[BasicGates] Add task 2.5 to the workbook (#758)

Author: tonyholdroyd112

BasicGates/BasicGates.ipynb

@@ -590,6 +590,13 @@
     "    // ...\n",
     "}"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "*Can't come up with a solution? See the explained solution in the [Basic Gates Workbook](./Workbook_BasicGates.ipynb#Task-2.5.-Fredkin-gate).*"
+   ]
   }
  ],
  "metadata": {

BasicGates/Workbook_BasicGates.ipynb

@@ -1652,6 +1652,142 @@
    "source": [
     "[Return to Task 2.4 of the Basic Gates kata](./BasicGates.ipynb#Task-2.4.-Toffoli-gate)."
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Task 2.5. Fredkin gate\n",
+    "\n",
+    "**Input:** Three qubits (stored in an array of length 3) in an arbitrary three-qubit state \n",
+    "$\\alpha |000\\rangle + \\beta |001\\rangle + \\gamma |010\\rangle + \\delta |011\\rangle + \\epsilon |100\\rangle + \\color{blue}\\zeta|101\\rangle + \\color{blue}\\eta|110\\rangle + \\theta|111\\rangle$.\n",
+    "\n",
+    "**Goal:** Swap the states of second and third qubit if and only if the state of the first qubit is $|1\\rangle$, i.e., change the three-qubit state to $\\alpha |000\\rangle + \\beta |001\\rangle + \\gamma |010\\rangle + \\delta |011\\rangle + \\epsilon |100\\rangle + \\color{red}\\eta|101\\rangle + \\color{red}\\zeta|110\\rangle + \\theta|111\\rangle$.\n",
+    "\n",
+    "\n",
+    "### Solution\n",
+    "\n",
+    "Again this is essentially bookwork, because there is only one gate that performs this state change (and the task title already gave it away!)  \n",
+    "The Fredkin gate is also known as the controlled swap gate (Controlled SWAP):"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    " \\begin{bmatrix}\n",
+    "    1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n",
+    "    0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n",
+    "  \\end{bmatrix}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "and our initial state is:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\begin{bmatrix}\n",
+    "   \\alpha\\\\\n",
+    "   \\beta\\\\\n",
+    "   \\gamma\\\\\n",
+    "   \\delta\\\\\n",
+    "   \\epsilon\\\\\n",
+    "   \\zeta\\\\\n",
+    "   \\eta\\\\\n",
+    "   \\theta\\\\       \n",
+    "\\end{bmatrix}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "So we have:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    " \\begin{bmatrix}\n",
+    "    1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n",
+    "    0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n",
+    "  \\end{bmatrix}\n",
+    " \\begin{bmatrix}\n",
+    "   \\alpha\\\\\n",
+    "   \\beta\\\\\n",
+    "   \\gamma\\\\\n",
+    "   \\delta\\\\\n",
+    "   \\epsilon\\\\\n",
+    "  \\color{blue} \\zeta\\\\\n",
+    "  \\color{blue} \\eta\\\\\n",
+    "   \\theta\\\\       \n",
+    "\\end{bmatrix}\n",
+    "=\n",
+    " \\begin{bmatrix}\n",
+    "   \\alpha\\\\\n",
+    "   \\beta\\\\\n",
+    "   \\gamma\\\\\n",
+    "   \\delta\\\\\n",
+    "   \\epsilon\\\\\n",
+    "  \\color{red} \\eta\\\\\n",
+    "  \\color{red} \\zeta\\\\\n",
+    "   \\theta\\\\       \n",
+    "\\end{bmatrix}\n",
+    "=\n",
+    "\\alpha |000\\rangle + \\beta |001\\rangle + \\gamma |010\\rangle + \\delta |011\\rangle + \\epsilon |100\\rangle + \\color{red}\\eta|101\\rangle + \\color{red}\\zeta|110\\rangle + \\theta|111\\rangle\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Notice carefully how the qubits are passed to the gate: `[qs[0]], (qs[1], [qs[2])`. The `Controlled` functor produces an operation that takes two parameters: the first one is an array of control qubits (in this case a single-element array consisting of the first qubit), and the second parameter is a tuple of all parameters you'd pass to the original gate (in this gate two single-qubit parameters that would be arguments to a SWAP gate)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "%kata T205_FredkinGate\n",
+    "\n",
+    "operation FredkinGate (qs : Qubit[]) : Unit is Adj+Ctl {\n",
+    "    Controlled SWAP([qs[0]], (qs[1], qs[2]));\n",
+    "}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "[Return to Task 2.5 of the Basic Gates kata](./BasicGates.ipynb#Task-2.5.-Fredkin-gate)."
+   ]
   }
  ],
  "metadata": {