docs (challenge): minor changes to the writing

docs/challenges/dynamical_decoupling.ipynb

@@ -32,11 +32,7 @@
     "<ul>\n",
     "    <li>In which scenario applying DD leads to effective error suppression</li>\n",
     "    <li>Which DD sequence is most effective</li>\n",
-    "</ul>\n",
-    "\n",
-    "For this study we shall select a TwoLocal circuit which is often used in problems related to quantum chemistry and quantum machine learning. Moreover, we shall use a compute-uncompute version of the circuit. In this, the circuit unitary $U$ is followed by $U^{\\dagger}$ so that the ideal noiseless outcome is $|0\\rangle^{\\otimes n}$ for an $n$-qubit circuit.\n",
-    "\n",
-    "The advantage of such a circuit is that the ideal expectation values for $Z$ type observables are known without explicit simulation which is not possible for utility-scale circuits. The disadvantage is that having an uncompute unitary doubles the depth of the base circuit."
+    "</ul>"
    ]
   },
   {
@@ -64,7 +60,7 @@
     "\n",
     "We start by declaring an example circuit. Feel free to change it to something else.<br>\n",
     "\n",
-    "In this study, we shall consider a circuit in which one or more chains of CNOTs are sandwitched between two layers of Hadamard gates acting on all the qubits. The number of chains will allow us the vary the `density` of the circuit. We shall discuss about `density` more later.<br>\n",
+    "In this study, we shall consider a circuit in which one or more ladders of CNOTs are sandwitched between two layers of Hadamard gates acting on all the qubits. By construction, it is a square circuit with the 2-qubit depth scaling linearly with the number of qubits. Therefore, it intentionally leads to a lot of idling time in the qubits. The number of ladders will allow us to vary the `density` of the circuit, which changes the idling time of each qubit. We shall discuss more about `density` later.\n",
     "\n",
     "Let us first try to argue about the outcome of such a circuit. We start with an $n$-qubit system in state $|00..0\\rangle$. Applying Hadamard on all the qubits creates an equal superposition of the form\n",
     "\n",
@@ -181,7 +177,7 @@
    "id": "d3a940fa-6b30-4ef4-935e-fbf2df0a5ec0",
    "metadata": {},
    "source": [
-    "Let us select the average of all weight-1 Pauli-Z observables for this example. Note that by nature of the compute-uncompute circuit the ideal value is exactly 1."
+    "Let us select the average of all weight-1 Pauli-Z observables for this example."
    ]
   },
   {
@@ -520,7 +516,7 @@
    "id": "7dd6a3dc-e8f3-4557-b890-f95bee9a0724",
    "metadata": {},
    "source": [
-    "In the above circuit, it is possible to add layers of CNOT gates on the left and right of the main diagonal consisting of CNOTs. For example, if the number of diagonals is $2k+1$, the circuit contains $k$ diagonals each to the upper and lower triangle, along with the main diagonal. We can visualize this below with `num_diagonals = 3` which will add one diagonal to the left and right of the main diagonal."
+    "In the above circuit, it is possible to add ladders of CNOT gates to the left and right of the main diagonal consisting of CNOTs. For example, if the number of diagonals is $2k+1$, the circuit contains $k$ ladders of CNOTs each to the upper and lower triangle, along with the main diagonal. We can visualize this below with `num_diagonals = 3` which will add one diagonal to the left and right of the main diagonal."
    ]
   },
   {
@@ -555,14 +551,14 @@
     "\n",
     "<center>density = num_diagonals/(num_qubits-1)</center><br>\n",
     "\n",
-    "The total number of diagonals can vary from `0` to `num_qubits - 1`. Therefore, the `density` of the circuit varies from `0` to `1`. The goal of this challenge is to understand the utility of DD with variation in the `density` of the circuit.<br>\n",
+    "The total number of diagonals can vary from `1` to `num_qubits - 1`. Therefore, the `density` of the circuit varies from `0` to `1`. The goal of this challenge is to understand the utility of DD with variation in the `density` of the circuit.<br>\n",
     "\n",
-    "**Question**: For each num_qubit $\\in \\{8, 16, 32, 64\\}$, vary the number of diagonals as $\\{1, \\frac{num\\_qubits}{2}-1, \\frac{num\\_qubits}{4}-1 \\}$. In other words, the `density`, for each number of qubits, is varied between $\\{\\frac{1}{num\\_qubits-1}, \\frac{num\\_qubits-2}{2num\\_qubits-2}, \\frac{num\\_qubits-4}{4num\\_qubits-4} \\}$.\n",
+    "**Question**: For each num_qubit $\\in \\{8, 16, 32, 64\\}$, vary the number of diagonals as $\\{1, \\frac{num\\_qubits}{2}-1, num\\_qubits-1 \\}$. In other words, the `density`, for each number of qubits, is varied between $\\{\\frac{1}{num\\_qubits-1}, \\frac{num\\_qubits-2}{2num\\_qubits-2}, 1 \\}$.\n",
     "\n",
     "For each instance\n",
     "\n",
     "<ol>\n",
-    "    <li>determine the best DD pulse</li>\n",
+    "    <li>determine the best DD sequence</li>\n",
     "    <li>determine whether applying DD improves the outcome over no DD</li>\n",
     "    <li>plot the variation in the performance improvement obtained by DD over no DD with respect to density</li>\n",
     "</ol>"
@@ -581,7 +577,7 @@
    "id": "1bd78683-4503-49c5-ba5e-0d2bf5002ca8",
    "metadata": {},
    "source": [
-    "Vary the number of qubits and the density of the circuit. For each case, apply DD pulse and use measurement error mitigation by setting `resilience_level` to 1. For each instance \n",
+    "Vary the number of qubits and the density of the circuit. For each case, apply DD pulse and use measurement error mitigation by setting `resilience_level` to 1 (measurement error mitigation) and 2 (zero noise extrapolation). For each instance \n",
     "\n",
     "<ol>\n",
     "    <li>determine whether the best DD pulse changes from the previous scenario with no mitigation</li>\n",