descriptor is represented by two qubits and is measured  after  applying the function $rules2$. No solution is indicated, why? There are two marked solutions in four states, we cannot apply Grover's amplification! \n",
    "The marked solution cannot be bigger than the one fourth of the present states, which is present during a deeper search. \n",
    "In our example of four states ond two marked solutions we a guaranted 50% success if we measure. "
   ]
  }
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