Fix typos in advanced/aqua tutorials

Author: mrossinek

qiskit/advanced/aqua/amplitude_estimation.ipynb

@@ -28,7 +28,7 @@
    "source": [
     "### Introduction\n",
     "<br>\n",
-    "This notebook illustrates amplitude estimation in the simplest case, where the (assumed to be unkown) success probability $p$ of a Bernoulli random variable is estimated.\n",
+    "This notebook illustrates amplitude estimation in the simplest case, where the (assumed to be unknown) success probability $p$ of a Bernoulli random variable is estimated.\n",
     "In other words, we assume a qubit is prepared in a state $\\sqrt{1-p}\\,\\big|0\\rangle + \\sqrt{p}\\,\\big|1\\rangle$, i.e., the probability of measuring $\\big|1\\rangle$ equals $p$.\n",
     "This matches the results that have been demonstrated on real hardware in [1].\n",
     "<br>\n",

qiskit/advanced/aqua/generating_random_variates.ipynb

@@ -34,7 +34,7 @@
     "For example, the measurement of a quantum superposition is intrinsically random,\n",
     "as suggested by Born's rule.\n",
     "Consequently, some of the\n",
-    "best random-number generators are based on such quantum-mechanical effects. (See the \n",
+    "best random-number generators are based on such quantum-mechanical effects.\n",
     "Further, with a logarithmic amount of random bits, quantum computers can produce\n",
     "linearly many more bits, which is known as \n",
     "randomness expansion protocols. \n",
@@ -46,7 +46,7 @@
     "\n",
     "## Random Bits and the Bernoulli distribution\n",
     "\n",
-    "It is clear that there are many options for generating random bits (i.e., Bernoulli-distributed scalars, taking values either 0 or 1). Starting from a simple circuit such as a Hadamard gate followed by measurement, one can progress to vectors of Bernoulli-distributed elements. By addition of such random variates, we could get binomial distributions. By multiplication we could get geometric distributions, although perhaps leading to a circuit depth that may be impratical at the moment, though.\n",
+    "It is clear that there are many options for generating random bits (i.e., Bernoulli-distributed scalars, taking values either 0 or 1). Starting from a simple circuit such as a Hadamard gate followed by measurement, one can progress to vectors of Bernoulli-distributed elements. By addition of such random variates, we could get binomial distributions. By multiplication we could get geometric distributions, although perhaps leading to a circuit depth that may be impractical at the moment, though.\n",
     "\n",
     "Let us start by importing the basic modules and creating a quantum circuit for generating random bits:"
    ]

qiskit/advanced/aqua/linear_systems_of_equations.ipynb

@@ -27,7 +27,7 @@
     "The HHL algorithm (after the author’s surnames Harrow-Hassidim-Lloyd) [1] is a quantum algorithm to solve systems of linear equations $A \\vec{x} = \\vec{b}$. To perform this calculation quantum mechanically, we need in general 4 main steps requiring three qubit registers:\n",
     "<ol>\n",
     "<li>First, we have to express the vector $\\vec{b}$ as a quantum state $|b\\rangle$ on a quantum register.</li>\n",
-    "<li>Now, we have to decompose $\\vec{b}$ into a superposition of eigenvectors of A remembering on the linear combination of the vector $\\vec{b}$. We achieve this using the Quantum Phase Estimation algorithm (Quantum Phase Estimation (QPE)). Since the matrix is hereby diagonalized wherefore $A$ is easily invertible.</li>\n",
+    "<li>Now, we have to decompose $\\vec{b}$ into a superposition of eigenvectors of $A$ remembering on the linear combination of the vector $\\vec{b}$. We achieve this using the Quantum Phase Estimation algorithm (Quantum Phase Estimation (QPE)). Since the matrix is hereby diagonalized wherefore $A$ is easily invertible.</li>\n",
     "<li>The inversion of the eigenvector base of $A$ is achieved by rotating an ancillary qubit by an angle $\\arcsin \\left( \\frac{C}{\\lambda _{\\text{i}}} \\right)$ around the y-axis where $\\lambda_{\\text{i}}$ are the eigenvalues of $A$. Now, we obtain the state $A^{-1}|b\\rangle = |x \\rangle$.</li>\n",
     "<li>We need to uncompute the register storing the eigenvalues using the inverse QPE. We measure the ancillary qubit whereby the measurement of 1 indicates that the matrix inversion was successful. The inverse QPE leaves the system in a state proportional to the solution vector $|x\\rangle$. In many cases one is not interested in the single vector elements of $|x\\rangle$ but only on certain properties. These are accessible by applying a problem-specific operator $M$ to the state $|x\\rangle$. Another use-case of the HHL algorithm is the implementation in a larger quantum program.</li>\n",
     "</ol>\n",