correct some typo in lecture 0

lectures/0_qc_ingredients.ipynb

@@ -204,7 +204,7 @@
     ".\n",
     "$$\n",
     "\n",
-    "2. $Z$: the phaseflip gate which puts a $−$ in front of $\\vert 1\\rangle,\n",
+    "2. $Z$: the phaseflip gate which puts a $−$ in front of $\\vert 1\\rangle$,\n",
     "\n",
     "$$\n",
     "Z=\\begin{pmatrix}\n",
@@ -310,7 +310,7 @@
     "\\end{array}\\right).\n",
     "$$\n",
     "\n",
-    "4. SWAP gate which swaps the state in the two qubit as $\\text{SWAP}\\vert b_1b_2\\rangle = \\vert b_2b_1\\rangle$. The matrix form is\n",
+    "4. SWAP gate: which swaps the state in the two qubit as $\\text{SWAP}\\vert b_1b_2\\rangle = \\vert b_2b_1\\rangle$. The matrix form is\n",
     "\n",
     "$$\n",
     "\\text{SWAP} = \\frac{1}{2}(I\\otimes I + X\\otimes X + Y\\otimes Y + Z\\otimes Z)=\\begin{pmatrix}1&0&0&0\\\\0&0&1&0\\\\0&1&0&0\\\\0&0&0&1\\end{pmatrix}.\n",
@@ -449,7 +449,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "**Measurements on full computational basis:** The results of measuring a quantum state follow Born rule: for $\\vert \\psi\\rangle $, when measuring on computational basis, we obtain $\\vert i\\rangle$ with probability $|\\langle i\\vert \\psi\\rangle|^2$. And the state is collpased to $\\vert i\\rangle$ after measurement.\n",
+    "**Measurements on full computational basis:** The results of measuring a quantum state follow Born rule: for $\\vert \\psi\\rangle $, when measuring on computational basis, we obtain $\\vert i\\rangle$ with probability $|\\langle i\\vert \\psi\\rangle|^2$. And the state is collapsed to $\\vert i\\rangle$ after measurement.\n",
     "\n",
     "**Measurements on partial computational basis:** We can also measure $m$ qubits for $n$-qubits states, the probability we get $\\vert i\\rangle_m$ is $|\\vert i\\rangle \\langle i\\vert \\psi\\rangle|^2$, and the state left is now $\\vert i\\rangle \\langle i\\vert \\psi\\rangle$ (unnormalized for simplicity).\n",
     "\n",