[BasicGates] Add task 2.4 to the workbook (#756)

microsoft · Feb 26, 2022 · 12661af · 12661af
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diff --git a/BasicGates/BasicGates.ipynb b/BasicGates/BasicGates.ipynb
@@ -559,6 +559,13 @@
     "}"
    ]
   },
+  {
+   "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-2.4.-Toffoli-gate).*"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},

diff --git a/BasicGates/Workbook_BasicGates.ipynb b/BasicGates/Workbook_BasicGates.ipynb
@@ -1525,6 +1525,133 @@
    "source": [
     "[Return to Task 2.3 of the Basic Gates kata](./BasicGates.ipynb#Task-2.3.-Two-qubit-gate---3)."
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Task 2.4. Toffoli gate\n",
+    "\n",
+    "**Input:** Three qubits (stored in an array of length 3) in an arbitrary three-qubit state \n",
+    "$\\alpha |000\\rangle + \\beta |001\\rangle + \\gamma |010\\rangle + \\delta |011\\rangle + \\epsilon |100\\rangle + \\zeta|101\\rangle + \\color{blue}\\eta|110\\rangle + \\color{blue}\\theta|111\\rangle$.\n",
+    "\n",
+    "**Goal:** Flip the state of the third qubit if the state of the first two is $|11\\rangle$, i.e., change the three-qubit state to $\\alpha |000\\rangle + \\beta |001\\rangle + \\gamma |010\\rangle + \\delta |011\\rangle + \\epsilon |100\\rangle + \\zeta|101\\rangle + \\color{red}\\theta|110\\rangle + \\color{red}\\eta|111\\rangle$.\n",
+    "\n",
+    "### Solution\n",
+    "\n",
+    "This is essentially bookwork, because there is only one gate that performs this state change (and the task title already gave it away!) The Toffoli gate is:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    " \\begin{bmatrix}\n",
+    "    1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n",
+    "    0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n",
+    "  \\end{bmatrix}\n",
+    " $$"
+   ]
+  },
+  {
+   "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": [
+    "$$\n",
+    " \\begin{bmatrix}\n",
+    "    1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n",
+    "    0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n",
+    "    0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n",
+    "  \\end{bmatrix}\n",
+    " \\begin{bmatrix}\n",
+    "   \\alpha\\\\\n",
+    "   \\beta\\\\\n",
+    "   \\gamma\\\\\n",
+    "   \\delta\\\\\n",
+    "   \\epsilon\\\\\n",
+    "   \\zeta\\\\\n",
+    "   \\color{blue}\\eta\\\\\n",
+    "   \\color{blue}\\theta\\\\       \n",
+    "\\end{bmatrix}\n",
+    "=\n",
+    " \\begin{bmatrix}\n",
+    "   \\alpha\\\\\n",
+    "   \\beta\\\\\n",
+    "   \\gamma\\\\\n",
+    "   \\delta\\\\\n",
+    "   \\epsilon\\\\\n",
+    "   \\zeta\\\\\n",
+    "   \\color{red}\\theta\\\\\n",
+    "   \\color{red}\\eta\\\\       \n",
+    "\\end{bmatrix}\n",
+    "=\n",
+    "\\alpha |000\\rangle + \\beta |001\\rangle + \\gamma |010\\rangle + \\delta |011\\rangle + \\epsilon |100\\rangle + \\zeta|101\\rangle + \\color{red}\\theta|110\\rangle + \\color{red}\\eta|111\\rangle\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "%kata T204_ToffoliGate\n",
+    "\n",
+    "operation ToffoliGate (qs : Qubit[]) : Unit is Adj+Ctl {\n",
+    "    CCNOT(qs[0], qs[1], qs[2]);\n",
+    "}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "[Return to Task 2.4 of the Basic Gates kata](./BasicGates.ipynb#Task-2.4.-Toffoli-gate)."
+   ]
   }
  ],
  "metadata": {