controlled_gates.ipynb

{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Controlled Gates \n",
    "<br/>\n",
    "\n",
    "<center> \n",
    "\n",
    "Write a function controlled which takes a 2×2 matrix U representing a single qubit operator, and makes a 4×4 matrix which is a controlled variant of U, with the first argument being the control qubit.\n",
    "Write a Quil program to define a controlled-Y gate in this manner. Find the wavefunction when applying this gate to qubit 1 controlled by qubit 0.\n",
    "\n",
    "</center>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import math\n",
    "import numpy as np\n",
    "from pyquil.quil import Program\n",
    "from pyquil.api import QVMConnection\n",
    "from pyquil.gates import H, X\n",
    "\n",
    "qvm = QVMConnection()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### What is a controlled gate? \n",
    "<br/>\n",
    "Controlled gates act on 2 or more qubits, where one or more qubits act as a control for some operation.  So if the controlled qubit, $q_0$, is in state $\\lvert 1 \\rangle$, it preforms the operation to $q_1$.  Else do nothing.\n",
    "<br/>\n",
    "### How to create a controlled gate given gate U?\n",
    "<br/>\n",
    "Given U which looks as follows...\n",
    "<br/>\n",
    "\n",
    "$$\n",
    "U =  \\left(\\begin{array}{cc} \n",
    "u_{00} & u_{01}\\\\\n",
    "u_{10} & u_{11}\n",
    "\\end{array}\\right)\n",
    "$$\n",
    "<br/>\n",
    "\n",
    "The controlled version looks as follows...\n",
    "<br/>\n",
    "\n",
    "$$\n",
    "C(U) =  \\left(\\begin{array}{cc} \n",
    "1 & 0 & 0 & 0\\\\\n",
    "0 & 1 & 0 & 0 \\\\\n",
    "0 & 0 & u_{00} & u_{01}\\\\\n",
    "0 & 0 & u_{10} & u_{11}\n",
    "\\end{array}\\right)\n",
    "$$\n",
    "<br/>\n",
    "\n",
    "This can be shown using kronecker products which can be found here https://en.wikipedia.org/wiki/Quantum_logic_gate#Controlled_(cX_cY_cZ)_gates\n",
    "but for simplistic purposes will only show the end result.\n",
    "Now we have all we need to create the function to make a controlled U gate!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def make_u_gate(u00, u01, u10, u11):\n",
    "    \n",
    "    #makes numPY array\n",
    "    U = np.array(([u00, u01], \n",
    "                  [u10, u11]))\n",
    "    return U;\n",
    "\n",
    "def gates(gate_name):\n",
    "    \n",
    "    if(gate_name == \"CX\"):\n",
    "        u_gate = make_u_gate(0.0, 1.0, 1.0, 0.0)\n",
    "        \n",
    "    elif(gate_name == \"CY\"):\n",
    "        u_gate = make_u_gate(0.0, -1.0j, 1.0j, 0.0)\n",
    "        \n",
    "    elif(gate_name == \"CZ\"):\n",
    "        u_gate = make_u_gate(1.0, 0.0, 0.0, -1.0)\n",
    "        \n",
    "    else:\n",
    "        print(\"Must be a unitary matrix\")\n",
    "        u00 = float(input(\"u00: \"))\n",
    "        u01 = float(input(\"u01: \"))\n",
    "        u10 = float(input(\"u10: \"))\n",
    "        u11 = float(input(\"u11: \"))\n",
    "        u_gate = make_u_gate(u00, u01, u10, u11)\n",
    "        \n",
    "    return u_gate\n",
    "\n",
    "def controlled(U):\n",
    "    \n",
    "    #create cU array\n",
    "    cU = np.array(([1.0, 0.0, 0.0, 0.0], \n",
    "                   [0.0, 1.0, 0.0, 0.0],\n",
    "                   [0.0, 0.0, U[0,0], U[0,1]],\n",
    "                   [0.0, 0.0, U[1,0], U[1,1]]))\n",
    "    \n",
    "    #define the gate\n",
    "    p = Program().defgate(\"c-u\", cU)\n",
    "    \n",
    "    #put the control, q0, in 1's state \n",
    "    #you can change this to see what results you recieve for controlled gates \n",
    "    p.inst(X(0))\n",
    "   \n",
    "    #apply C(U)\n",
    "    p.inst((\"c-u\", 0, 1))\n",
    "    #calculate and print wavefucntion\n",
    "    wavefunction = qvm.wavefunction(p)\n",
    "    print(\"Wavefunction: \", wavefunction)\n",
    "    return wavefunction.plot()    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Gate name(CX, CY, CZ, other): CY\n",
      "Wavefunction:  1j|11>\n"
     ]
    },
    {
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gate_name = input(\"Gate name(CX, CY, CZ, other): \")\n",
    "U = gates(gate_name)\n",
    "controlled(U)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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