Discrete Quantum Walks suggested English changes

Author: AnnePicus

tutorials/advanced_tutorials/discrete_quantum_walk/discrete_quantum_walk.ipynb

@@ -15,7 +15,10 @@
    "source": [
     "\"Quantum Walk\" is an approach for developing and designing quantum algorithms. Consider it as a quantum analogy to the classical random walk. This generic technique underlies many quantum algorithms. In particular, the Grover's search algorithm can be viewed as a quantum walk.\n",
     "\n",
-    "Similarly to classical random walks, quantum walks are divided into the discrete case and the continuous one. This notebook focuses on discrete quantum walks, and is organized as follows: first, [one to one](##Classical-Random-Walks-vs.-Quantum-Walks) compares classical random walks with quantum walks via a specific example: a walk on a circle. Then, a [general quantum model](##How-to-Build-a-general-Quantum-Walk-with-Classiq) studies quantum walks and subsequently applies it to the [circle case](##Example:-Symmetric-Quantum-Walk-on-a-Circle) and to a [hypercube graph](##Example:-4D-Hypercube-with-a-Grover-Coin). "
+    "Similarly to classical random walks, quantum walks are divided into the discrete and continuous cases. This notebook focuses on discrete quantum walks, and is organized as follows: \n",
+    "\n",
+    "1. [One to one](##Classical-Random-Walks-vs.-Quantum-Walks) compares classical random walks with quantum walks via a specific example: a walk on a circle.\n",
+    "2. A [general quantum model](##How-to-Build-a-general-Quantum-Walk-with-Classiq) studies quantum walks and subsequently applies it to the [circle case](##Example:-Symmetric-Quantum-Walk-on-a-Circle) and to a [hypercube graph](##Example:-4D-Hypercube-with-a-Grover-Coin). "
    ]
   },
   {
@@ -24,6 +27,7 @@
    "metadata": {},
    "source": [
     "This tutorial demonstrates the following concepts of Classiq:\n",
+    "\n",
     "* The resemblance between classical and quantum programming with Classiq.\n",
     "* Classiq's built-in constructs:\n",
     "    * `control`: general controlled-logic\n",
@@ -51,6 +55,7 @@
    "metadata": {},
    "source": [
     "Specify and implement a simple example: a discrete walk on a circle, or more precisely, on a regular polygon with $2^N$ nodes (see Figure 1). The idea behind the random/quantum walk:\n",
+    "\n",
     "1. Start at some initial point, e.g., the zeroth node.\n",
     "2. Flip a coin. Heads moves one step clockwise; tails moves counterclockwise.\n",
     "3. Repeat steps 1-2 for a given total number of steps $T$."
@@ -101,6 +106,7 @@
    "metadata": {},
    "source": [
     "Define a function to flip a coin:\n",
+    "\n",
     "* Classical: the coin is either 0 or 1. To flip the coin, draw a random number from the set $\\{0,1\\}$.\n",
     "* Quantum: the coin is represented by a qubit and a \"flip\" is defined by some unitary operation on it. Choose the Hadamard gate, which sends the $|0\\rangle$ state into an equal superposition of $|0\\rangle$ and $|1\\rangle$."
    ]
@@ -133,7 +139,8 @@
    "id": "d10a317a-0267-4b51-b230-3d3cd393d3fd",
    "metadata": {},
    "source": [
-    "Next, define a function for moving clockwise and counterclockwise. This operation is a modular addition by $\\pm 1$.\n",
+    "Next, define a function for moving clockwise and counterclockwise. This operation is a modular addition by $\\pm 1$:\n",
+    "\n",
     "* Classical: the position is an integer in $[-2^{N-1}, 2^{N-1}-1]$. Use basic arithmetic operations.\n",
     "* Quantum: the position is an $N$-qubits state. Build an in-place modular addition by 1 (see an explanation at [the end of this notebook](#Technical-Notes)). Note that since quantum operations are reversible, you can define a counterclockwise step as the inverse of the clockwise step."
    ]
@@ -237,7 +244,7 @@
    "id": "ea9266b8-60a0-4f93-bc6b-942cd6d85cb7",
    "metadata": {},
    "source": [
-    "Define and run a specific example, taking a circle of size $2^7$, a total time of 50 steps, and 10000 samples. "
+    "Define and run a specific example, taking a circle of size $2^7$, a total time of 50 steps, and 10,000 samples. "
    ]
   },
   {
@@ -323,7 +330,7 @@
    "source": [
     "<center>\n",
     "<img src=\"https://docs.classiq.io/resources/quantum_walk_circle_circuit.png\" style=\"width:100%\">\n",
-    "<figcaption align = \"middle\"> Figure 2. The circuit for a quantum walk on a circle with $2^7$ nodes. The last three blocks are repeated `t` times. </figcaption>\n",
+    "<figcaption align = \"middle\"> Figure 2. The circuit for a quantum walk on a circle with $2^7$ nodes. The last three blocks repeat `t` times. </figcaption>\n",
     "</center>"
    ]
   },
@@ -399,15 +406,15 @@
    "id": "922b31e2-d1c4-4d7d-afd1-c633b5981441",
    "metadata": {},
    "source": [
-    "There is a clear difference between the two distributions. The classical distribution is symmetric around the zero position, whereas the quantum example is asymmetric with a peak far from 0. This is a small example of the different behaviors of classical random walks and quantum walks. More details and examples are in Ref. [[1](#review)]."
+    "There is a clear difference between the two distributions: the classical distribution is symmetric around the zero position, whereas the quantum example is asymmetric with a peak far from 0. This is a small example of the different behaviors of classical random walks and quantum walks. More details and examples are in Ref. [[1](#review)]."
    ]
   },
   {
    "cell_type": "markdown",
    "id": "d0115edd-4b37-4482-8a30-c80bc0e50bac",
    "metadata": {},
    "source": [
-    "## How to Build a General Quantum Walk with Classiq"
+    "## Building a General Quantum Walk with Classiq"
    ]
   },
   {
@@ -416,6 +423,7 @@
    "metadata": {},
    "source": [
     "Define a quantum function for a discrete quantum walk. The arguments of the function:\n",
+    "\n",
     "* `time`: an integer for the number of walking steps.\n",
     "* `coin_flip_qfunc`: the quantum function for \"flipping\" the coin.\n",
     "* `walks_qfuncs`: a list of quantum functions for all possible transitions at a given point.\n",
@@ -633,7 +641,7 @@
    "id": "7dc4fde2-43ee-4249-84ee-909c3b694ce0",
    "metadata": {},
    "source": [
-    "Define a model for quantum walk on a hypercube. Two nodes in the hypercube are connected to each other if their Hamming distance is 1. Thus, a step along a hypercube is given by moving \"1 Hamming distance away\". For a $d$-dimentional hypercube, at each node there are $d$ possible directions to move. Each of them is given by applying a bit flip on one of the bits. In the quantum case, this is obtained by applying an X gate on one of the $N$ qubits.\n",
+    "Define a model for quantum walk on a hypercube. Two nodes in the hypercube are connected to each other if their Hamming distance is 1. Thus, a step along a hypercube is given by moving \"1 Hamming distance away\". For a $d$-dimensional hypercube, at each node there are $d$ possible directions to move. Each of them is given by applying a bit flip on one of the bits. In the quantum case, this is obtained by applying an X gate on one of the $N$ qubits.\n",
     "\n",
     "For the coin operator, take the \"Grover diffuser\" function. This choice refers to a symmetric quantum walk [[1](#review)]. The Grover diffuser operator is a reflection around a given state $|\\psi\\rangle$:\n",
     "$$\n",
@@ -767,7 +775,7 @@
    "source": [
     "<center>\n",
     "<img src=\"https://docs.classiq.io/resources/quantum_walk_hypercube_circuit.png\" style=\"width:100%\">\n",
-    "<figcaption align = \"middle\"> Figure 4. The circuit for a quantum walk on a 4D cube with a Grover coin. The last two blocks are repeated `t` times. </figcaption>\n",
+    "<figcaption align = \"middle\"> Figure 4. The circuit for a quantum walk on a 4D cube with a Grover coin. The last two blocks repeat `t` times. </figcaption>\n",
     "</center>"
    ]
   },
@@ -821,7 +829,7 @@
    "id": "3cca3d17-4ec3-4cba-b770-13b38e58faa4",
    "metadata": {},
    "source": [
-    "We found that at $T=4$ the probability to measure the walker at the opposite corner is larger than $1/2$. We can check an analogous question in the classical random walk, where the distribution is taken over an ensemble of independent experiments):"
+    "At $T=4$, the probability to measure the walker at the opposite corner is larger than $1/2$. Check an analogous question in the classical random walk, where the distribution is taken over an ensemble of independent experiments:"
    ]
   },
   {
@@ -876,11 +884,9 @@
    "id": "916c1dd1-3f08-417e-a7f7-adbaa33b60d4",
    "metadata": {},
    "source": [
-    "## Technical Notes\n",
-    "\n",
-    "### QFT modular adder\n",
+    "## Note Regarding QFT Modular Adder\n",
     "\n",
-    "Below we explain the `quantum_step_clockwise` function defined for the quantum walk on a circle. The unitary matrix that represents walking on a circle operates as follows:\n",
+    "This section explains the `quantum_step_clockwise` function defined for the quantum walk on a circle. The unitary matrix that represents walking on a circle operates as follows:\n",
     "$$\n",
     "U_{+}|i\\rangle =\n",
     "\\left\\{\n",
@@ -902,7 +908,7 @@
     "\t 1 & 0 & 0 & \\cdots & 0 & 0\n",
     "\\end{pmatrix}.\n",
     "$$\n",
-    "In Fourier space this matrix is diagonal, namely, the Fourier matrix $\\mathcal{FT} $ diagonalizes it. Moreover, the diagonal entries forms a goemetric series:\n",
+    "In Fourier space, this matrix is diagonal; namely, the Fourier matrix $\\mathcal{FT} $ diagonalizes it. Moreover, the diagonal entries forms a goemetric series:\n",
     "$$\n",
     "U_+  = \\mathcal{FT} \\cdot \\begin{pmatrix}\n",
     "\t\\alpha^0 &  0 & \\cdots& \\cdots  & 0 \\\\\n",
@@ -912,15 +918,15 @@
     "\t 0 & \\cdots & 0 & \\cdots & \\alpha^{2^N-1}\n",
     "\\end{pmatrix} \\cdot \\mathcal{FT}^{\\dagger},\n",
     "$$\n",
-    "with $\\alpha=e^{2\\pi i /2^N }$. We can implement both the $\\mathcal{FT}$ matrix and the diagonal matrix efficiently on a quantum computer, the former by a QFT and the latter by applying a series of $N$ RZ rotations."
+    "with $\\alpha=e^{2\\pi i /2^N }$. You can implement both the $\\mathcal{FT}$ matrix and the diagonal matrix efficiently on a quantum computer; the former by a QFT and the latter by applying a series of $N$ RZ rotations."
    ]
   },
   {
    "cell_type": "markdown",
    "id": "88b24611-5d16-4fa0-a014-d192459198d3",
    "metadata": {},
    "source": [
-    "<a id='review'>[1]</a>: [Kempe, J. \"Quantum random walks: an introductory overview.\" Contemporary Physics 44, 307 (2003)](https://arxiv.org/abs/quant-ph/0303081).\n",
+    "<a id='review'>[1]</a>: [Kempe, J. \"Quantum random walks: An introductory overview.\" Contemporary Physics 44, 307 (2003)](https://arxiv.org/abs/quant-ph/0303081).\n",
     "\n",
     "<a id='hypercube'>[2]</a>: [Kempe, J. \"Discrete quantum walks hit exponentially faster.\" Probability theory and related fields 133, 215 (2005)](https://arxiv.org/abs/quant-ph/0205083)."
    ]