Fix renderin in MultiGubitGates for CNOT gate

Author: remilvus

tutorials/MultiQubitGates/MultiQubitGates.ipynb

@@ -153,10 +153,10 @@
     "    <tr>\n",
     "        <td style=\"text-align:center; border:1px solid\">$\\text{CNOT}$</td>\n",
     "        <td style=\"text-align:center; border:1px solid\">$\\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 \\end{bmatrix}$</td>\n",
-    "        <td style=\"text-align:center; border:1px solid\">$\\text{CNOT}|\\psi\\rangle = \\alpha|00\\rangle + \\beta|01\\rangle + \\color{red}\\delta|10\\rangle + \\color{red}\\gamma|11\\rangle$</td>\n",
-    "        <td style=\"text-align:center; border:1px solid\">$\\text{CNOT}|00\\rangle = |00\\rangle \\\\\n",
-    "        \\text{CNOT}|01\\rangle = |01\\rangle \\\\\n",
-    "        \\text{CNOT}|10\\rangle = |11\\rangle \\\\\n",
+    "        <td style=\"text-align:center; border:1px solid\">$\\text{CNOT}|\\psi\\rangle = \\alpha|00\\rangle + \\beta|01\\rangle + {\\color{red}\\delta}|10\\rangle + {\\color{red}\\gamma}|11\\rangle$</td>\n",
+    "        <td style=\"text-align:center; border:1px solid\">$\\text{CNOT}|00\\rangle = |00\\rangle$<\\br>$\n",
+    "        \\text{CNOT}|01\\rangle = |01\\rangle$<\\br>$\n",
+    "        \\text{CNOT}|10\\rangle = |11\\rangle$<\\br>$\n",
     "        \\text{CNOT}|11\\rangle = |10\\rangle$</td>\n",
     "        <td style=\"text-align:center; border:1px solid\"><a href=\"https://docs.microsoft.com/qsharp/api/qsharp/microsoft.quantum.intrinsic.cnot\">CNOT</a></td>\n",
     "    </tr>\n",
@@ -225,12 +225,12 @@
     "\n",
     "Let's consider ket-bra representation of the $\\text{CNOT}$ gate:\n",
     "\n",
-    "$$\\text{CNOT} = |00\\rangle\\langle00| + |01\\rangle\\langle01| + |10\\rangle\\langle11| + |11\\rangle\\langle10| = \\\\\n",
-    "= \\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{bmatrix}\\begin{bmatrix} 1 & 0 & 0 & 0 \\end{bmatrix} +\n",
+    "$$\\text{CNOT} = |00\\rangle\\langle00| + |01\\rangle\\langle01| + |10\\rangle\\langle11| + |11\\rangle\\langle10| =$$\n",
+    "$$= \\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{bmatrix}\\begin{bmatrix} 1 & 0 & 0 & 0 \\end{bmatrix} +\n",
     "\\begin{bmatrix} 0 \\\\ 1 \\\\ 0 \\\\ 0 \\end{bmatrix}\\begin{bmatrix} 0 & 1 & 0 & 0 \\end{bmatrix} +\n",
     "\\begin{bmatrix} 0 \\\\ 0 \\\\ 1 \\\\ 0 \\end{bmatrix}\\begin{bmatrix} 0 & 0 & 0 & 1 \\end{bmatrix} +\n",
-    "\\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 1 \\end{bmatrix}\\begin{bmatrix} 0 & 0 & 1 & 0 \\end{bmatrix} = \\\\ =\n",
-    "\\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ \\end{bmatrix} + \n",
+    "\\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 1 \\end{bmatrix}\\begin{bmatrix} 0 & 0 & 1 & 0 \\end{bmatrix} =$$ \n",
+    "$$=\\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ \\end{bmatrix} + \n",
     "\\begin{bmatrix} 0 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ \\end{bmatrix} + \n",
     "\\begin{bmatrix} 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 \\\\ \\end{bmatrix} + \n",
     "\\begin{bmatrix} 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ \\end{bmatrix} =\n",
@@ -239,10 +239,10 @@
     "This representation can be used to carry out calculations in Dirac notation without ever switching back to matrix representation:\n",
     "\n",
     "$$\\text{CNOT}|10\\rangle \n",
-    "= \\big(|00\\rangle\\langle00| + |01\\rangle\\langle01| + |10\\rangle\\langle11| + |11\\rangle\\langle10|\\big)|10\\rangle = \\\\\n",
-    "= |00\\rangle\\langle00|10\\rangle + |01\\rangle\\langle01|10\\rangle + |10\\rangle\\langle11|10\\rangle + |11\\rangle\\langle10|10\\rangle = \\\\\n",
-    "= |00\\rangle\\big(\\langle00|10\\rangle\\big) + |01\\rangle\\big(\\langle01|10\\rangle\\big) + |10\\rangle\\big(\\langle11|10\\rangle\\big) + |11\\rangle\\big(\\langle10|10\\rangle\\big) = \\\\\n",
-    "= |00\\rangle(0) + |01\\rangle(0) + |10\\rangle(0) + |11\\rangle(1) = |11\\rangle$$\n",
+    "= \\big(|00\\rangle\\langle00| + |01\\rangle\\langle01| + |10\\rangle\\langle11| + |11\\rangle\\langle10|\\big)|10\\rangle =$$\n",
+    "$$= |00\\rangle\\langle00|10\\rangle + |01\\rangle\\langle01|10\\rangle + |10\\rangle\\langle11|10\\rangle + |11\\rangle\\langle10|10\\rangle =$$\n",
+    "$$= |00\\rangle\\big(\\langle00|10\\rangle\\big) + |01\\rangle\\big(\\langle01|10\\rangle\\big) + |10\\rangle\\big(\\langle11|10\\rangle\\big) + |11\\rangle\\big(\\langle10|10\\rangle\\big) =$$\n",
+    "$$= |00\\rangle(0) + |01\\rangle(0) + |10\\rangle(0) + |11\\rangle(1) = |11\\rangle$$\n",
     "\n",
     "> Notice how a lot of the inner product terms turn out to equal 0, and our expression is easily simplified. We have expressed the CNOT gate in terms of outer product of computational basis states, which are orthonormal, and apply it to another computational basis state, so the individual inner products are going to always be 0 or 1. "
    ]
@@ -294,10 +294,10 @@
     "> * Two, as we can clearly see, are computational basis states $|00\\rangle$ and $|01\\rangle$ with eigen values $1$ and $1$, respectively (the basis states that are not affected by the gate). \n",
     "> * The other two are $|1\\rangle \\otimes |+\\rangle = \\frac{1}{\\sqrt{2}}\\big(|10\\rangle + |11\\rangle\\big)$ and $|1\\rangle \\otimes |-\\rangle = \\frac{1}{\\sqrt{2}}\\big(|10\\rangle - |11\\rangle\\big)$ with eigenvalues $1$ and $-1$, respectively:\n",
     ">\n",
-    "> $$\\text{CNOT}|0\\rangle \\otimes |0\\rangle = |0\\rangle \\otimes |1\\rangle \\\\\n",
-    "\\text{CNOT}|0\\rangle \\otimes |1\\rangle = |0\\rangle \\otimes |1\\rangle \\\\\n",
-    "\\text{CNOT}|1\\rangle \\otimes |+\\rangle = |1\\rangle \\otimes |+\\rangle \\\\\n",
-    "\\text{CNOT}|1\\rangle \\otimes |-\\rangle = -|1\\rangle \\otimes |-\\rangle$$\n",
+    "> $$\\text{CNOT}|0\\rangle \\otimes |0\\rangle = |0\\rangle \\otimes |1\\rangle$$\n",
+    "$$\\text{CNOT}|0\\rangle \\otimes |1\\rangle = |0\\rangle \\otimes |1\\rangle$$\n",
+    "$$\\text{CNOT}|1\\rangle \\otimes |+\\rangle = |1\\rangle \\otimes |+\\rangle$$\n",
+    "$$\\text{CNOT}|1\\rangle \\otimes |-\\rangle = -|1\\rangle \\otimes |-\\rangle$$\n",
     ">\n",
     "> Here's what the decomposition looks like:\n",
     ">\n",
@@ -306,7 +306,7 @@
     "= |00\\rangle\\langle00| + |01\\rangle\\langle01| +  \n",
     "\\frac{1}{2}\\big[\\big(|10\\rangle + |11\\rangle\\big)\\big(\\langle10| + \\langle11|\\big) - \\big(|10\\rangle - |11\\rangle\\big)\\big(\\langle10| - \\langle11|\\big)\\big] = \\\\\n",
     "= |00\\rangle\\langle00| + |01\\rangle\\langle01| +\n",
-    "\\frac{1}{2}\\big(\\color{red}{|10\\rangle\\langle10|} + |10\\rangle\\langle11| + |11\\rangle\\langle10| + \\color{red}{|11\\rangle\\langle11|} - \\color{red}{|10\\rangle\\langle10|} + |10\\rangle\\langle11| + |11\\rangle\\langle10| - \\color{red}{|11\\rangle\\langle11|}\\big) = \\\\\n",
+    "\\frac{1}{2}\\big({\\color{red}{|10\\rangle\\langle10|}} + |10\\rangle\\langle11| + |11\\rangle\\langle10| + {\\color{red}{|11\\rangle\\langle11|}} - {\\color{red}{|10\\rangle\\langle10|}} + |10\\rangle\\langle11| + |11\\rangle\\langle10| - {\\color{red}{|11\\rangle\\langle11|}}\\big) = \\\\\n",
     "= |00\\rangle\\langle00| + |01\\rangle\\langle01| + \\frac{1}{2}\\big(2|10\\rangle\\langle11| + 2|11\\rangle\\langle10|\\big) = \\\\ \n",
     "= |00\\rangle\\langle00| + |01\\rangle\\langle01| + |10\\rangle\\langle11| + |11\\rangle\\langle10|$$"
    ]