Swap test English suggestions

Author: AnnePicus

algorithms/swap_test/swap_test.ipynb

@@ -5,17 +5,17 @@
    "id": "e865ba60-b8b6-4dbb-b1d9-fd6883d15d86",
    "metadata": {},
    "source": [
-    "# The Swap Test Algorithm\n",
+    "# Swap Test Algorithm\n",
     "\n",
-    "The Swap test is a quantum function that checks the overlap between two quantum states: the inputs of the function are two quantum registers of the same size, $|\\psi_1\\rangle, \\,|\\psi_2\\rangle$, and it returns as an output a single test qubit whose state encodes the overlap between the two inputs: $|q\\rangle_{\\rm test} = \\alpha|0\\rangle + \\sqrt{1-\\alpha^2}|1\\rangle$, with\n",
+    "The swap test is a quantum function that checks the overlap between two quantum states. The inputs of the function are two quantum registers of the same size, $|\\psi_1\\rangle, \\,|\\psi_2\\rangle$, and it returns as output a single test qubit whose state encodes the overlap between the two inputs: $|q\\rangle_{\\rm test} = \\alpha|0\\rangle + \\sqrt{1-\\alpha^2}|1\\rangle$, with\n",
     "$$\n",
     "\\alpha^2 = \\frac{1}{2}\\left(1+|\\langle \\psi_1 |\\psi_2 \\rangle |^2\\right).\n",
     "$$\n",
     "Thus, the probability of measuring the test qubit at state $|0\\rangle$ is $1$ if the states are identical (up to a global phase) and 0.5 if the states are orthogonal to each other.\n",
     "\n",
-    "The quantum model starts with an $H$ gate on the test qubit, followed by swapping between the two states controlled on the test qubit (a controlled-SWAP gate for each of the qubits in the two states), and a final $H$ gate on the test qubit\n",
+    "The quantum model starts with an $H$ gate on the test qubit, followed by swapping between the two states controlled on the test qubit (a controlled-SWAP gate for each of the qubits in the two states) and a final $H$ gate on the test qubit.\n",
     "\n",
-    "A general scheme of the Swap test algorithm:\n",
+    "A general scheme of the swap test algorithm:\n",
     "\n",
     "<center>\n",
     "<div style=\"text-align:center;\">\n",
@@ -99,9 +99,9 @@
    "id": "eeaa11bb-4a3a-4172-bc29-1bb8860f232a",
    "metadata": {},
    "source": [
-    "## Swap Test - Qmod Implementations\n",
-    "The Swap test is defined as a library function in the qmod language (definition can be found on our [public github](https://github.com/Classiq/classiq-library/blob/main/functions/open_library_definitions/swap_test.qmod)).\n",
-    "Users can easily add their own functions."
+    "## Swap Test Qmod Implementations\n",
+    "The swap test is defined as a library function in the Qmod language. The definition is in the [public GitHub](https://github.com/Classiq/classiq-library/blob/main/functions/open_library_definitions/swap_test.qmod).\n",
+    "You can add your own functions."
    ]
   },
   {
@@ -127,13 +127,13 @@
    "id": "959995ad-aa6b-4b31-a25b-95396f8ffa7f",
    "metadata": {},
    "source": [
-    "## Comparing Measured and Exact Overlap\n",
-    "Using the expected probability of measuring the state $|0\\rangle$ as defined above:\n",
+    "## Comparing Measured with Exact Overlap\n",
+    "Using the expected probability of measuring the state $|0\\rangle$ as defined above,\n",
     "$$\n",
-    "\\alpha^2 = \\frac{1}{2}\\left(1+|\\langle \\psi_1 |\\psi_2 \\rangle |^2\\right).\n",
+    "\\alpha^2 = \\frac{1}{2}\\left(1+|\\langle \\psi_1 |\\psi_2 \\rangle |^2\\right),\n",
     "$$\n",
-    "we extract the overlap $|\\langle \\psi_1 |\\psi_2 \\rangle |=\\sqrt{2 P\\left(q_{\\text{test}}=|0\\rangle\\right)-1}$  \n",
-    "The exact overlap is computed with the dot product of the two state-vectors.\n",
+    "we extract the overlap $|\\langle \\psi_1 |\\psi_2 \\rangle |=\\sqrt{2 P\\left(q_{\\text{test}}=|0\\rangle\\right)-1}$. \n",
+    "The exact overlap is computed with the dot product of the two state vectors.\n",
     "Note that for the sake of this demonstration we execute this circuit $100,000$ times to improve the precision of the probability estimate. This is usually not required in actual programs."
    ]
   },
@@ -188,7 +188,7 @@
     "assert np.isclose(\n",
     "    overlap_from_swap_test, exact_overlap, RTOL\n",
     "), f\"\"\"\n",
-    "The quantum result is too far than classical one, by a relative tolerance of {RTOL}. Please verify your parameters\"\"\""
+    "The quantum result is too far from the classical one by a relative tolerance of {RTOL}. Please verify your parameters\"\"\""
    ]
   },
   {