English suggestions for hybrid QNN

Author: AnnePicus

algorithms/qml/hybrid_qnn/hybrid_qnn_for_subset_majority.ipynb

@@ -21,20 +21,20 @@
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-    "Neural networks is one of the major branches in machine learning, with wide usa in applications and research. A neural network, or more generally a deep neural network, is a parametric function of a specific structure (inspired from biological neural networks in biology), which is trained to capture specific functionality.\n",
+    "Neural networks is one of the major branches in machine learning, with wide use in applications and research. A neural network—or, more generally, a deep neural network—is a parametric function of a specific structure (inspired by neural networks in biology), which is trained to capture specific functionality.\n",
     "\n",
     "In its most basic form, a neural network for learning a function $\\vec{f}: \\mathbb{R}^N\\rightarrow \\mathbb{R}^M$ looks as follows:\n",
     "1. There is an input vector of size $N$ (red circles in Fig. 1).\n",
-    "2. Each entry of the input goes into a hidden layer of size $K$, where each neuron (blue circles in Fig. 1) is defined with some \"activation function\" $y^{k}(\\vec{w}^{(1)}; \\vec{x})$ for $k=1,\\dots,K$, and $\\vec{w}^{(1)}$ are parameters.\n",
-    "3. Finally, the output of the hidden layer is sent to the output layer (green circles in Fig. 1) $\\tilde{f}^{m}(\\vec{w}^{(2)};\\vec{y})$ for $m=1,\\dots,M$, and $\\vec{w}^{(2)}$ are parameters.\n",
+    "2. Each entry of the input goes into a hidden layer of size $K$, where each neuron (blue circles in Fig. 1) is defined with an \"activation function\" $y^{k}(\\vec{w}^{(1)}; \\vec{x})$ for $k=1,\\dots,K$, and $\\vec{w}^{(1)}$ are parameters.\n",
+    "3. The output of the hidden layer is sent to the output layer (green circles in Fig. 1) $\\tilde{f}^{m}(\\vec{w}^{(2)};\\vec{y})$ for $m=1,\\dots,M$, and $\\vec{w}^{(2)}$ are parameters.\n",
     "\n",
     "The output $\\vec{\\tilde{f}}$ is thus a parametric function (in $\\vec{w}^{(1)},\\,\\vec{w}^{(2)}$), which can be trained to capture the target function $\\vec{f}$.\n",
     "\n",
     "\n",
     "![png](neural_network.png)\n",
     "\n",
     "<center>\n",
-    "<figcaption align = \"middle\"> Figure 1. A single layer classical neural network (taken from Wikipedia). Here the input size is $N=3$, the output size is $M=3$, and the hidden layer has $L=4$ neurons. </figcaption>\n",
+    "<figcaption align = \"middle\"> Figure 1. A single layer classical neural network (from Wikipedia). Here, the input size is $N=3$, the output size is $M=3$, and the hidden layer has $L=4$ neurons. </figcaption>\n",
     "</center>"
    ]
   },
@@ -45,7 +45,7 @@
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    },
    "source": [
-    "**Deep neural networks** - Deep neural networks are similar to the description above, having more than one hidden layer. This provides a more complex structure that can capture more complex functionalities."
+    "**Deep neural networks** are similar to the description above, having more than one hidden layer. This provides a more complex structure that can capture more complex functionalities."
    ]
   },
   {
@@ -55,16 +55,16 @@
     "jp-MarkdownHeadingCollapsed": true
    },
    "source": [
-    "### Quantum Neural Networks\n",
+    "## Quantum Neural Networks\n",
     "\n",
-    "The idea of a quantum neural network refers to combining parametric circuits as a replacement to all, or part of, classical layers in classical neural networks. The basic object in QNN is thus a **quantum layer**. A quantum layer has a classical input and it returns a classical output. The output is obtained by running a quantum program. A quantum layer is thus composed of three parts:\n",
+    "The idea of a quantum neural network refers to combining parametric circuits as a replacement for all or some of the classical layers in classical neural networks. The basic object in QNN is thus a **quantum layer**, which has a classical input and returns a classical output. The output is obtained by running a quantum program. A quantum layer is thus composed of three parts:\n",
     "1. A quantum part that encodes the input: This is a parametric quantum function for representing the entries of a single data point. There are three canonical ways to encode a data vector of size $N$: [angle-encoding](https://github.com/Classiq/classiq-library/blob/main/functions/qmod_library_reference/classiq_open_library/variational_data_encoding/variational_data_encoding.ipynb) using $N$ qubits, [dense angle-encoding](https://github.com/Classiq/classiq-library/blob/main/functions/qmod_library_reference/classiq_open_library/variational_data_encoding/variational_data_encoding.ipynb) using $\\lceil N/2\\rceil$ qubits, and amplitude-encoding using $\\lceil\\log_2N\\rceil$ qubits.\n",
     "2. A quantum ansatz part: This is a parametric quantum function, whose parameters are trained as the weights in classical layers.\n",
     "3. A postprocess classical part, for returning an output classical vector.\n",
     "\n",
     "The integration of quantum layers in classical neural networks may offer reduction in resources for a given functionality, as the network (or part of it) is expressed via the Hilbert space, providing different expressibility compared to classical networks.\n",
     "\n",
-    "This notebook demonstrates QNN by treating a specific function, the subset majority, for which we construct, train, and verify a hybrid classical-quantum neural network. The notebook assumes familiarity with Classiq and NN with PyTorch. See [QML guide with Classiq](https://github.com/Classiq/classiq-library/blob/main/tutorials/documentation_materials/user_guide/qml_with_classiq_guide/qml_with_classiq_guide.ipynb) ."
+    "This notebook demonstrates QNN by treating a specific function—the subset majority—for which we construct, train, and verify a hybrid classical-quantum neural network. The notebook assumes familiarity with Classiq and NN with PyTorch. See the [QML guide with Classiq](https://github.com/Classiq/classiq-library/blob/main/tutorials/documentation_materials/user_guide/qml_with_classiq_guide/qml_with_classiq_guide.ipynb)."
    ]
   },
   {
@@ -75,7 +75,7 @@
     "## Example: Hybrid Neural Network for the Subset Majority Function\n",
     "\n",
     "\n",
-    "For an integer $N$ and a given subset of indices $S \\subset \\{0,1,\\dots,N\\}$ we define the subset majority function, $M_{S}:\\{0,1\\}^{\\times N}\\rightarrow \\{0,1\\}$ that acts on binary strings of size $N$ as follows: it returns 1 if the number of ones within the a substring according to $S$ are larger than $|S|//2$, and 0 otherwise,\n",
+    "For an integer $N$ and a given subset of indices $S \\subset \\{0,1,\\dots,N\\}$ we define the subset majority function, $M_{S}:\\{0,1\\}^{\\times N}\\rightarrow \\{0,1\\}$ that acts on binary strings of size $N$ as follows: it returns 1 if the number of ones within the substring according to $S$ is larger than $|S|//2$, and 0 otherwise,\n",
     "$$\n",
     "M_S(\\vec{b}) = \\left\\{ \\begin{array}{l l }\n",
     "1 & \\text{if } \\sum_{j\\in S} b_{j}>|S|//2, \\\\\n",
@@ -84,7 +84,7 @@
     "\\right .\n",
     "$$\n",
     "\n",
-    "For example, we consider $N=7$ and $S=\\{0,1,4\\}$.\n",
+    "For example, we consider $N=7$ and $S=\\{0,1,4\\}$:\n",
     "* The string 0101110 corresponds to the substring 011, for which the number of ones is 2(>1). Therefore, $M_S(0101110)=1$.\n",
     "* The string 0011111 corresponds to the substring 001, for which the number of ones is 1(=1). Therefore, $M_S(0101110)=0$."
    ]
@@ -94,7 +94,7 @@
    "id": "ce5f70ee-11d1-44f9-9f12-219515f1250e",
    "metadata": {},
    "source": [
-    "### Generating data for a specific example\n",
+    "### Generating Data for a Specific Example\n",
     "\n",
     "Let us consider a specific example for our demonstration. We choose $N=10$ and generate all possible data of $2^N$ bit strings. We also take a specific subset $S=\\{1, 3, 4, 6, 7, 9\\}$."
    ]
@@ -208,15 +208,15 @@
    "id": "a5dd682a-73c5-4954-9fc2-934ae187b920",
    "metadata": {},
    "source": [
-    "### Constructing a hybrid network"
+    "### Constructing a Hybrid Network"
    ]
   },
   {
    "cell_type": "markdown",
    "id": "583a05af-fde0-4bc1-9d03-6574cd96cc8a",
    "metadata": {},
    "source": [
-    "We will build the following hybrid neural network:\n",
+    "We build the following hybrid neural network:\n",
     "\n",
     "**Data flattening $\\rightarrow$ A classical linear layer of size 10 to 4 with `Tanh` activation $\\rightarrow$ A qlayer of size 4 to 2 $\\rightarrow$ a classical linear layer of size 2 to 1 with `ReLU` activation.**"
    ]
@@ -226,15 +226,21 @@
    "id": "371cb77f-efac-4203-976a-94d426d9826c",
    "metadata": {},
    "source": [
-    "The classical layers can be defined with PyTorch built-in functions. The quantum layer is constructed with (1) a [dense angle-encoding](https://github.com/Classiq/classiq-library/blob/main/functions/qmod_library_reference/classiq_open_library/variational_data_encoding/variational_data_encoding.ipynb) function, (2) a simple ansatz with RY and RZZ rotations, and (3) a postprocess that is based on a measurement per qubit."
+    "The classical layers can be defined with PyTorch built-in functions. The quantum layer is constructed with \n",
+    "\n",
+    "(1) a [dense angle-encoding](https://github.com/Classiq/classiq-library/blob/main/functions/qmod_library_reference/classiq_open_library/variational_data_encoding/variational_data_encoding.ipynb) function \n",
+    "\n",
+    "(2) a simple ansatz with RY and RZZ rotations \n",
+    "\n",
+    "(3) a postprocess that is based on a measurement per qubit"
    ]
   },
   {
    "cell_type": "markdown",
    "id": "e8b2e803-7d75-474e-ae31-bfb4de049ae4",
    "metadata": {},
    "source": [
-    "#### The quantum layer"
+    "#### The Quantum Layer"
    ]
   },
   {
@@ -328,7 +334,7 @@
    "id": "88b4321f-ccfe-4b99-b480-475720157356",
    "metadata": {},
    "source": [
-    "#### The full hybrid network"
+    "#### The Full Hybrid Network"
    ]
   },
   {
@@ -391,15 +397,15 @@
    "id": "d822e3b0-3ee7-49a7-86ac-86d2cd6fbc93",
    "metadata": {},
    "source": [
-    "### Training and verifying the networks"
+    "### Training and Verifying the Networks"
    ]
   },
   {
    "cell_type": "markdown",
    "id": "fcfcb0fe-65f0-426a-a5b8-6c3ca56f33f2",
    "metadata": {},
    "source": [
-    "We define some hyperparameters such as loss function and optimization method, and a training function."
+    "We define some hyperparameters such as loss function and optimization method, and a training function:"
    ]
   },
   {
@@ -532,7 +538,7 @@
    "id": "c7307920-333a-4fc1-8a26-b76e266925ec",
    "metadata": {},
    "source": [
-    "### Training and verifying the network"
+    "### Training and Verifying the Network"
    ]
   },
   {