samples/QSharp/Chapter3-samples.ipynb

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    "![(book cover)](https://covers.oreillystatic.com/images/0636920167433/cat.gif \"(book cover)\")\n",
    "# \"Programming Quantum Computers\" by O'Reilly Media -  [book info](http://shop.oreilly.com/product/0636920167433.do)  - [all code samples](https://oreilly-qc.github.io)\n",
    "\n",
    "## Code samples for Chapter 3\n",
    "These code samples were written by Mariia Mykhailova."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 3-1: Creating a multi-qubit state that can be expressed in terms of its qubits"
   ]
  },
  {
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       "<ul><li>SeparableState</li></ul>"
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       "SeparableState"
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   "source": [
    "// Example 3-1: Creating a multi-qubit state that can be expressed in terms of its qubits\n",
    "\n",
    "// open namespace which defines diagnostic routines\n",
    "open Microsoft.Quantum.Diagnostics;\n",
    "\n",
    "operation SeparableState () : Unit {\n",
    "    // allocate the qubits\n",
    "    use (q1, q2, q3) = (Qubit(), Qubit(), Qubit());\n",
    "    // put each of the qubits q2 and q3 into superposition of 0 and 1\n",
    "    H(q2);\n",
    "    H(q3);\n",
    "\n",
    "    // output the wave function of the three-qubit state\n",
    "    DumpMachine();\n",
    "\n",
    "    // make sure the qubits are back to the 0 state\n",
    "    ResetAll([q1, q2, q3]);\n",
    "}"
   ]
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       "\r\n",
       "                    <table style=\"table-layout: fixed; width: 100%\">\r\n",
       "                        <thead>\r\n",
       "                            \r\n",
       "                        <tr>\r\n",
       "                            <th>Qubit IDs</th>\r\n",
       "                            <td span=\"3\">0, 1, 2</td>\r\n",
       "                        </tr>\r\n",
       "                    \r\n",
       "                            <tr>\r\n",
       "                                <th style=\"width: 20ch)\">Basis state (little endian)</th>\r\n",
       "                                <th style=\"width: 20ch\">Amplitude</th><th style=\"width: calc(100% - 26ch - 20ch)\">Meas. Pr.</th><th style=\"width: 6ch\">Phase</th>\r\n",
       "                            </tr>\r\n",
       "                        </thead>\r\n",
       "                        <tbody>\r\n",
       "                        \r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|0\\right\\rangle$</td>\r\n",
       "                                <td>$0.5000 + 0.0000 i$</td>\r\n",
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       "                                <td>$\\left|1\\right\\rangle$</td>\r\n",
       "                                <td>$0.0000 + 0.0000 i$</td>\r\n",
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       "                                <td>$\\left|2\\right\\rangle$</td>\r\n",
       "                                <td>$0.5000 + 0.0000 i$</td>\r\n",
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       "                                <td>$\\left|3\\right\\rangle$</td>\r\n",
       "                                <td>$0.0000 + 0.0000 i$</td>\r\n",
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       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|4\\right\\rangle$</td>\r\n",
       "                                <td>$0.5000 + 0.0000 i$</td>\r\n",
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       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|5\\right\\rangle$</td>\r\n",
       "                                <td>$0.0000 + 0.0000 i$</td>\r\n",
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       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|6\\right\\rangle$</td>\r\n",
       "                                <td>$0.5000 + 0.0000 i$</td>\r\n",
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       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|7\\right\\rangle$</td>\r\n",
       "                                <td>$0.0000 + 0.0000 i$</td>\r\n",
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      "text/plain": [
       "|0⟩\t0.5000000000000001 + 0𝑖\n",
       "|1⟩\t0 + 0𝑖\n",
       "|2⟩\t0.5000000000000001 + 0𝑖\n",
       "|3⟩\t0 + 0𝑖\n",
       "|4⟩\t0.5000000000000001 + 0𝑖\n",
       "|5⟩\t0 + 0𝑖\n",
       "|6⟩\t0.5000000000000001 + 0𝑖\n",
       "|7⟩\t0 + 0𝑖"
      ]
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       "()"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%simulate SeparableState"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 3-2: Make a Bell pair"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/json": "[\"PrepareAndMeasureBellPair\",\"PrepareMultipleBellPairs\"]",
      "text/html": [
       "<ul><li>PrepareAndMeasureBellPair</li><li>PrepareMultipleBellPairs</li></ul>"
      ],
      "text/plain": [
       "PrepareAndMeasureBellPair, PrepareMultipleBellPairs"
      ]
     },
     "execution_count": 3,
     "metadata": {},
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    }
   ],
   "source": [
    "// Example 3-2: Make a Bell pair\n",
    "\n",
    "operation PrepareAndMeasureBellPair () : Unit {\n",
    "    // allocate the qubits\n",
    "    use (a, b) = (Qubit(), Qubit());\n",
    "    // put qubit a in superposition\n",
    "    H(a);\n",
    "\n",
    "    // entangle qubits a and b\n",
    "    CNOT(a, b);\n",
    "\n",
    "    // measure both qubits and output the results\n",
    "    Message($\"Measurement results: {M(a)}, {M(b)}\");\n",
    "\n",
    "    // make sure the qubits are back to the 0 state\n",
    "    ResetAll([a, b]);\n",
    "}\n",
    "\n",
    "operation PrepareMultipleBellPairs () : Unit {\n",
    "    // repeat the experiment multiple times to observe the correlation between measurement results:\n",
    "    // the two bits will be random but always the same\n",
    "    for i in 1..10 {\n",
    "        PrepareAndMeasureBellPair();\n",
    "    }\n",
    "}"
   ]
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Measurement results: Zero, Zero\n",
      "Measurement results: One, One\n",
      "Measurement results: One, One\n",
      "Measurement results: Zero, Zero\n",
      "Measurement results: One, One\n",
      "Measurement results: Zero, Zero\n",
      "Measurement results: Zero, Zero\n",
      "Measurement results: Zero, Zero\n",
      "Measurement results: Zero, Zero\n",
      "Measurement results: Zero, Zero\n"
     ]
    },
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       "()"
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    }
   ],
   "source": [
    "%simulate PrepareMultipleBellPairs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 3-3: Phase kickback"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/json": "[\"PhaseKickback\"]",
      "text/html": [
       "<ul><li>PhaseKickback</li></ul>"
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       "PhaseKickback"
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   ],
   "source": [
    "// Example 3-3: Phase kickback\n",
    "\n",
    "// open namespace which defines diagnostic routines\n",
    "open Microsoft.Quantum.Diagnostics;\n",
    "\n",
    "operation PhaseKickback () : Unit {\n",
    "    // allocate two registers, control and target\n",
    "    use (reg1, reg2) = (Qubit[2], Qubit());\n",
    "    // put each qubit of the control register into superposition of 0 and 1\n",
    "    ApplyToEach(H, reg1);\n",
    "\n",
    "    // initialize the target qubit into 1 state\n",
    "    X(reg2);\n",
    "\n",
    "    // output the wave function of the three-qubit state BEFORE phase kickback\n",
    "    DumpMachine();\n",
    "\n",
    "    // apply phase rotations controlled on the first register\n",
    "    Controlled T(reg1[0..0], reg2);\n",
    "    Controlled S(reg1[1..1], reg2);\n",
    "\n",
    "    // output the wave function of the three-qubit state AFTER phase kickback\n",
    "    DumpMachine();\n",
    "\n",
    "    // make sure the qubits are back to the 0 state\n",
    "    ResetAll(reg1 + [reg2]);\n",
    "}"
   ]
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      "application/json": "{\"div_id\":\"dump-machine-div-c2039c8d-22a6-4cd4-9cd9-7ffc912681a7\",\"qubit_ids\":[0,1,2],\"n_qubits\":3,\"amplitudes\":[{\"Real\":0.0,\"Imaginary\":0.0,\"Magnitude\":0.0,\"Phase\":0.0},{\"Real\":0.0,\"Imaginary\":0.0,\"Magnitude\":0.0,\"Phase\":0.0},{\"Real\":0.0,\"Imaginary\":0.0,\"Magnitude\":0.0,\"Phase\":0.0},{\"Real\":0.0,\"Imaginary\":0.0,\"Magnitude\":0.0,\"Phase\":0.0},{\"Real\":0.5000000000000001,\"Imaginary\":0.0,\"Magnitude\":0.5000000000000001,\"Phase\":0.0},{\"Real\":0.5000000000000001,\"Imaginary\":0.0,\"Magnitude\":0.5000000000000001,\"Phase\":0.0},{\"Real\":0.5000000000000001,\"Imaginary\":0.0,\"Magnitude\":0.5000000000000001,\"Phase\":0.0},{\"Real\":0.5000000000000001,\"Imaginary\":0.0,\"Magnitude\":0.5000000000000001,\"Phase\":0.0}]}",
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       "\r\n",
       "                    <table style=\"table-layout: fixed; width: 100%\">\r\n",
       "                        <thead>\r\n",
       "                            \r\n",
       "                        <tr>\r\n",
       "                            <th>Qubit IDs</th>\r\n",
       "                            <td span=\"3\">0, 1, 2</td>\r\n",
       "                        </tr>\r\n",
       "                    \r\n",
       "                            <tr>\r\n",
       "                                <th style=\"width: 20ch)\">Basis state (little endian)</th>\r\n",
       "                                <th style=\"width: 20ch\">Amplitude</th><th style=\"width: calc(100% - 26ch - 20ch)\">Meas. Pr.</th><th style=\"width: 6ch\">Phase</th>\r\n",
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       "                        </thead>\r\n",
       "                        <tbody>\r\n",
       "                        \r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|0\\right\\rangle$</td>\r\n",
       "                                <td>$0.0000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"0\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-75ecefe5-80e0-43e7-8ad2-d0ed416df4ec\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 0;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-75ecefe5-80e0-43e7-8ad2-d0ed416df4ec\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|1\\right\\rangle$</td>\r\n",
       "                                <td>$0.0000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"0\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-f1d767b2-cf59-4372-b448-4b873bb7a154\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 0;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-f1d767b2-cf59-4372-b448-4b873bb7a154\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|2\\right\\rangle$</td>\r\n",
       "                                <td>$0.0000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"0\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-afeb6816-608a-4d09-8b76-7199a83f45e5\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 0;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-afeb6816-608a-4d09-8b76-7199a83f45e5\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|3\\right\\rangle$</td>\r\n",
       "                                <td>$0.0000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"0\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-70139933-7dec-4b65-ab56-a59ef87c8322\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 0;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-70139933-7dec-4b65-ab56-a59ef87c8322\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|4\\right\\rangle$</td>\r\n",
       "                                <td>$0.5000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-732833de-297e-4c08-b990-3c532748edb6\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-732833de-297e-4c08-b990-3c532748edb6\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|5\\right\\rangle$</td>\r\n",
       "                                <td>$0.5000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-97b57afb-98a0-4e78-84c3-3862f03a622e\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-97b57afb-98a0-4e78-84c3-3862f03a622e\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|6\\right\\rangle$</td>\r\n",
       "                                <td>$0.5000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-c8940ff9-d291-455d-8d1f-291c3a9fcad6\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-c8940ff9-d291-455d-8d1f-291c3a9fcad6\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|7\\right\\rangle$</td>\r\n",
       "                                <td>$0.5000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-224ca01f-866e-4a4a-a524-7eafdf95d13e\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-224ca01f-866e-4a4a-a524-7eafdf95d13e\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \r\n",
       "                        </tbody>\r\n",
       "                    </table>"
      ],
      "text/plain": [
       "|0⟩\t0 + 0𝑖\n",
       "|1⟩\t0 + 0𝑖\n",
       "|2⟩\t0 + 0𝑖\n",
       "|3⟩\t0 + 0𝑖\n",
       "|4⟩\t0.5000000000000001 + 0𝑖\n",
       "|5⟩\t0.5000000000000001 + 0𝑖\n",
       "|6⟩\t0.5000000000000001 + 0𝑖\n",
       "|7⟩\t0.5000000000000001 + 0𝑖"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/json": "{\"div_id\":\"dump-machine-div-2bc96a0e-e9c0-43d0-b77e-d7c15657bb04\",\"qubit_ids\":[0,1,2],\"n_qubits\":3,\"amplitudes\":[{\"Real\":0.0,\"Imaginary\":0.0,\"Magnitude\":0.0,\"Phase\":0.0},{\"Real\":0.0,\"Imaginary\":0.0,\"Magnitude\":0.0,\"Phase\":0.0},{\"Real\":0.0,\"Imaginary\":0.0,\"Magnitude\":0.0,\"Phase\":0.0},{\"Real\":0.0,\"Imaginary\":0.0,\"Magnitude\":0.0,\"Phase\":0.0},{\"Real\":0.5000000000000001,\"Imaginary\":0.0,\"Magnitude\":0.5000000000000001,\"Phase\":0.0},{\"Real\":0.35355339059327384,\"Imaginary\":0.35355339059327384,\"Magnitude\":0.5000000000000001,\"Phase\":0.7853981633974483},{\"Real\":0.0,\"Imaginary\":0.5000000000000001,\"Magnitude\":0.5000000000000001,\"Phase\":1.5707963267948966},{\"Real\":-0.35355339059327384,\"Imaginary\":0.35355339059327384,\"Magnitude\":0.5000000000000001,\"Phase\":2.356194490192345}]}",
      "text/html": [
       "\r\n",
       "                    <table style=\"table-layout: fixed; width: 100%\">\r\n",
       "                        <thead>\r\n",
       "                            \r\n",
       "                        <tr>\r\n",
       "                            <th>Qubit IDs</th>\r\n",
       "                            <td span=\"3\">0, 1, 2</td>\r\n",
       "                        </tr>\r\n",
       "                    \r\n",
       "                            <tr>\r\n",
       "                                <th style=\"width: 20ch)\">Basis state (little endian)</th>\r\n",
       "                                <th style=\"width: 20ch\">Amplitude</th><th style=\"width: calc(100% - 26ch - 20ch)\">Meas. Pr.</th><th style=\"width: 6ch\">Phase</th>\r\n",
       "                            </tr>\r\n",
       "                        </thead>\r\n",
       "                        <tbody>\r\n",
       "                        \r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|0\\right\\rangle$</td>\r\n",
       "                                <td>$0.0000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"0\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-8c36d309-716d-401c-a821-10938688c299\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 0;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-8c36d309-716d-401c-a821-10938688c299\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|1\\right\\rangle$</td>\r\n",
       "                                <td>$0.0000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"0\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-39efb122-04e8-4b7c-bbdc-3ba7bb18fa05\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 0;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-39efb122-04e8-4b7c-bbdc-3ba7bb18fa05\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|2\\right\\rangle$</td>\r\n",
       "                                <td>$0.0000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"0\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-ef36375e-70fb-43fc-af82-c576e25bac40\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 0;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-ef36375e-70fb-43fc-af82-c576e25bac40\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|3\\right\\rangle$</td>\r\n",
       "                                <td>$0.0000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"0\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-f575054e-0713-492d-a504-38878e2bbd84\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 0;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-f575054e-0713-492d-a504-38878e2bbd84\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|4\\right\\rangle$</td>\r\n",
       "                                <td>$0.5000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-4ce15585-3365-4106-88c5-5e188878665a\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-4ce15585-3365-4106-88c5-5e188878665a\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|5\\right\\rangle$</td>\r\n",
       "                                <td>$0.3536 + 0.3536 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-bb65f5a6-ded4-491d-99f8-daeb051bf813\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-bb65f5a6-ded4-491d-99f8-daeb051bf813\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(45deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|6\\right\\rangle$</td>\r\n",
       "                                <td>$0.0000 + 0.5000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-b49ded5e-c02b-4899-899e-8700c94924e4\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-b49ded5e-c02b-4899-899e-8700c94924e4\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(90deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|7\\right\\rangle$</td>\r\n",
       "                                <td>$-0.3536 + 0.3536 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-58bf7b34-48ed-4424-a9e3-2a5688a1eefa\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-58bf7b34-48ed-4424-a9e3-2a5688a1eefa\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(135deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \r\n",
       "                        </tbody>\r\n",
       "                    </table>"
      ],
      "text/plain": [
       "|0⟩\t0 + 0𝑖\n",
       "|1⟩\t0 + 0𝑖\n",
       "|2⟩\t0 + 0𝑖\n",
       "|3⟩\t0 + 0𝑖\n",
       "|4⟩\t0.5000000000000001 + 0𝑖\n",
       "|5⟩\t0.35355339059327384 + 0.35355339059327384𝑖\n",
       "|6⟩\t0 + 0.5000000000000001𝑖\n",
       "|7⟩\t-0.35355339059327384 + 0.35355339059327384𝑖"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/json": "{\"@type\":\"tuple\"}",
      "text/plain": [
       "()"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%simulate PhaseKickback"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 3-4: The swap test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/json": "[\"SwapTest\",\"RunSwapTest\"]",
      "text/html": [
       "<ul><li>SwapTest</li><li>RunSwapTest</li></ul>"
      ],
      "text/plain": [
       "SwapTest, RunSwapTest"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "// Example 3-4: The swap test\n",
    "\n",
    "// open namespace which defines type conversion functions\n",
    "open Microsoft.Quantum.Convert;\n",
    "// open namespace which defines MResetZ\n",
    "open Microsoft.Quantum.Measurement;\n",
    "\n",
    "// Returns True if the states of the input qubits are equal\n",
    "operation SwapTest (input1 : Qubit, input2 : Qubit) : Bool {\n",
    "    // allocate the output qubit\n",
    "    use output = Qubit();\n",
    "    // initialize the output qubit\n",
    "    H(output);\n",
    "\n",
    "    // swap the input states conditioned on the output state\n",
    "    Controlled SWAP([output], (input1, input2));\n",
    "\n",
    "    // extract the result and return the output qubit back to 0 state\n",
    "    H(output);\n",
    "    X(output);\n",
    "\n",
    "    return M(output) == One;\n",
    "}\n",
    "\n",
    "operation RunSwapTest () : Unit {\n",
    "    let attempts = 100;\n",
    "    mutable reportedEqual = 0;\n",
    "    // repeat the test multiple times to observe the probability of the states being reported equal\n",
    "    for i in 1..attempts {\n",
    "        // allocate qubits to be tested\n",
    "        use (input1, input2) = (Qubit(), Qubit());\n",
    "        // initialize the qubits in the states we want to compare\n",
    "        // leave input1 in the |0⟩ state and rotate input2 - the larger the angle, the further away are the states\n",
    "        // try replacing 0.1 with 0.0 (when the states will be equal) \n",
    "        // and with 0.2, 0.3, ... up to 1.0 when the state is rotated all the way to the -|0⟩\n",
    "        Ry(2.0 * 3.14 * 0.1, input2);\n",
    "\n",
    "        // some other suggested state initializations:\n",
    "        // X(input1); H(input1); H(input2); Z(input2);    // same state |-⟩, prepared in different ways\n",
    "        // X(input1); H(input1); H(input2);               // orthogonal states: input1 in the |-⟩ state, input2 in the |+⟩ state\n",
    "\n",
    "        if (SwapTest(input1, input2)) {\n",
    "            set reportedEqual += 1;\n",
    "        }\n",
    "\n",
    "        ResetAll([input1, input2]);\n",
    "    }\n",
    "    Message($\"The states were reported equal {IntAsDouble(reportedEqual) / IntAsDouble(attempts) * 100.0 }% of the time\");\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The states were reported equal 97% of the time\r\n"
     ]
    },
    {
     "data": {
      "application/json": "{\"@type\":\"tuple\"}",
      "text/plain": [
       "()"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%simulate RunSwapTest"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 3-5: Custom conditional phase"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/json": "[\"CustomConditionalPhase\"]",
      "text/html": [
       "<ul><li>CustomConditionalPhase</li></ul>"
      ],
      "text/plain": [
       "CustomConditionalPhase"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "// Example 3-5: Custom conditional phase\n",
    "\n",
    "// open namespace which defines diagnostic routines\n",
    "open Microsoft.Quantum.Diagnostics;\n",
    "\n",
    "operation CustomConditionalPhase () : Unit {\n",
    "    // allocate two qubits\n",
    "    use (q1, q2) = (Qubit(), Qubit());\n",
    "    // put each of the qubits into superposition of 0 and 1\n",
    "    H(q1);\n",
    "    H(q2);\n",
    "\n",
    "    // apply phases\n",
    "    T(q2);\n",
    "    CNOT(q1, q2);\n",
    "    Adjoint T(q2);\n",
    "    CNOT(q1, q2);\n",
    "    T(q1);\n",
    "\n",
    "    // output the wave function of the two-qubit state\n",
    "    DumpMachine();\n",
    "\n",
    "    // apply a single gate equivalent to the earlier sequence of gates\n",
    "    Controlled S([q1], q2);\n",
    "\n",
    "    // output the wave function of the two-qubit state\n",
    "    DumpMachine();\n",
    "\n",
    "    // make sure the qubits are back to the 0 state\n",
    "    ResetAll([q1, q2]);\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "application/json": "{\"div_id\":\"dump-machine-div-d5e4c49e-48e6-4551-b3d6-2bb3598e6087\",\"qubit_ids\":[0,1],\"n_qubits\":2,\"amplitudes\":[{\"Real\":0.5000000000000001,\"Imaginary\":0.0,\"Magnitude\":0.5000000000000001,\"Phase\":0.0},{\"Real\":0.5000000000000001,\"Imaginary\":0.0,\"Magnitude\":0.5000000000000001,\"Phase\":0.0},{\"Real\":0.5000000000000001,\"Imaginary\":0.0,\"Magnitude\":0.5000000000000001,\"Phase\":0.0},{\"Real\":0.0,\"Imaginary\":0.5000000000000001,\"Magnitude\":0.5000000000000001,\"Phase\":1.5707963267948966}]}",
      "text/html": [
       "\r\n",
       "                    <table style=\"table-layout: fixed; width: 100%\">\r\n",
       "                        <thead>\r\n",
       "                            \r\n",
       "                        <tr>\r\n",
       "                            <th>Qubit IDs</th>\r\n",
       "                            <td span=\"3\">0, 1</td>\r\n",
       "                        </tr>\r\n",
       "                    \r\n",
       "                            <tr>\r\n",
       "                                <th style=\"width: 20ch)\">Basis state (little endian)</th>\r\n",
       "                                <th style=\"width: 20ch\">Amplitude</th><th style=\"width: calc(100% - 26ch - 20ch)\">Meas. Pr.</th><th style=\"width: 6ch\">Phase</th>\r\n",
       "                            </tr>\r\n",
       "                        </thead>\r\n",
       "                        <tbody>\r\n",
       "                        \r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|0\\right\\rangle$</td>\r\n",
       "                                <td>$0.5000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-8c9773c2-633a-45b4-a478-e09dd6e58b04\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-8c9773c2-633a-45b4-a478-e09dd6e58b04\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|1\\right\\rangle$</td>\r\n",
       "                                <td>$0.5000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-0e1f981a-9eb0-4dc0-8c92-7f313c757184\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-0e1f981a-9eb0-4dc0-8c92-7f313c757184\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|2\\right\\rangle$</td>\r\n",
       "                                <td>$0.5000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-19336f53-0e1d-4cc0-9a75-0f9ed500e4e4\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-19336f53-0e1d-4cc0-9a75-0f9ed500e4e4\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|3\\right\\rangle$</td>\r\n",
       "                                <td>$0.0000 + 0.5000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-8d03fe74-8014-4bb3-843b-a2948f01e383\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-8d03fe74-8014-4bb3-843b-a2948f01e383\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(90deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \r\n",
       "                        </tbody>\r\n",
       "                    </table>"
      ],
      "text/plain": [
       "|0⟩\t0.5000000000000001 + 0𝑖\n",
       "|1⟩\t0.5000000000000001 + 0𝑖\n",
       "|2⟩\t0.5000000000000001 + 0𝑖\n",
       "|3⟩\t0 + 0.5000000000000001𝑖"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/json": "{\"div_id\":\"dump-machine-div-19a81b5e-cba8-4cac-8ad0-b92cb7746ede\",\"qubit_ids\":[0,1],\"n_qubits\":2,\"amplitudes\":[{\"Real\":0.5000000000000001,\"Imaginary\":0.0,\"Magnitude\":0.5000000000000001,\"Phase\":0.0},{\"Real\":0.5000000000000001,\"Imaginary\":0.0,\"Magnitude\":0.5000000000000001,\"Phase\":0.0},{\"Real\":0.5000000000000001,\"Imaginary\":0.0,\"Magnitude\":0.5000000000000001,\"Phase\":0.0},{\"Real\":-0.5000000000000001,\"Imaginary\":0.0,\"Magnitude\":0.5000000000000001,\"Phase\":3.141592653589793}]}",
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       "\r\n",
       "                    <table style=\"table-layout: fixed; width: 100%\">\r\n",
       "                        <thead>\r\n",
       "                            \r\n",
       "                        <tr>\r\n",
       "                            <th>Qubit IDs</th>\r\n",
       "                            <td span=\"3\">0, 1</td>\r\n",
       "                        </tr>\r\n",
       "                    \r\n",
       "                            <tr>\r\n",
       "                                <th style=\"width: 20ch)\">Basis state (little endian)</th>\r\n",
       "                                <th style=\"width: 20ch\">Amplitude</th><th style=\"width: calc(100% - 26ch - 20ch)\">Meas. Pr.</th><th style=\"width: 6ch\">Phase</th>\r\n",
       "                            </tr>\r\n",
       "                        </thead>\r\n",
       "                        <tbody>\r\n",
       "                        \r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|0\\right\\rangle$</td>\r\n",
       "                                <td>$0.5000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-f3a01d67-7e32-43b8-9b6d-7296fd024358\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-f3a01d67-7e32-43b8-9b6d-7296fd024358\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|1\\right\\rangle$</td>\r\n",
       "                                <td>$0.5000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-6800c201-ec88-43b0-9d1f-730ceec96647\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-6800c201-ec88-43b0-9d1f-730ceec96647\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|2\\right\\rangle$</td>\r\n",
       "                                <td>$0.5000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-e2790e84-4343-465b-8df9-0da1df7b3ad3\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-e2790e84-4343-465b-8df9-0da1df7b3ad3\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(0deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \n",
       "\r\n",
       "                            <tr>\r\n",
       "                                <td>$\\left|3\\right\\rangle$</td>\r\n",
       "                                <td>$-0.5000 + 0.0000 i$</td>\r\n",
       "                                \r\n",
       "                                <td>\r\n",
       "                                    <progress\r\n",
       "                                        max=\"100\"\r\n",
       "                                        value=\"25.00000000000001\"\r\n",
       "                                        style=\"width: 100%;\"\r\n",
       "                                    > \r\n",
       "                                    <td>\r\n",
       "                                    <p id=\"round-875d02b8-780d-4873-9f3d-894507aa8c7e\"> \r\n",
       "                                    <script>\r\n",
       "                                    var num = 25.00000000000001;\r\n",
       "                                    num = num.toFixed(4);\r\n",
       "                                    var num_string = num + \"%\";\r\n",
       "                                     document.getElementById(\"round-875d02b8-780d-4873-9f3d-894507aa8c7e\").innerHTML = num_string;\r\n",
       "                                    </script> </p>\r\n",
       "                                    </td>\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                                \r\n",
       "                                <td style=\"transform: rotate(180deg);\r\n",
       "                   text-align: center;\">\r\n",
       "                                 ↑\r\n",
       "                                </td>\r\n",
       "                            \r\n",
       "                            </tr>\r\n",
       "                        \r\n",
       "                        </tbody>\r\n",
       "                    </table>"
      ],
      "text/plain": [
       "|0⟩\t0.5000000000000001 + 0𝑖\n",
       "|1⟩\t0.5000000000000001 + 0𝑖\n",
       "|2⟩\t0.5000000000000001 + 0𝑖\n",
       "|3⟩\t-0.5000000000000001 + 0𝑖"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/json": "{\"@type\":\"tuple\"}",
      "text/plain": [
       "()"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%simulate CustomConditionalPhase"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 3-6: Remote-controlled randomness"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/json": "[\"RemoteControlledRandomness\"]",
      "text/html": [
       "<ul><li>RemoteControlledRandomness</li></ul>"
      ],
      "text/plain": [
       "RemoteControlledRandomness"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "// Example 3-6: Remote-controlled randomness\n",
    "\n",
    "// open namespace which defines type conversion functions\n",
    "open Microsoft.Quantum.Convert;\n",
    "// open namespace which defines MResetZ\n",
    "open Microsoft.Quantum.Measurement;\n",
    "\n",
    "operation RemoteControlledRandomness () : Unit {\n",
    "    let attempts = 1000;\n",
    "    mutable result = [0, 0, 0, 0];\n",
    "    for i in 1 .. attempts {\n",
    "        // allocate two qubits\n",
    "        use (a, b) = (Qubit(), Qubit());\n",
    "        H(a);\n",
    "        // measuring a now will give us 0 or 1 with 50% probability\n",
    "\n",
    "        H(b);\n",
    "        T(b);\n",
    "        H(b);\n",
    "        // measuring b now will give us 0 with 85% probability and 1 with 15% probability\n",
    "\n",
    "        // entangle a and b\n",
    "        CNOT(a, b);\n",
    "        // now, you can read *either* qubit and get 0 or 1 with 50% probability;\n",
    "        // if the result is 0, then the probability of the *remaining* qubit measuring 1 is 15%, else it's 85%;\n",
    "        // if the result is 1, the probabilities are the other way around\n",
    "\n",
    "        // let's measure qubit a first\n",
    "        // (we'll convert measurement results into integer to keep statistics in an array)\n",
    "        let index = (M(a) == One ? 1 | 0) * 2 + (M(b) == One ? 1 | 0);\n",
    "        set result w/= index <- result[index] + 1;\n",
    "    }\n",
    "    Message($\"Overall measurement counts (out of {attempts}): {result}\");\n",
    "    let a0b0_percentage = IntAsDouble(result[0]) / IntAsDouble(result[0] + result[1]) * 100.0;\n",
    "    Message($\"When a was measured to be 0, b was measured 0 {a0b0_percentage}% of times\");\n",
    "    let a1b0_percentage = IntAsDouble(result[2]) / IntAsDouble(result[2] + result[3]) * 100.0;\n",
    "    Message($\"When a was measured to be 1, b was measured 0 {a1b0_percentage}% of times\");\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Overall measurement counts (out of 1000): [446,73,73,408]\n",
      "When a was measured to be 0, b was measured 0 85.9344894026975% of times\n",
      "When a was measured to be 1, b was measured 0 15.176715176715177% of times\n"
     ]
    },
    {
     "data": {
      "application/json": "{\"@type\":\"tuple\"}",
      "text/plain": [
       "()"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%simulate RemoteControlledRandomness"
   ]
  }
 ],
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