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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "713028f0",
+   "metadata": {},
+   "source": [
+    "# QoolQit Exercises — Module 1\n",
+    "## Graphs and Embedding\n",
+    "\n",
+    "Welcome to this hands-on introduction to **QoolQit**, the Python library for\n",
+    "algorithm development in the Rydberg analog model. Across four self-contained\n",
+    "modules you will learn all the building blocks of a QoolQit application, and\n",
+    "assemble them in a quantum application in the final module:\n",
+    "\n",
+    "| Module | Topic |\n",
+    "|--------|-------|\n",
+    "| 1 | Graphs and Embedding |\n",
+    "| 2 | Register, Drive and Quantum Programs |\n",
+    "| 3 | Compilation and Execution |\n",
+    "| 4 | Putting it all together: solving a QUBO |\n",
+    "\n",
+    "### In this module you will learn\n",
+    "- How to create graphs with the `DataGraph` class (pre-defined layouts,\n",
+    "  random graphs, graphs from raw data)\n",
+    "- The difference between *abstract* graphs and graphs *with coordinates*\n",
+    "- How to give coordinates to an abstract graph.\n",
+    "\n",
+    "\n",
+    "> **How to use this notebook.** \n",
+    "> - Cells marked **✏️ Exercise** contain gaps\n",
+    "> indicated by `...` or `# TODO` — replace them with working code following\n",
+    "> the instructions. \n",
+    "> - Cells marked **✅ Check** verify your answer: run them\n",
+    "> after completing the exercise. Everything else is provided and runs as-is.\n",
+    "> A separate **solution notebook** will be published.\n",
+    ">\n",
+    "> **API note:** we use qoolqit version 1.4"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6226deb9",
+   "metadata": {},
+   "source": [
+    "## 1. The `DataGraph` class\n",
+    "\n",
+    "In QoolQit, problems and atom layouts are described by graphs. The\n",
+    "`DataGraph` class (a subclass of `networkx.Graph`) is the central data\n",
+    "structure: it can hold **connectivity** (edges), **coordinates** (positions\n",
+    "of the nodes in the plane), and node and edge **weights**."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f89ebcdc",
+   "metadata": {},
+   "source": [
+    "### ✏️ Exercise 1.1 — Pre-defined graph layouts\n",
+    "\n",
+    "Create and draw three graphs:\n",
+    "1. `g_line`: a **line** graph with 5 nodes;\n",
+    "2. `g_circle`: a **circle** graph with 6 nodes and `spacing=1.0`;\n",
+    "3. `g_square`: a **square** grid graph with `m=3` rows and `n=3` columns.\n",
+    "\n",
+    "Use each graph's `.draw()` method to visualize it."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "af9b7821",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# TODO: create the three graphs\n",
+    "g_line = ...\n",
+    "g_circle = ...\n",
+    "g_square = ..."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "e506aeb8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(12, 4))\n",
+    "\n",
+    "# TODO: print the graphs on the subplots\n",
+    "# ..."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "eca26aea",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# ✅ Check\n",
+    "assert g_line.number_of_nodes() == 5\n",
+    "assert g_circle.number_of_nodes() == 6\n",
+    "assert g_square.number_of_nodes() == 9\n",
+    "print(\"All three graphs look right!\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a9b0848c",
+   "metadata": {},
+   "source": [
+    "## 2. Abstract graphs vs. graphs with coordinates\n",
+    "\n",
+    "Not every graph has coordinates. What is required is only their connectivity: edges and nodes. \n",
+    "A **random Erdős–Rényi graph**, for\n",
+    "instance, is purely *abstract*: it defines which nodes are connected, but\n",
+    "says nothing about where the nodes sit in the plane. Since neutral atoms live\n",
+    "in real space, sooner or later every graph needs coordinates. That is the\n",
+    "job of *embedding* (Section 4).\n",
+    "\n",
+    "Useful properties to interrogate a graph: `has_coords`, `has_edges`,\n",
+    "`has_node_weights`, `has_edge_weights`."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e3ffa662",
+   "metadata": {},
+   "source": [
+    "### ✏️ Exercise 1.2 — An abstract random graph\n",
+    "\n",
+    "1. Create `g_er`, an Erdős–Rényi random graph with `n=8` nodes, edge\n",
+    "   probability `p=0.4` and `seed=3` (for reproducibility).\n",
+    "2. Print whether it has coordinates (`has_coords`) and how many edges it has.\n",
+    "3. Try to compute `g_er.min_distance()` inside a `try/except Exception` block\n",
+    "   and print the error — distances make no sense without coordinates!"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "61d92e10",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# TODO: create the random ER graph\n",
+    "g_er = ...\n",
+    "\n",
+    "print(\"Has coordinates:\", ...)\n",
+    "print(\"Number of edges:\", g_er.number_of_edges())\n",
+    "\n",
+    "try:\n",
+    "    g_er.min_distance()\n",
+    "except AttributeError as err:\n",
+    "    print(\"As expected, this fails:\", err)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "724d4d0a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# ✅ Check\n",
+    "assert g_er.number_of_nodes() == 8\n",
+    "assert not g_er.has_coords, \"An ER graph should be abstract (no coordinates)\"\n",
+    "print(\"Correct — g_er is an abstract graph.\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2a2264fd",
+   "metadata": {},
+   "source": [
+    "## 3. Embedding: assign coordinates to an abstract graph or interaction matrix\n",
+    "\n",
+    "**Embedding** is the process of assigning coordinates to a graph. QoolQit can take an\n",
+    "  abstract graph and return the same graph *with coordinates* or take a symmetric matrix of *desired interactions* and return a graph whose node positions physically realize them (next section).\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d7a6df28",
+   "metadata": {},
+   "source": [
+    "### ✏️ Exercise 1.4 — Spring-layout embedding of the random graph\n",
+    "\n",
+    "1. Import `SpringLayoutEmbedder` from `qoolqit.embedding` and instantiate it.\n",
+    "2. Embed the abstract graph `g_er` from Exercise 1.2 into `g_er_embedded`.\n",
+    "3. Verify the embedded graph now has coordinates, print its `min_distance()`,\n",
+    "   and draw it."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "b0314e34",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# TODO: instantiate the embedder and embed g_er\n",
+    "embedder = ...\n",
+    "g_er_embedded = ...\n",
+    "\n",
+    "print(\"Has coordinates:\", g_er_embedded.has_coords)\n",
+    "print(\"Minimum distance:\", g_er_embedded.min_distance())\n",
+    "g_er_embedded.draw()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0f8d3fcb",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# ✅ Check\n",
+    "assert g_er_embedded.has_coords, \"The embedded graph should have coordinates\"\n",
+    "print(\"Spring-layout embedding successful!\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "de059158",
+   "metadata": {},
+   "source": [
+    "### ✏️ Exercise 1.5 — Embed a target interaction matrix\n",
+    "\n",
+    "1. Define the symmetric 3×3 target matrix\n",
+    "   `M = [[0, 1, 0.3], [1, 0, 0.5], [0.3, 0.5, 0]]` as a NumPy array.\n",
+    "2. Instantiate an `InteractionEmbedder` (from `qoolqit.embedding`) and embed\n",
+    "   `M` into `g_int`.\n",
+    "3. Draw `g_int`, then print its `interactions()` dictionary side by side with\n",
+    "   the corresponding entries of `M`. How close did the embedder get?\n",
+    "\n",
+    "> 💡 The `interactions()` method computes $1/r^6$ for each pair of nodes from\n",
+    "> the coordinates."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6b2d9b52",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "\n",
+    "# TODO: define the target interaction matrix\n",
+    "M = np.array(...)\n",
+    "\n",
+    "# TODO: embed it\n",
+    "g_int = ...\n",
+    "\n",
+    "g_int.draw()\n",
+    "\n",
+    "for (i, j), J in g_int.interactions().items():\n",
+    "    print(f\"pair ({i},{j}):  J = {J:.4f}   target M = {M[i, j]:.4f}\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "881fa9d1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# ✅ Check — every realized interaction within 5% of its target\n",
+    "for (i, j), J in g_int.interactions().items():\n",
+    "    assert abs(J - M[i, j]) < 0.05, f\"pair ({i},{j}) is off: {J} vs {M[i, j]}\"\n",
+    "print(\"Interaction embedding matches the target matrix!\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f8f2abe6",
+   "metadata": {},
+   "source": [
+    "### Next module\n",
+    "Graphs describe *data* and *geometry*. In **Module 2** we turn geometry into\n",
+    "physics: the `Register` of atoms, the time-dependent `Drive`, and the\n",
+    "`QuantumProgram` that combines them."
+   ]
+  }
+ ],
+ "metadata": {
+  "language_info": {
+   "name": "python"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}