From 149bc11b1d359efe9ed15d95489c1cc905959e29 Mon Sep 17 00:00:00 2001
From: Adrien Devolder <adridevolder@hotmail.com>
Date: Tue, 30 Jun 2026 14:44:52 -0400
Subject: [PATCH 01/21] Adding demo related to QMapDataset

---
 .../QMapDataset_demo/graph_circuit.png        | Bin 0 -> 239012 bytes
 .../QMapDataset_demo/graph_hardware.png       | Bin 0 -> 452901 bytes
 .../QMapDataset_demo/metadata.json            |  26 ++++++++++++++++++
 .../QMapDataset_demo/requirements.in          |   5 ++++
 4 files changed, 31 insertions(+)
 create mode 100644 demonstrations_v2/QMapDataset_demo/graph_circuit.png
 create mode 100644 demonstrations_v2/QMapDataset_demo/graph_hardware.png
 create mode 100644 demonstrations_v2/QMapDataset_demo/metadata.json
 create mode 100644 demonstrations_v2/QMapDataset_demo/requirements.in

diff --git a/demonstrations_v2/QMapDataset_demo/graph_circuit.png b/demonstrations_v2/QMapDataset_demo/graph_circuit.png
new file mode 100644
index 0000000000000000000000000000000000000000..846883711348174c58d328672fae5025884c1236
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diff --git a/demonstrations_v2/QMapDataset_demo/metadata.json b/demonstrations_v2/QMapDataset_demo/metadata.json
new file mode 100644
index 0000000000..97b480b0ae
--- /dev/null
+++ b/demonstrations_v2/QMapDataset_demo/metadata.json
@@ -0,0 +1,26 @@
+{
+  "title": "Solving the qubit maping problem with machine learning",
+  "authors": [
+    {
+      "username": "rscadrien"
+    }
+  ],
+  "executable_stable": true,
+  "executable_latest": true,
+  "dateOfPublication": "2026-06-30T00:00:00+00:00",
+  "dateOfLastModification": "2026-06-30T00:00:00+00:00",
+  "categories": [],
+  "tags": [],
+  "previewImages": [
+    {
+      "type": "thumbnail",
+      "uri": "/_static\\demo_thumbnails\\regular_demo_thumbnails\\thumbnail_placeholder.png"
+    }
+  ],
+  "seoDescription": "The demo introduces the qubit mapping problem and explores machine learning-based approaches for solving it.",
+  "doi": "",
+  "references": [],
+  "basedOnPapers": [],
+  "referencedByPapers": [],
+  "relatedContent": []
+}
\ No newline at end of file
diff --git a/demonstrations_v2/QMapDataset_demo/requirements.in b/demonstrations_v2/QMapDataset_demo/requirements.in
new file mode 100644
index 0000000000..1549d5a756
--- /dev/null
+++ b/demonstrations_v2/QMapDataset_demo/requirements.in
@@ -0,0 +1,5 @@
+### Please add any additional requirements that are needed for your demo here ###
+### If you are curious about global dependencies, check the `/dependencies` directory ###
+torch
+torch_geometric
+networkx

From a368a60b85494fd05526af08a8c4fd33aba19b43 Mon Sep 17 00:00:00 2001
From: Adrien Devolder <adridevolder@hotmail.com>
Date: Tue, 30 Jun 2026 15:31:56 -0400
Subject: [PATCH 02/21] Add Thumbnail sketch

---
 demonstrations_v2/QMapDataset_demo/demo.ipynb | 1527 +++++++++++++++++
 1 file changed, 1527 insertions(+)
 create mode 100644 demonstrations_v2/QMapDataset_demo/demo.ipynb

diff --git a/demonstrations_v2/QMapDataset_demo/demo.ipynb b/demonstrations_v2/QMapDataset_demo/demo.ipynb
new file mode 100644
index 0000000000..96f96d2fa7
--- /dev/null
+++ b/demonstrations_v2/QMapDataset_demo/demo.ipynb
@@ -0,0 +1,1527 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "c-eY0ni42lzT"
+   },
+   "source": [
+    "# **Solving the qubit mapping problem with machine learning**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The demo introduces the qubit mapping problem, an NP-hard challenge in quantum computing systems with limited hardware connectivity, and explores machine learning-based approaches for solving it. It presents the QMapDataset available in PennyLane and shows how it can be used to train a Graph Neural Network to tackle the qubit mapping problem."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "kuwdpYNS2oGq"
+   },
+   "source": [
+    "## **1. What is the qubit mapping problem?**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "WEALQzoz2pbU"
+   },
+   "source": [
+    "When we write quantum circuits (with Pennylane, of course!), we usually assume that all the qubits are connected to each other, or more precisely, that two-qubit gates can be applied between any qubits. While some infrastructures have all-to-all connectivity (like the photonic devices developed by Xanadu, the ion or the neutral atom platforms), superconducting quantum processors have fixed connectivity between qubits. Only physically connected qubits can directly interact. For example, the first generation of superconducting devices had 5 qubits with a simple linear connectivity:\n",
+    "\n",
+    "$pq_0-pq_1-pq_2-pq_3-pq_4$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "7PBZHOPq2ygy"
+   },
+   "source": [
+    "Physical qubit 0 and physical qubit 3  are not directly connected. If a two-qubit gate is required in your code between qubits 0 and 3, it is necessary to apply SWAP gates. SWAP gates exchange the states of two neighboring qubits for all basis states in the superposition (more details here [What is a SWAP gate? | PennyLane](https://pennylane.ai/qml/glossary/what-is-a-swap-gate)).In our example, we would need to apply two SWAP gates: one between $pq_0$ and $pq_1$, and one between $pq_1$ and $pq_2$. If the state was $\\frac{1}{\\sqrt{2}}(|10000\\rangle + |00001\\rangle)$ before the SWAP gates, it becomes $\\frac{1}{\\sqrt{2}}(|00100\\rangle + |00001\\rangle)$. Then, we can apply the CNOT gate between $pq_2$ and $pq_3$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "lmp4qozV2rDT"
+   },
+   "source": [
+    "The lack of full connectivity between physical qubits does not alter the universality of quantum computation in these devices but leads to an increase in the depth of the effective quantum circuit. A SWAP gate corresponds to 3 CNOT gates. Therefore, instead of 1 CNOT gate, we have 7 CNOT gates. This increase can be fatal for your quantum computation, especially on a NISQ device."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "q5b4849Pihlp"
+   },
+   "source": [
+    "Indeed, NISQ devices are sensitive to noise. In particular, two-qubit gates are noisy. Therefore, when a large number of two-qubit gates is applied on a NISQ device, the quantum information is progressively lost and randomized. Therefore, when running a quantum circuit on a NISQ device, we try to minimize the number of two-qubit gates. Obviously, the increase by a factor of 7 in our example is terrible for the noise robustness of our quantum computation. Even for fault-tolerant quantum computers, reducing the number of CNOT gates would help to reduce the time of computation and therefore the computational cost.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "sZ2I1TJTihjo"
+   },
+   "source": [
+    "It is here that qubit mapping can help mitigate the number of SWAP gates. There is no rule that constrains the first qubit of your quantum circuit to be the first physical qubit, the second qubit of your quantum circuit to be the second physical qubit, etc. We can separate the two sets of qubits: on one side, we have the qubits of our quantum circuit, called logical qubits and denoted $lq_0$, $lq_1$, etc., and on the other side, the physical qubits of the hardware denoted $pq_0$, $pq_1$ etc.. There does not necessarily have to be an identity mapping between the two. Instead, we can define any mapping: $\\{lq\\}\\rightarrow \\{pq\\}$. The qubit mapping problem corresponds to finding the initial mapping between the logical qubits and the physical qubits that minimizes the number of SWAP gates during the execution of the quantum circuit by the hardware. In our simple example, if we assign $lq_0\\rightarrow pq_0$ and $lq_3\\rightarrow pq_1$, a CNOT gate between physical qubits 0 and 1 corresponds to a CNOT gate between logical qubits 0 and 3, as desired."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "51F9nSFTihgr"
+   },
+   "source": [
+    "Let's consider a slightly more complex quantum circuit:\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "fnHdlxsoi5bt"
+   },
+   "outputs": [],
+   "source": [
+    "import pennylane as qp\n",
+    "dev = qp.device(\"default.qubit\", wires=5)\n",
+    "@qp.qnode(dev)\n",
+    "def  circuit():\n",
+    "    qp.CNOT(wires=[0,4])\n",
+    "    qp.CNOT(wires=[1,3])\n",
+    "    qp.CNOT(wires=[0,2])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "ST0kkexFiheg"
+   },
+   "source": [
+    "So, there are 3 CNOT gates: one between $lq_0$ and $lq_4$, one between $lq_1$ and $lq_3$, and one between $lq_0$ and $lq_2$. Let’s first examine the number of SWAP gates with the identity mapping ($lq_0\\rightarrow pq_0$, $lq_1\\rightarrow pq_1$, etc.):\n",
+    "\n",
+    "$pq_0(lq_0)-pq_1(lq_1)-pq_2(lq_2)-pq_3(lq_3)-pq_4(lq_4)$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "WeSqQqCEihcq"
+   },
+   "source": [
+    "Here, in parentheses, the logical qubit mapped to the corresponding physical qubit is shown.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "pUaHUc5zihar"
+   },
+   "source": [
+    "For the first CNOT gate between $lq_0$ and $lq_4$, we need to apply 3 SWAP gates (SWAP($pq_0$, $pq_1$), SWAP($pq_1$, $pq_2$), and SWAP($pq_2$, $pq_3$)). The mapping becomes:\n",
+    "\n",
+    "$pq_0(lq_1)-pq_1(lq_2)-pq_2(lq_3)-pq_3(lq_0)-pq_4(lq_4)$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "Dy7H4lyZihYw"
+   },
+   "source": [
+    "Afterward, for the CNOT gate between $lq_1$ and $lq_3$, we need a SWAP gate between $pq_1$ and $pq_2$:\n",
+    "\n",
+    "$pq_0(lq_1)-pq_1(lq_3)-pq_2(lq_2)-pq_4(lq_0)-pq_4(lq_4)$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "JFzvFKx5ihVq"
+   },
+   "source": [
+    "Finally, for the CNOT gate between $lq_0$ and $lq_2$, we do not need any additional SWAP gates. Therefore, in total, there are 4 SWAP gates."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "3NOJYBf1ihTm"
+   },
+   "source": [
+    "Now, let’s consider the following initial mapping:\n",
+    "\n",
+    "$pq_0(lq_1)-pq_1(lq_3)-pq_2(lq_2)-pq_3(lq_0)-pq_4(lq_4)$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "FX_QRYLRihRg"
+   },
+   "source": [
+    "In this case, there is no need for SWAP gates. The three CNOT gates can be applied directly. We go from 15 CNOT gates to 3 CNOT gates, a reduction by a factor of 5 just by changing the initial mapping!"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "aZhhKNRLkDJV"
+   },
+   "source": [
+    "Now that the qubit mapping problem is clear from these 5-qubit examples, let’s give a generalized definition. We start with:\n",
+    "\n",
+    "* A quantum circuit C defined on n logical qubits $lq_0, lq_1, \\dots, lq_{n-1}$\n",
+    "* A quantum device with m physical qubits $pq_0, pq_1, \\dots, pq_{m-1}$ connected by some graph $G_D$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "q1Ygx0QskDGA"
+   },
+   "source": [
+    "The goal of the qubit mapping problem is to find the initial mapping $\\pi_0: \\{lq\\}\\rightarrow \\{pq\\}$ in order to minimize the number of SWAP gates during the execution of the quantum circuit C on the quantum device. Note that in general, the number of logical qubits $n$ is smaller than the number of physical qubits $m$ (and cannot be bigger)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "fbEgFJoXkDEK"
+   },
+   "source": [
+    "The initial qubit mapping problem can be represented as an assignment matrix A of dimension $n\\times m$:\n",
+    "\n",
+    "* $A_{ij}=1$ if logical qubit $lq_i$ is mapped to the physical qubit $pq_j$.\n",
+    "* $A_{ij}=0$ otherwise\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "WSFJAehjkDCK"
+   },
+   "source": [
+    "There are two constraints on the assignment matrix A :\n",
+    "* Each logical qubit is assigned exactly one physical qubit $\\sum_{j=1}^m A_{ij} = 1, ~\\forall i$\n",
+    "\n",
+    "* Each physical qubit holds at most one logical qubit $\\sum_{i=1}^n A_{ij}\\leq 1, ~\\forall j$\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "RzC9Q3GekDAH"
+   },
+   "source": [
+    "An important quantity is the distance between two physical qubits $D_{ij}=d(pq_i, pq_j)$. The distance is the shortest-path length between two physical qubits in the hardware connectivity graph, and it directly estimates how many SWAPs are needed to make them interact. Of course, the distance would depend on the hardware connectivity graph. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "2jRa1MbckC-c"
+   },
+   "source": [
+    "We will explore in the next section the actual methods used to find the optimal assignment matrix A and therefore to solve the qubit mapping problem.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "2-izzdBQkC8P"
+   },
+   "source": [
+    "## **2.How to solve the qubit mapping problem?**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "6xsztWspkC54"
+   },
+   "source": [
+    "The qubit mapping problem is unfortunately a NP problem and is exponentially hard to find the optimal solution when the number of logical and physical qubits increases. However, there are some heuristic methods that allow us to solve it with a reasonable computational time. The most famous heuristic algorithm is the SABRE algorithm, used for example by Qiskit in its compilation.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "iKXYaZzlkC3_"
+   },
+   "source": [
+    "The SABRE algorithm works by iteratively updating the qubit mapping guided by a cost function based on the total distance.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "s0RGC159kC1-"
+   },
+   "source": [
+    "**Forward propagation**\n",
+    "1. **Initialize the mapping**: The algorithm starts by initializing a single mapping between logical and physical qubits (typically random or identity).\n",
+    "Example (5-qubit device): $pq_0(lq_0)-pq_1(lq_2)-pq_2(lq_3)-pq_3(lq_4)-pq_4(lq_1)$\n",
+    "\n",
+    "2. **Define the front layer**: The front layer is the set of gates that are ready to be executed, meaning all their dependencies in the circuit have been satisfied. For the quantum circuit (CNOT(0,4), CNOT(1,3) and CNOT(0,2), the front layer is {CNOT(0,4), CNOT(1,3)}. CNOT(0,2) is not included because it depends on the execution of CNOT(0,4).\n",
+    "\n",
+    "3. **Check gate executability**: For each gate in the front layer, the algorithm checks whether the mapped physical qubits are connected in the hardware graph:\n",
+    "If they are adjacent, the gate “is executed” and is withdrawn from the front layer.\n",
+    "If not, SWAP gate is required and step 4 is applied.\n",
+    "In our case, the CNOT(0,4) and CNOT(1,3) cannot be executed.\n",
+    "\n",
+    "4. **Compute routing cost**: The algorithm calculates the sum of the distance of the operations in the front layer:\n",
+    "$H = \\sum_{(a,b)} d(\\pi(a), \\pi(b))$\n",
+    "\n",
+    "5. **Evaluate SWAP candidates**: The algorithm considers SWAPs on edges of the hardware graph. For each candidate SWAP, they recompute the cost H and select the SWAP that minimizes the cost.\n",
+    "In our case, the SWAP(2,3) gives the distance $H=2+1=3$, which is better than other options.  So this SWAP is applied.\n",
+    "\n",
+    "6. **Iterate forward pass**: Steps 2–5 are repeated. The front layer is updated dynamically as gates become executable. Gates are executed whenever possible. SWAPs are inserted when necessary. At the end of this process, the algorithm produces a final mapping  for the forward pass.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "wo5jtgGJkC0A"
+   },
+   "source": [
+    "**Backward propagation**\n",
+    "\n",
+    "7. **Reverse circuit execution**: The algorithm then repeats the same procedure, but on the reversed circuit (executing gates in reverse order), starting from the final mapping obtained in the forward pass.\n",
+    "\n",
+    "8. **Improved initial mapping**: At the end of the backward pass, a new initial mapping $\\pi_0$ is obtained. This initial mapping is typically better than the original random initialization."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "fsVnXeFCkCx2"
+   },
+   "source": [
+    "In practice, SABRE may repeat multiple forward and backward passes (usually around 3 iterations), gradually improving the mapping until no significant improvement is observed.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "96vTzEyskCv4"
+   },
+   "source": [
+    "The SABRE algorithm has a polynomial computational complexity of approximately $\\mathcal{O}(Tm^2 n)$ where $T$ is the total number of two-qubit gates in the quantum circuit. This is significantly better than the exponential scaling of an exact solution. However, as the number of qubits in superconducting processors continues to increase, the runtime of the SABRE algorithm also grows accordingly.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "mOyw8ouzkCt6"
+   },
+   "source": [
+    "Machine learning models could offer better scalability while maintaining high accuracy.\n",
+    "In this demo, we consider supervised learning. Supervised learning requires a dataset with input data and corresponding labels to train the model. This is the purpose of QMapDataset, which provides datasets and tools to generate them for solving the qubit mapping problem using supervised machine learning models. QMapDataset is available in PennyLane. We will explore its features in the next section.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "uEh-QmbwkCsD"
+   },
+   "source": [
+    "## **3. QMapDataset**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "C6PRla4WmADC"
+   },
+   "source": [
+    "QMapDataset is a dataset generator for the qubit mapping problem, designed for IBM quantum devices. The dataset is built from three main components: the hardware, the circuits, and the mapping.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "DQav8AHGmABJ"
+   },
+   "source": [
+    "**For the hardware**, First, we choose an IBM quantum processor. To make the dataset more diverse, the code can randomly permute the properties of the qubits and gates (data augmentation). We can control how often these permutations are applied.  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "U7WANYQZl__0"
+   },
+   "source": [
+    "Then, the code collects all the relevant hardware information:\n",
+    "\n",
+    "* physical qubit properties ($T_1$, $T_2$, readout error)\n",
+    "* single-qubit gate errors for each qubit\n",
+    "* two-qubit gate errors for each connected pair of qubits\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "dySVIh7Pl_9Z"
+   },
+   "source": [
+    "**For the circuits**, These are sampled either from:\n",
+    "\n",
+    "* random quantum circuits\n",
+    "* well-known circuits (QPE, Grover, QFT, GHZ state preparation etc.)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "sWa_g0APl_7y"
+   },
+   "source": [
+    "By default, the choice is split 50–50 between random and well-known circuits.\n",
+    "\n",
+    "For random quantum circuits, the code also sample:\n",
+    "* the number of logical qubits\n",
+    "* the circuit depth\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "vvrkGrV-l_5l"
+   },
+   "source": [
+    "The number of logical qubits is chosen between 3 and the number of physical qubits available on the hardware. The depth follows this distribution:  25% shallow (5–15 layers), 40% medium (16–50), 30% deep (51–150), and 5% very deep circuits (151–400)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "ts3eQZQxl_3o"
+   },
+   "source": [
+    "To further enrich the dataset, we apply random permutations of qubits to the well-known circuits (since there are only a few of them). After that, the code collects:\n",
+    "\n",
+    "* Single-qubit gate counts for each logical qubit\n",
+    "* Two-qubit gate counts for each pair of logical qubits\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "7JvG3xxLl_1U"
+   },
+   "source": [
+    "Finally, **for the mapping**, the code uses the generated hardware and circuits to compute the qubit mapping with an advanced version of the SABRE algorithm (optimization_level = 3 in Qiskit)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "kS5ZsjiLkCqF"
+   },
+   "source": [
+    "If you want more details, you can have a look at the GitHub repo: [rscadrien/QMapDataset: Generator of dataset for qubit mapping](https://github.com/rscadrien/QMapDataset)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "CnOk-TREkCHV"
+   },
+   "source": [
+    "The good news is that you do not need to generate this dataset using QMapDataset yourself! We have already prepared datasets for three different types of IBM quantum processors: Eagle 3, Heron 1, and Heron 2. Each dataset contains 10,000 samples, where each sample consists of a circuit, hardware description, and mapping.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "xXjty4hDkCJv"
+   },
+   "source": [
+    "Let’s see how to use these datasets with PennyLane and extract their information.\n",
+    "\n",
+    "The first step is to load a dataset from the Xanadu server. For example, to load the dataset for the Eagle 3 architecture:\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "cFmt01_Om4QP"
+   },
+   "outputs": [],
+   "source": [
+    "import pennylane as qp\n",
+    "[ds] = qp.data.load(\"other\", name=\"qubit-mapping-eagle3\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "tH1A-toZm48N"
+   },
+   "source": [
+    "For the Heron 1 and Heron 2 architectures, you only need to replace the name argument with \"qubit-mapping-heron1\" or \"qubit-mapping-heron2\", respectively.\n",
+    "\n",
+    "Each dataset is divided into three attributes: circuits, hardware, and mappings, following the structure discussed earlier. For example, to access the first sample:\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "M90bkqSPm6sV"
+   },
+   "outputs": [],
+   "source": [
+    "sample0_circuit = ds.circuits[0]\n",
+    "sample0_hardware = ds.hardware[0]\n",
+    "sample0_mapping = ds.mappings[0]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "J5wH6Bd1m4wz"
+   },
+   "source": [
+    "Each of these is a dictionary containing the features of the circuit, hardware, or mapping. For example, to access the single-qubit and two-qubit gate counts of the quantum circuit:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "kw_78wVvm9cD"
+   },
+   "outputs": [],
+   "source": [
+    "Single_qubit_count = sample0_circuit[\"single_qubit_counts\"]\n",
+    "Two_qubit_count = sample0_circuit[\"two_qubit_counts\"]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "DXjNk5vNm-WP"
+   },
+   "source": [
+    "To retrieve $T_1$, $T_2$, and the two-qubit gate error from the hardware data:\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "Klm95l0LnBOw"
+   },
+   "outputs": [],
+   "source": [
+    "T_1 = sample0_hardware[\"T1\"]\n",
+    "T_2 = sample0_hardware[\"T2\"]\n",
+    "Two_qubit_error = sample0_hardware[\"ecr\"]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "mrQ1aFYXnBrh"
+   },
+   "source": [
+    "Finally, to obtain the mapping (which can serve as the label in a supervised learning model):\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "nnj5S3fJnDFU"
+   },
+   "outputs": [],
+   "source": [
+    "mapping = sample0_mapping[\"final_mapping\"]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "F6SduSotnJJ8"
+   },
+   "source": [
+    "Detailed information about all available features and how to access them can be found in the “Data” section of the Qubit Mapping Eagle 3 page on the PennyLane website. ([Qubit Mapping Eagle 3](https://pennylane.ai/datasets/qubit-mapping-eagle3) )."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "R16FjUi_nQQs"
+   },
+   "source": [
+    "## **4.Building a machine learning model to solve the qubit mapping problem**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "RQLfsWGbnRFc"
+   },
+   "source": [
+    "In this session, we will show an example of how to preprocess data from QMapDataset and how to build a machine learning model that can solve the qubit mapping problem."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "HhmHF0bRnYq3"
+   },
+   "source": [
+    "### **4.1 Preprocessing**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "LZ-DWZAKnZQp"
+   },
+   "source": [
+    "The qubit mapping problem strongly depends on the connectivity between the physical qubits in the given hardware, as well as the connectivity between the logical qubits in the circuits (represented by two-qubit gate counts). Therefore, a graph is a natural representation for solving the qubit mapping problem. We will see how to construct graphs for both the circuits and the hardware using the QMapDataset available in PennyLane.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "NabNn5wKnak5"
+   },
+   "source": [
+    "A graph is a way of showing relationships. It is made up of points, called nodes (or vertices), that are connected by lines, called edges. Both nodes and edges can also have extra information attached to them, called features.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "GXahu_-pnbgd"
+   },
+   "source": [
+    "For the **circuit**, the nodes represent the logical qubits, and edges connect pairs of logical qubits between which two-qubit gates are applied in the quantum circuit. The features of each node are the counts of different single-qubit gates applied to the corresponding logical qubit. The features of each edge are the counts of different two-qubit gates applied between the two logical qubits. Since the relative values of these features are what matter, we standardize them. This is the code to build the graph for the quantum circuit of the QMapDataset:\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "dUql_BFlneiy"
+   },
+   "outputs": [],
+   "source": [
+    "import torch\n",
+    "from torch_geometric.data import Data\n",
+    "def build_circuit_graph (circuit_data):\n",
+    "    # -----------------------------\n",
+    "    # STEP 1: Extract and scale single-qubit gate counts\n",
+    "    # Each function returns a vector of size (n_qubits,)\n",
+    "    # representing how often each gate was applied per qubit\n",
+    "    # -----------------------------\n",
+    "    rz_counts_scaled = get_single_qubit_counts(circuit_data, \"rz\")\n",
+    "    sx_counts_scaled = get_single_qubit_counts(circuit_data, \"sx\")\n",
+    "    x_counts_scaled = get_single_qubit_counts(circuit_data, \"x\")\n",
+    "\n",
+    "    # -----------------------------\n",
+    "    # STEP 2: Extract and scale two-qubit gate counts\n",
+    "    # These describe interactions between pairs of qubits\n",
+    "    # -----------------------------\n",
+    "    ecr_scaled = get_two_qubit_counts(circuit_data,'ecr')\n",
+    "\n",
+    "    # -----------------------------\n",
+    "    # STEP 3: Build node feature matrix\n",
+    "    # Each node = a logical qubit\n",
+    "    # Features = [rz count, sx count, x count]\n",
+    "    # Shape: (n_qubits, 3)\n",
+    "    # -----------------------------\n",
+    "    X =torch.tensor(\n",
+    "    list(zip(rz_counts_scaled, sx_counts_scaled, x_counts_scaled)),\n",
+    "    dtype=torch.float\n",
+    "    )\n",
+    "    # -----------------------------\n",
+    "    # STEP 4: Build graph connectivity (edges)\n",
+    "    # edge_index: which qubits are connected\n",
+    "    # edge_attr: weight/feature of each edge (scaled gate counts)\n",
+    "    # -----------------------------\n",
+    "    ecr_index, ecr_attr = edge_info(ecr_scaled)\n",
+    "    # -----------------------------\n",
+    "    # STEP 5: Create PyTorch Geometric graph object\n",
+    "    # x -> node features\n",
+    "    # edge_index -> connectivity (COO format)\n",
+    "    # edge_attr -> edge features\n",
+    "    # -----------------------------\n",
+    "    G_circuit = Data(\n",
+    "    x=X,\n",
+    "    edge_index=ecr_index,\n",
+    "    edge_attr=ecr_attr,\n",
+    "    )\n",
+    "    return G_circuit\n",
+    "\n",
+    "def get_single_qubit_counts(data, gate_name):\n",
+    "    # Number of logical qubits in the circuit\n",
+    "    n = data[\"n_logical_qubits\"]\n",
+    "    # Try to retrieve counts for the given gate type\n",
+    "    counts = data.get(\"single_qubit_counts\", {}).get(gate_name, None)\n",
+    "    # If no data exists, return a zero vector\n",
+    "    if counts is None:\n",
+    "        return np.zeros(n)\n",
+    "    else:\n",
+    "    # Otherwise normalize counts across qubits (z-score normalization)\n",
+    "        return standard_scaling_across_qubits(np.array(counts))\n",
+    "\n",
+    "def get_two_qubit_counts(data, gate_name):\n",
+    "    # Safely retrieve list of two-qubit gate interactions\n",
+    "    counts = data.get(\"two_qubit_counts\", {}).get(gate_name, [])\n",
+    "\n",
+    "    # If no interactions exist, return empty list\n",
+    "    if not counts:\n",
+    "        return counts\n",
+    "    else:\n",
+    "    # Otherwise scale the edge weights\n",
+    "      return scaled_two_qubit_list(counts)\n",
+    "\n",
+    "def standard_scaling_across_qubits(vector):\n",
+    "    # Compute mean and standard deviation for normalization\n",
+    "    mean = vector.mean()\n",
+    "    std = vector.std()\n",
+    "    # Avoid division by zero if all values are identical\n",
+    "    if std == 0 :\n",
+    "        vector_scaled = vector\n",
+    "    else:\n",
+    "        vector_scaled = (vector - mean)/std\n",
+    "    return vector_scaled\n",
+    "\n",
+    "def scaled_two_qubit_list(two_data):\n",
+    "    # Extract raw edge weights from dictionary list\n",
+    "    values = np.array([d['value']for d in two_data], dtype=float)\n",
+    "    # Normalize weights\n",
+    "    scaled_values = standard_scaling_across_qubits(values)\n",
+    "    # Reattach scaled values back into original edge dictionaries\n",
+    "    two_gate_scaled = [\n",
+    "    {**d, 'value':v_scaled}\n",
+    "    for d, v_scaled in zip(two_data, scaled_values)\n",
+    "    ]\n",
+    "    return two_gate_scaled\n",
+    "\n",
+    "def edge_info(gate_dict):\n",
+    "    # Lists to store graph connectivity\n",
+    "    edge_index = []\n",
+    "    edge_attr = []\n",
+    "    # Convert list of edge dictionaries into tensor format\n",
+    "    for gate in gate_dict:\n",
+    "        i,j,w = gate[\"row\"], gate[\"col\"], gate[\"value\"]\n",
+    "        edge_index.append([i,j])\n",
+    "        edge_attr.append([w])\n",
+    "    # Convert to PyTorch tensors\n",
+    "    # edge_index shape: [2, num_edges]\n",
+    "    edge_index = torch.tensor(edge_index, dtype=torch.long).T\n",
+    "    edge_attr = torch.tensor(edge_attr, dtype=torch.float)\n",
+    "    return edge_index, edge_attr"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "Q3NbrOeVnfQ5"
+   },
+   "source": [
+    "For example, if this code is applied for the circuit of the 27th sample, we obtain:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "TNXkhztEngJd"
+   },
+   "outputs": [],
+   "source": [
+    "Graph_circuit = build_circuit_graph(ds.circuits[26])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "EZiBosOnnhGZ"
+   },
+   "source": [
+    "Using the networkx package, we can visualize the graph using the following graph:\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "vik1-T78nkY-"
+   },
+   "outputs": [],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import networkx as nx\n",
+    "from torch_geometric.utils import to_networkx\n",
+    "# Convert PyG Data object to NetworkX graph\n",
+    "G = to_networkx(\n",
+    "    test,\n",
+    "    edge_attrs=[\"edge_attr\"],  # keep CX counts\n",
+    "    node_attrs=[\"x\"],          # keep node features\n",
+    "    to_undirected=False\n",
+    ")\n",
+    "edge_labels = {\n",
+    "    (u, v): f\"{d['edge_attr'][0]:.2f}\"\n",
+    "    for u, v, d in G.edges(data=True)\n",
+    "}\n",
+    "nx.draw(\n",
+    "    G,\n",
+    "    pos,\n",
+    "    with_labels=True,\n",
+    "    node_size=800,\n",
+    ")\n",
+    "\n",
+    "nx.draw_networkx_edge_labels(\n",
+    "    G,\n",
+    "    pos,\n",
+    "    edge_labels=edge_labels\n",
+    ")\n",
+    "\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "-74N13M3nmml"
+   },
+   "source": [
+    "We obtain the following graph:\n",
+    "![My graph](graph_circuit.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "DeAOr5biodV6"
+   },
+   "source": [
+    "For the **hardware**, the nodes represent the physical qubits, and edges connect pairs of physical qubits that are directly connected in the device architecture. The features of each node are the $T_1$ and $T_2$ coherence times, readout error, and single-qubit gate errors for each physical qubit. The features of each edge are the errors of the different two-qubit gates applied between connected physical qubits. As with the circuits, the feature values are standardized. The code to build the graph for the hardware is pretty similar to the one for the circuit:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "mAFplmqGn6tl"
+   },
+   "outputs": [],
+   "source": [
+    "def build_hardware_graph(hardware_data):\n",
+    "    # Scaling single qubit properties and gate\n",
+    "    T1_scaled = standard_scaling_across_qubits(np.array(hardware_data['T1']))\n",
+    "    T2_scaled = standard_scaling_across_qubits(np.array(hardware_data['T2']))\n",
+    "    readout_error_scaled = standard_scaling_across_qubits(np.array(hardware_data['readout_error']))\n",
+    "    sx_error_scaled = standard_scaling_across_qubits(np.array(hardware_data['sx_error']))\n",
+    "    x_error_scaled = standard_scaling_across_qubits(np.array(hardware_data['x_error']))\n",
+    "    #Scaling two qubit gate properties\n",
+    "    cx_error_scaled = scaled_two_qubit_list(hardware_data['cx'])\n",
+    "    #Build the node feature matrix\n",
+    "    X = torch.tensor(\n",
+    "    list(zip(T1_scaled, T2_scaled, readout_error_scaled, sx_error_scaled, x_error_scaled)),\n",
+    "    dtype=torch.float\n",
+    "    )\n",
+    "    #Build the edge index and attributes matrix\n",
+    "    cx_index, cx_attr = edge_info(cx_error_scaled)\n",
+    "    #Build the graph\n",
+    "    G_hardware = Data(\n",
+    "    x=X,\n",
+    "    edge_index=cx_index,\n",
+    "    edge_attr=cx_attr,\n",
+    "    )\n",
+    "    return G_hardware"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "gUAwb-arokXE"
+   },
+   "source": [
+    "We can again visualize the graph using networkx:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Draw graph\n",
+    "\n",
+    "plt.figure(figsize=(10, 10))\n",
+    "\n",
+    "pos = nx.spring_layout(\n",
+    "    G_hardware,\n",
+    "    seed=42,\n",
+    "    k=1 / np.sqrt(len(G_hardware)),\n",
+    "    iterations=200\n",
+    ")\n",
+    "\n",
+    "nx.draw_networkx_edges(G_hardware, pos, alpha=0.2, width=0.5)\n",
+    "nx.draw_networkx_nodes(G_hardware, pos, node_size=40)\n",
+    "\n",
+    "plt.axis(\"off\")\n",
+    "plt.savefig(\"graph_hardware.png\", dpi=300, bbox_inches=\"tight\")\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "![My graph](graph_hardware.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "gu9V-yu3o1Sg"
+   },
+   "source": [
+    "### **4.2 Machine learning model**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "ub3VapYRo1PD"
+   },
+   "source": [
+    "We will show how to build a model that can learn qubit mapping. For this model, we will use an architecture adapted to the graph structure of the input: a Graph Neural Network (GNN). The key principle behind GNNs is that a node learns by aggregating information from its neighbors. A GNN typically works in layers, and each layer is mostly composed of three steps:\n",
+    "\n",
+    "* Message passing : Each node receives information from its neighbors\n",
+    "* Aggregation: Combine neighbor information (e.g., sum, mean, max)\n",
+    "* Update: Use a neural network to update the node’s embedding.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "bcHuT4jWo1NN"
+   },
+   "source": [
+    "In our case, we will use two separate GNNs: one for the quantum circuit and one for the hardware. After several GNN layers, we obtain updated embedding vectors for the nodes of both graphs, $h_L^{(i)}$ and $h_P^{(j)}$. ($h_L^{(i)}$ is the vector of the i th node of the quantum circuit graph while $h_P^{(j)}$ is the vector of the jth node of the hardware graph).\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "4cfX4ur8o1LG"
+   },
+   "source": [
+    " We then compare these embeddings using a dot product:\n",
+    "\n",
+    "$S_{ij}=h_L^{(i)}\\cdot h_P^{(j)}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "LtUfHGxXo1JC"
+   },
+   "source": [
+    "The matrix  element $S_{ij}$ represents the logit that the logical qubit $lq_i$ is mapped to the physical qubit $pq_j$. The probability that logical qubit k maps to physical qubit j is calculated from the logit:\n",
+    "\n",
+    "$p_{ij}=\\frac{\\text{exp}(S_{ij})}{\\sum_{j^\\prime} \\text{exp}(S_{ij}^\\prime)}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "G1KJt8P9o0le"
+   },
+   "source": [
+    " During training, the matrix $p_{ij}$ should ideally converge to the assignment matrix $A_{ij}$ presented in Section 2, where each row contains exactly one nonzero element equal to 1. After training, during inference, the strict assignment constraints are enforced by applying the Hungarian algorithm.  Unfortunately, this algorithm is not differentiable and therefore cannot be used during training.The code for this architecture is as follows :\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "NlJMmFyzpkdB"
+   },
+   "outputs": [],
+   "source": [
+    "class QubitMapping_dot_Model(nn.Module):\n",
+    "    def __init__(self, node_c_dim, node_h_dim, edge_c_dim, edge_h_dim, hidden_dim):\n",
+    "        super().__init__()\n",
+    "\n",
+    "        # === GNN for the LOGICAL CIRCUIT graph ===\n",
+    "        # These layers learn embeddings for logical qubits based on circuit structure\n",
+    "\n",
+    "        # First layer: edge-conditioned convolution\n",
+    "        # Edge features (e.g., gate types) generate weights dynamically\n",
+    "        self.nn_c1 = make_edge_mlp(edge_c_dim, node_c_dim * hidden_dim)\n",
+    "        self.gnn_c1 = NNConv(node_c_dim, hidden_dim, nn=self.nn_c1, aggr='add')\n",
+    "\n",
+    "        # Second layer: refines embeddings\n",
+    "        self.nn_c2 = make_edge_mlp(edge_c_dim, hidden_dim * hidden_dim)\n",
+    "        self.gnn_c2 = NNConv(hidden_dim, hidden_dim, nn=self.nn_c2, aggr='add')\n",
+    "\n",
+    "        # === GNN for the HARDWARE graph ===\n",
+    "        # These layers learn embeddings for physical qubits based on hardware connectivity\n",
+    "\n",
+    "        self.nn_h1 = make_edge_mlp(edge_h_dim, node_h_dim * hidden_dim)\n",
+    "        self.gnn_h1 = NNConv(node_h_dim, hidden_dim, nn=self.nn_h1, aggr='add')\n",
+    "\n",
+    "        self.nn_h2 = make_edge_mlp(edge_h_dim, hidden_dim * hidden_dim)\n",
+    "        self.gnn_h2 = NNConv(hidden_dim, hidden_dim, nn=self.nn_h2, aggr='add')\n",
+    "\n",
+    "\n",
+    "    def forward(self, G_circuit, G_hardware):\n",
+    "\n",
+    "        # === Step 1: Compute embeddings for LOGICAL qubits ===\n",
+    "        # Each node becomes a vector summarizing its role in the circuit\n",
+    "        x_c = F.relu(self.gnn_c1(\n",
+    "            G_circuit.x,              # node features\n",
+    "            G_circuit.edge_index,    # connectivity\n",
+    "            G_circuit.edge_attr      # edge features (e.g., gate info)\n",
+    "        ))\n",
+    "\n",
+    "        h_L = F.relu(self.gnn_c2(\n",
+    "            x_c,\n",
+    "            G_circuit.edge_index,\n",
+    "            G_circuit.edge_attr\n",
+    "        ))\n",
+    "\n",
+    "        # === Step 2: Compute embeddings for PHYSICAL qubits ===\n",
+    "        # Each node becomes a vector describing hardware properties\n",
+    "        x_h = F.relu(self.gnn_h1(\n",
+    "            G_hardware.x,\n",
+    "            G_hardware.edge_index,\n",
+    "            G_hardware.edge_attr\n",
+    "        ))\n",
+    "\n",
+    "        h_P = F.relu(self.gnn_h2(\n",
+    "            x_h,\n",
+    "            G_hardware.edge_index,\n",
+    "            G_hardware.edge_attr\n",
+    "        ))\n",
+    "\n",
+    "\n",
+    "        # === Step 3: Compute compatibility between logical and physical qubits ===\n",
+    "\n",
+    "        prob_list = []\n",
+    "\n",
+    "        # Number of graphs in the batch\n",
+    "        num_graphs = G_circuit.batch.max().item() + 1\n",
+    "\n",
+    "        # Loop over each graph in the batch\n",
+    "        for i in range(num_graphs):\n",
+    "\n",
+    "            # --- Select nodes belonging to graph i ---\n",
+    "            # (because batching mixes all graphs together)\n",
+    "            mask_c = (G_circuit.batch == i)\n",
+    "            mask_h = (G_hardware.batch == i)\n",
+    "\n",
+    "            # Extract embeddings for this specific graph\n",
+    "            h_L_i = h_L[mask_c]  # logical qubits\n",
+    "            h_P_i = h_P[mask_h]  # physical qubits\n",
+    "\n",
+    "            # --- Compute compatibility matrix ---\n",
+    "            # Dot product measures how well each pair matches\n",
+    "            # Shape: [num_logical_qubits, num_physical_qubits]\n",
+    "            S_i = torch.matmul(h_L_i, h_P_i.T)\n",
+    "\n",
+    "            if self.training:\n",
+    "                # During training:\n",
+    "                # return raw scores (used in loss function)\n",
+    "                prob_i = S_i\n",
+    "                prob_list.append(prob_i)\n",
+    "\n",
+    "            else:\n",
+    "                # During inference:\n",
+    "                # find the BEST one-to-one assignment\n",
+    "\n",
+    "                # Move to CPU + numpy for Hungarian algorithm\n",
+    "                S_cpu = S_i.detach().cpu().numpy()\n",
+    "\n",
+    "                # Hungarian algorithm finds assignment maximizing total score\n",
+    "                row_ind, col_ind = linear_sum_assignment(-S_cpu)\n",
+    "\n",
+    "                # Build mapping: logical qubit -> physical qubit\n",
+    "                mapping_i = torch.zeros(h_L_i.size(0), dtype=torch.long)\n",
+    "                mapping_i[row_ind] = torch.tensor(col_ind, dtype=torch.long)\n",
+    "\n",
+    "                prob_list.append(mapping_i)\n",
+    "\n",
+    "        # === Step 4: Merge results from all graphs ===\n",
+    "\n",
+    "        if self.training:\n",
+    "            # Concatenate score matrices\n",
+    "            # Shape: [total_logical_nodes_in_batch, num_physical_nodes_per_graph]\n",
+    "            prob = torch.cat(prob_list, dim=0)\n",
+    "            return prob\n",
+    "        else:\n",
+    "            # Concatenate mappings\n",
+    "            # Shape: [total_logical_nodes_in_batch]\n",
+    "            mapping = torch.cat(prob_list, dim=0)\n",
+    "            return mapping"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "tXekkRp2plFf"
+   },
+   "source": [
+    "### **4.3 Loss function and metric**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "U_IriuM0pn7x"
+   },
+   "source": [
+    "We need to have a loss function to minimize for the training of the model."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "P7S0CBd8pn4j"
+   },
+   "source": [
+    "For each logical qubit k, we know the correct assignment (from the training data), which we denote as $k_c$. This is the physical qubit that the logical qubit should be mapped to."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "1YHatNH6pn2b"
+   },
+   "source": [
+    "The loss for the logical qubit k is then defined as:\n",
+    "\n",
+    "$L_k=\\log(p_{k,k_c})$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "3lmNMNC_pn0c"
+   },
+   "source": [
+    "Finally, the total loss is computed by averaging over all logical qubits :\n",
+    "\n",
+    "$L = \\frac{1}{n} \\sum_{k=1}^n -\\log(p_{k,k_c})$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "I0Ds5dOJpnHU"
+   },
+   "source": [
+    "This is known as the cross-entropy loss. With Pytorch, the cross entropy loss can be calculated by using nn.CrossEntropyLoss():\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "y8nGNCcKp7xZ"
+   },
+   "outputs": [],
+   "source": [
+    "loss_fn = nn.CrossEntropyLoss()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "7Vb-3Td7p-L8"
+   },
+   "source": [
+    "This loss encourages the model to assign high probability to the correct qubit mapping and to assign low probability to incorrect mappings. However, it does not enforce one-to-one mapping  and does not prevent collisions (two logical qubits choosing the same physical qubit). It is the reason that it is better to use the Hungarian algorithm during inference.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "H1YqZkAip-IX"
+   },
+   "source": [
+    "In addition to the loss function, there are some metrics that help to evaluate the quality of the model and are easier to interpret than the loss function. We consider two different metrics:\n",
+    "* logical qubit-level accuracy\n",
+    "* full mapping accuracy\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "vSNLcnh5p8y7"
+   },
+   "source": [
+    "To understand the difference between them, let us consider an example with 5 logical qubits and 5 physical qubits. The correct mapping (from the training data) is [2,1,4,3,0] and the model predicts [2,0,4,3,1].\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "IkeLRv2hqDzh"
+   },
+   "source": [
+    "For the logical qubit-level accuracy, we simply count how many individual assignments are correct. In this example, 3 out of 5 qubits are correctly mapped. So the accuracy is 60%.     \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "NBVjWv1DqE3Y"
+   },
+   "source": [
+    "For the full mapping accuracy, the prediction is considered correct only if the entire assignment is exactly correct. In this example, the predicted mapping is not identical to the ground truth. So the full mapping is incorrect. This metric is therefore much stricter and harder to improve.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "J8wtTMsaqFwP"
+   },
+   "source": [
+    "You could ask: Why use both metrics? These two metrics serve different goals. The logical qubit-level accuracy is useful to monitor progress during training. The full mapping accuracy reflects the actual goal of the problem: producing a correct global qubit mapping. During training, the logical-level accuracy improves first and helps track learning progress while the full-mapping accuracy becomes meaningful only later, when the model is already quite accurate. Note that the Hungarian algorithm significantly improves the full-mapping accuracy. As a result, the full-mapping accuracy is often higher on the validation set than on the training set.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "tShJDEofqG8H"
+   },
+   "source": [
+    "These metrics are calculated with the following code for the training and the validation dataset:\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "vuLpu-inqIdk"
+   },
+   "outputs": [],
+   "source": [
+    "def batch_graph_metrics_training(logit, qu_map, batch):\n",
+    "    \"\"\"\n",
+    "    Compute per-graph metrics in a single pass:\n",
+    "      - circuit-averaged node accuracy\n",
+    "      - exact-mapping accuracy\n",
+    "\n",
+    "    Args:\n",
+    "        logit: [N_total_nodes, N_P] predicted logit\n",
+    "        qu_map: [N_total_nodes] ground-truth physical qubits\n",
+    "        batch: [N_total_nodes] graph indices\n",
+    "\n",
+    "    Returns:\n",
+    "        node_acc_sum: sum of node-level accuracies per graph\n",
+    "        exact_sum: number of perfectly mapped graphs\n",
+    "        num_graphs: number of graphs in the batch\n",
+    "    \"\"\"\n",
+    "\n",
+    "    pred = logit.argmax(dim=1)\n",
+    "\n",
+    "    num_graphs = batch.max().item() + 1\n",
+    "    node_acc_sum = 0.0\n",
+    "    exact_sum = 0\n",
+    "\n",
+    "    for g in range(num_graphs):\n",
+    "        mask = (batch == g)\n",
+    "        correct_nodes = (pred[mask] == qu_map[mask]).sum().item()\n",
+    "        n_nodes = mask.sum().item()\n",
+    "\n",
+    "        node_acc_sum += correct_nodes / n_nodes\n",
+    "        if correct_nodes == n_nodes:\n",
+    "            exact_sum += 1\n",
+    "\n",
+    "    return node_acc_sum, exact_sum, num_graphs"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "P5qCPoLNqJnz"
+   },
+   "outputs": [],
+   "source": [
+    "def batch_graph_metrics_val(pred, qu_map, batch):\n",
+    "    \"\"\"\n",
+    "    Compute per-graph metrics in a single pass:\n",
+    "      - circuit-averaged node accuracy\n",
+    "      - exact-mapping accuracy\n",
+    "\n",
+    "    Args:\n",
+    "        pred: [N_total_nodes] predicted mapping\n",
+    "        qu_map: [N_total_nodes] ground-truth physical qubits\n",
+    "        batch: [N_total_nodes] graph indices\n",
+    "\n",
+    "    Returns:\n",
+    "        node_acc_sum: sum of node-level accuracies per graph\n",
+    "        exact_sum: number of perfectly mapped graphs\n",
+    "        num_graphs: number of graphs in the batch\n",
+    "    \"\"\"\n",
+    "\n",
+    "    num_graphs = batch.max().item() + 1\n",
+    "    node_acc_sum = 0.0\n",
+    "    exact_sum = 0\n",
+    "\n",
+    "    for g in range(num_graphs):\n",
+    "        mask = (batch == g)\n",
+    "        correct_nodes = (pred[mask] == qu_map[mask]).sum().item()\n",
+    "        n_nodes = mask.sum().item()\n",
+    "\n",
+    "        node_acc_sum += correct_nodes / n_nodes\n",
+    "        if correct_nodes == n_nodes:\n",
+    "            exact_sum += 1\n",
+    "\n",
+    "    return node_acc_sum, exact_sum, num_graphs"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "o-VijEzlqLKk"
+   },
+   "source": [
+    "### **4.3 Training Loop**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "rxj6X-7LqP6H"
+   },
+   "source": [
+    "We now have all the elements to build our training loop! Here it is the code:\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "RwSgfSsNqRpL"
+   },
+   "outputs": [],
+   "source": [
+    "def train_loop(model, optimizer, loss_fn, loader_train, loader_valid, n_epoch, device):\n",
+    "    \"\"\"\n",
+    "    Training and validation loop for qubit mapping model.\n",
+    "    Returns:\n",
+    "        loss_hist_train, accuracy_hist_train, exact_accuracy_hist_train,\n",
+    "        accuracy_hist_valid, exact_accuracy_hist_valid\n",
+    "    \"\"\"\n",
+    "    loss_hist_train = [0.0] * n_epoch\n",
+    "    accuracy_hist_train = [0.0] * n_epoch\n",
+    "    exact_accuracy_hist_train = [0.0] * n_epoch\n",
+    "    accuracy_hist_valid = [0.0] * n_epoch\n",
+    "    exact_accuracy_hist_valid = [0.0] * n_epoch\n",
+    "\n",
+    "    for epoch in range(n_epoch):\n",
+    "        # ------------ TRAIN ------------------\n",
+    "        model.train()\n",
+    "        total_loss = 0.0\n",
+    "        total_acc = 0.0\n",
+    "        total_exact = 0\n",
+    "        total_graphs = 0\n",
+    "        for G_c, G_h,  qu_map in loader_train:\n",
+    "            G_c = G_c.to(device)\n",
+    "            G_h = G_h.to(device)\n",
+    "\n",
+    "            qu_map = qu_map.to(device)\n",
+    "\n",
+    "            optimizer.zero_grad()\n",
+    "            logit = model(G_c, G_h)\n",
+    "            loss = loss_fn(logit, qu_map)\n",
+    "            loss.backward()\n",
+    "            optimizer.step()\n",
+    "            #Metrics\n",
+    "            acc_sum, exact_sum, n_graphs = batch_graph_metrics_training(\n",
+    "                logit, qu_map, G_c.batch\n",
+    "            )\n",
+    "\n",
+    "            total_loss += loss.item() * n_graphs\n",
+    "            total_acc += acc_sum\n",
+    "            total_exact += exact_sum\n",
+    "            total_graphs += n_graphs\n",
+    "        #Store epoch metrics\n",
+    "        loss_hist_train[epoch] = total_loss  / total_graphs\n",
+    "        accuracy_hist_train[epoch] = total_acc / total_graphs\n",
+    "        exact_accuracy_hist_train[epoch] = total_exact / total_graphs\n",
+    "\n",
+    "        #VALIDATION\n",
+    "        model.eval()\n",
+    "        total_acc = 0.0\n",
+    "        total_exact = 0\n",
+    "        total_graphs = 0\n",
+    "\n",
+    "        with torch.no_grad():\n",
+    "            for G_c_val, G_h_val, qu_map_val in loader_valid:\n",
+    "                G_c_val = G_c_val.to(device)\n",
+    "                G_h_val = G_h_val.to(device)\n",
+    "                qu_map_val = qu_map_val.to(device)\n",
+    "\n",
+    "                mapping = model(G_c_val, G_h_val)\n",
+    "\n",
+    "                acc_sum, exact_sum, n_graphs = batch_graph_metrics_val(\n",
+    "                mapping, qu_map_val, G_c_val.batch\n",
+    "                )\n",
+    "\n",
+    "                total_acc += acc_sum\n",
+    "                total_exact += exact_sum\n",
+    "                total_graphs += n_graphs\n",
+    "\n",
+    "        accuracy_hist_valid[epoch] = total_acc / total_graphs\n",
+    "        exact_accuracy_hist_valid[epoch] = total_exact / total_graphs\n",
+    "        print(\n",
+    "            f\"Epoch {epoch+1:03d} | \"\n",
+    "            f\"Train Loss: {loss_hist_train[epoch]:.4f} | \"\n",
+    "            f\"Train Acc: {accuracy_hist_train[epoch]:.4f} | \"\n",
+    "            f\"Train Exact Acc: {exact_accuracy_hist_train[epoch]:.4f} | \"\n",
+    "            f\"Val Acc: {accuracy_hist_valid[epoch]:.4f} | \"\n",
+    "            f\"Val Exact Acc: {exact_accuracy_hist_valid[epoch]:.4f}\"\n",
+    "        )\n",
+    "\n",
+    "    return (loss_hist_train,\n",
+    "            accuracy_hist_train, exact_accuracy_hist_train,\n",
+    "            accuracy_hist_valid, exact_accuracy_hist_valid)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "W2_PN36iqStx"
+   },
+   "source": [
+    "The training can be run by defining the model, the optimizer, the loss function, the sample loaders, the number of epochs and the device.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "rngz8XCLqTxs"
+   },
+   "outputs": [],
+   "source": [
+    "loss_hist_train, accuracy_hist_train, exact_accuracy_hist_train, \\\n",
+    "accuracy_hist_valid, exact_accuracy_hist_valid = train_loop(model, optimizer, loss_fn,\n",
+    "                                                            train_loader, val_loader,\n",
+    "                                                            n_epochs, device)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "Qr7-pybVqVSN"
+   },
+   "source": [
+    "For the optimizer, we can for example using an Adam optimizer which is an efficient optimizer which adapts the learning rate for each parameter based on estimates of the first and second moments (mean and variance) of gradients.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "e-s187KzqWkT"
+   },
+   "outputs": [],
+   "source": [
+    "optimizer = torch.optim.Adam(model.parameters(), lr=1e-3, weight_decay=1e-5)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "BarOe_RcqXDt"
+   },
+   "source": [
+    "With this architecture applied to a qubit mapping problem on hardware with 5 physical qubits (IBM Athens), a full-mapping accuracy of 90% can be obtained. This is not yet perfect, but it shows that a machine learning model can learn and solve the qubit mapping problem."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "mI5QzXpmqYrj"
+   },
+   "source": [
+    "## **5. Conclusion**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "6Y8u-rT7qaSS"
+   },
+   "source": [
+    "In this demo, we introduced the qubit mapping problem. Qubit mapping is a fundamental problem for hardwares with limited connectivity, such as superconducting devices. Unfortunately, the qubit mapping problem is NP-hard. One way to find approximate solutions is to use heuristic methods, such as the SABRE algorithm presented in this demo.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "aQNULJR0qbV2"
+   },
+   "source": [
+    "It remains an open question whether a machine learning model can efficiently solve this problem and have better scaling than the SABRE algorithm. To explore this idea, we need a dataset, which is the purpose of QMapDataset. Three datasets for the IBM architectures Eagle R3, Heron r1, and Heron r2, respectively, are now available in PennyLane, and we showed how to access them.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "pI9wFDStqcOr"
+   },
+   "source": [
+    "Finally, we presented a Graph Neural Network model to solve the qubit mapping problem. This model shows encouraging results on small hardware. The next challenge is how to scale it to larger hardware, such as Eagle R3 (127 qubits), Heron r1 (133 qubits), and Heron r2 (156 qubits). Maybe the person who finds the solution will be you, using the dataset available in PennyLane!\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "colab": {
+   "provenance": []
+  },
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.13.10"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}

From 0b2c70c257705e398b311e561547ce62d589c925 Mon Sep 17 00:00:00 2001
From: Adrien Devolder <adridevolder@hotmail.com>
Date: Tue, 30 Jun 2026 15:32:26 -0400
Subject: [PATCH 03/21] Chnage Thumbnail sketch

---
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 1 file changed, 0 insertions(+), 0 deletions(-)
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diff --git a/demonstrations_v2/QMapDataset_demo/Thumbnail_example.png b/demonstrations_v2/QMapDataset_demo/Thumbnail_example.png
new file mode 100644
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HcmV?d00001


From c7c34a2084f7f3f3db6befe16d071d3a1e82d47b Mon Sep 17 00:00:00 2001
From: drdren <104710745+drdren@users.noreply.github.com>
Date: Tue, 30 Jun 2026 16:50:00 -0400
Subject: [PATCH 04/21] Fix metadata

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>
---
 demonstrations_v2/QMapDataset_demo/metadata.json | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/demonstrations_v2/QMapDataset_demo/metadata.json b/demonstrations_v2/QMapDataset_demo/metadata.json
index 97b480b0ae..13c85fb84f 100644
--- a/demonstrations_v2/QMapDataset_demo/metadata.json
+++ b/demonstrations_v2/QMapDataset_demo/metadata.json
@@ -9,12 +9,12 @@
   "executable_latest": true,
   "dateOfPublication": "2026-06-30T00:00:00+00:00",
   "dateOfLastModification": "2026-06-30T00:00:00+00:00",
-  "categories": [],
+  "categories": ["Quantum Hardware", "Quantum Computing"],
   "tags": [],
   "previewImages": [
     {
       "type": "thumbnail",
-      "uri": "/_static\\demo_thumbnails\\regular_demo_thumbnails\\thumbnail_placeholder.png"
+      "uri": "/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_placeholder.png"
     }
   ],
   "seoDescription": "The demo introduces the qubit mapping problem and explores machine learning-based approaches for solving it.",

From ce575f01210839c9eaf8230219b8a74dd0f2fc80 Mon Sep 17 00:00:00 2001
From: Adrien Devolder <adridevolder@hotmail.com>
Date: Wed, 1 Jul 2026 09:02:23 -0400
Subject: [PATCH 05/21] Convert jupyter notebook to python file

---
 demonstrations_v2/QMapDataset_demo/demo.ipynb | 1527 -----------------
 demonstrations_v2/QMapDataset_demo/demo.py    | 1066 ++++++++++++
 2 files changed, 1066 insertions(+), 1527 deletions(-)
 delete mode 100644 demonstrations_v2/QMapDataset_demo/demo.ipynb
 create mode 100644 demonstrations_v2/QMapDataset_demo/demo.py

diff --git a/demonstrations_v2/QMapDataset_demo/demo.ipynb b/demonstrations_v2/QMapDataset_demo/demo.ipynb
deleted file mode 100644
index 96f96d2fa7..0000000000
--- a/demonstrations_v2/QMapDataset_demo/demo.ipynb
+++ /dev/null
@@ -1,1527 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "c-eY0ni42lzT"
-   },
-   "source": [
-    "# **Solving the qubit mapping problem with machine learning**"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The demo introduces the qubit mapping problem, an NP-hard challenge in quantum computing systems with limited hardware connectivity, and explores machine learning-based approaches for solving it. It presents the QMapDataset available in PennyLane and shows how it can be used to train a Graph Neural Network to tackle the qubit mapping problem."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "kuwdpYNS2oGq"
-   },
-   "source": [
-    "## **1. What is the qubit mapping problem?**"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "WEALQzoz2pbU"
-   },
-   "source": [
-    "When we write quantum circuits (with Pennylane, of course!), we usually assume that all the qubits are connected to each other, or more precisely, that two-qubit gates can be applied between any qubits. While some infrastructures have all-to-all connectivity (like the photonic devices developed by Xanadu, the ion or the neutral atom platforms), superconducting quantum processors have fixed connectivity between qubits. Only physically connected qubits can directly interact. For example, the first generation of superconducting devices had 5 qubits with a simple linear connectivity:\n",
-    "\n",
-    "$pq_0-pq_1-pq_2-pq_3-pq_4$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "7PBZHOPq2ygy"
-   },
-   "source": [
-    "Physical qubit 0 and physical qubit 3  are not directly connected. If a two-qubit gate is required in your code between qubits 0 and 3, it is necessary to apply SWAP gates. SWAP gates exchange the states of two neighboring qubits for all basis states in the superposition (more details here [What is a SWAP gate? | PennyLane](https://pennylane.ai/qml/glossary/what-is-a-swap-gate)).In our example, we would need to apply two SWAP gates: one between $pq_0$ and $pq_1$, and one between $pq_1$ and $pq_2$. If the state was $\\frac{1}{\\sqrt{2}}(|10000\\rangle + |00001\\rangle)$ before the SWAP gates, it becomes $\\frac{1}{\\sqrt{2}}(|00100\\rangle + |00001\\rangle)$. Then, we can apply the CNOT gate between $pq_2$ and $pq_3$."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "lmp4qozV2rDT"
-   },
-   "source": [
-    "The lack of full connectivity between physical qubits does not alter the universality of quantum computation in these devices but leads to an increase in the depth of the effective quantum circuit. A SWAP gate corresponds to 3 CNOT gates. Therefore, instead of 1 CNOT gate, we have 7 CNOT gates. This increase can be fatal for your quantum computation, especially on a NISQ device."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "q5b4849Pihlp"
-   },
-   "source": [
-    "Indeed, NISQ devices are sensitive to noise. In particular, two-qubit gates are noisy. Therefore, when a large number of two-qubit gates is applied on a NISQ device, the quantum information is progressively lost and randomized. Therefore, when running a quantum circuit on a NISQ device, we try to minimize the number of two-qubit gates. Obviously, the increase by a factor of 7 in our example is terrible for the noise robustness of our quantum computation. Even for fault-tolerant quantum computers, reducing the number of CNOT gates would help to reduce the time of computation and therefore the computational cost.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "sZ2I1TJTihjo"
-   },
-   "source": [
-    "It is here that qubit mapping can help mitigate the number of SWAP gates. There is no rule that constrains the first qubit of your quantum circuit to be the first physical qubit, the second qubit of your quantum circuit to be the second physical qubit, etc. We can separate the two sets of qubits: on one side, we have the qubits of our quantum circuit, called logical qubits and denoted $lq_0$, $lq_1$, etc., and on the other side, the physical qubits of the hardware denoted $pq_0$, $pq_1$ etc.. There does not necessarily have to be an identity mapping between the two. Instead, we can define any mapping: $\\{lq\\}\\rightarrow \\{pq\\}$. The qubit mapping problem corresponds to finding the initial mapping between the logical qubits and the physical qubits that minimizes the number of SWAP gates during the execution of the quantum circuit by the hardware. In our simple example, if we assign $lq_0\\rightarrow pq_0$ and $lq_3\\rightarrow pq_1$, a CNOT gate between physical qubits 0 and 1 corresponds to a CNOT gate between logical qubits 0 and 3, as desired."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "51F9nSFTihgr"
-   },
-   "source": [
-    "Let's consider a slightly more complex quantum circuit:\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "fnHdlxsoi5bt"
-   },
-   "outputs": [],
-   "source": [
-    "import pennylane as qp\n",
-    "dev = qp.device(\"default.qubit\", wires=5)\n",
-    "@qp.qnode(dev)\n",
-    "def  circuit():\n",
-    "    qp.CNOT(wires=[0,4])\n",
-    "    qp.CNOT(wires=[1,3])\n",
-    "    qp.CNOT(wires=[0,2])"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "ST0kkexFiheg"
-   },
-   "source": [
-    "So, there are 3 CNOT gates: one between $lq_0$ and $lq_4$, one between $lq_1$ and $lq_3$, and one between $lq_0$ and $lq_2$. Let’s first examine the number of SWAP gates with the identity mapping ($lq_0\\rightarrow pq_0$, $lq_1\\rightarrow pq_1$, etc.):\n",
-    "\n",
-    "$pq_0(lq_0)-pq_1(lq_1)-pq_2(lq_2)-pq_3(lq_3)-pq_4(lq_4)$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "WeSqQqCEihcq"
-   },
-   "source": [
-    "Here, in parentheses, the logical qubit mapped to the corresponding physical qubit is shown.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "pUaHUc5zihar"
-   },
-   "source": [
-    "For the first CNOT gate between $lq_0$ and $lq_4$, we need to apply 3 SWAP gates (SWAP($pq_0$, $pq_1$), SWAP($pq_1$, $pq_2$), and SWAP($pq_2$, $pq_3$)). The mapping becomes:\n",
-    "\n",
-    "$pq_0(lq_1)-pq_1(lq_2)-pq_2(lq_3)-pq_3(lq_0)-pq_4(lq_4)$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "Dy7H4lyZihYw"
-   },
-   "source": [
-    "Afterward, for the CNOT gate between $lq_1$ and $lq_3$, we need a SWAP gate between $pq_1$ and $pq_2$:\n",
-    "\n",
-    "$pq_0(lq_1)-pq_1(lq_3)-pq_2(lq_2)-pq_4(lq_0)-pq_4(lq_4)$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "JFzvFKx5ihVq"
-   },
-   "source": [
-    "Finally, for the CNOT gate between $lq_0$ and $lq_2$, we do not need any additional SWAP gates. Therefore, in total, there are 4 SWAP gates."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "3NOJYBf1ihTm"
-   },
-   "source": [
-    "Now, let’s consider the following initial mapping:\n",
-    "\n",
-    "$pq_0(lq_1)-pq_1(lq_3)-pq_2(lq_2)-pq_3(lq_0)-pq_4(lq_4)$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "FX_QRYLRihRg"
-   },
-   "source": [
-    "In this case, there is no need for SWAP gates. The three CNOT gates can be applied directly. We go from 15 CNOT gates to 3 CNOT gates, a reduction by a factor of 5 just by changing the initial mapping!"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "aZhhKNRLkDJV"
-   },
-   "source": [
-    "Now that the qubit mapping problem is clear from these 5-qubit examples, let’s give a generalized definition. We start with:\n",
-    "\n",
-    "* A quantum circuit C defined on n logical qubits $lq_0, lq_1, \\dots, lq_{n-1}$\n",
-    "* A quantum device with m physical qubits $pq_0, pq_1, \\dots, pq_{m-1}$ connected by some graph $G_D$."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "q1Ygx0QskDGA"
-   },
-   "source": [
-    "The goal of the qubit mapping problem is to find the initial mapping $\\pi_0: \\{lq\\}\\rightarrow \\{pq\\}$ in order to minimize the number of SWAP gates during the execution of the quantum circuit C on the quantum device. Note that in general, the number of logical qubits $n$ is smaller than the number of physical qubits $m$ (and cannot be bigger)."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "fbEgFJoXkDEK"
-   },
-   "source": [
-    "The initial qubit mapping problem can be represented as an assignment matrix A of dimension $n\\times m$:\n",
-    "\n",
-    "* $A_{ij}=1$ if logical qubit $lq_i$ is mapped to the physical qubit $pq_j$.\n",
-    "* $A_{ij}=0$ otherwise\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "WSFJAehjkDCK"
-   },
-   "source": [
-    "There are two constraints on the assignment matrix A :\n",
-    "* Each logical qubit is assigned exactly one physical qubit $\\sum_{j=1}^m A_{ij} = 1, ~\\forall i$\n",
-    "\n",
-    "* Each physical qubit holds at most one logical qubit $\\sum_{i=1}^n A_{ij}\\leq 1, ~\\forall j$\n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "RzC9Q3GekDAH"
-   },
-   "source": [
-    "An important quantity is the distance between two physical qubits $D_{ij}=d(pq_i, pq_j)$. The distance is the shortest-path length between two physical qubits in the hardware connectivity graph, and it directly estimates how many SWAPs are needed to make them interact. Of course, the distance would depend on the hardware connectivity graph. "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "2jRa1MbckC-c"
-   },
-   "source": [
-    "We will explore in the next section the actual methods used to find the optimal assignment matrix A and therefore to solve the qubit mapping problem.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "2-izzdBQkC8P"
-   },
-   "source": [
-    "## **2.How to solve the qubit mapping problem?**"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "6xsztWspkC54"
-   },
-   "source": [
-    "The qubit mapping problem is unfortunately a NP problem and is exponentially hard to find the optimal solution when the number of logical and physical qubits increases. However, there are some heuristic methods that allow us to solve it with a reasonable computational time. The most famous heuristic algorithm is the SABRE algorithm, used for example by Qiskit in its compilation.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "iKXYaZzlkC3_"
-   },
-   "source": [
-    "The SABRE algorithm works by iteratively updating the qubit mapping guided by a cost function based on the total distance.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "s0RGC159kC1-"
-   },
-   "source": [
-    "**Forward propagation**\n",
-    "1. **Initialize the mapping**: The algorithm starts by initializing a single mapping between logical and physical qubits (typically random or identity).\n",
-    "Example (5-qubit device): $pq_0(lq_0)-pq_1(lq_2)-pq_2(lq_3)-pq_3(lq_4)-pq_4(lq_1)$\n",
-    "\n",
-    "2. **Define the front layer**: The front layer is the set of gates that are ready to be executed, meaning all their dependencies in the circuit have been satisfied. For the quantum circuit (CNOT(0,4), CNOT(1,3) and CNOT(0,2), the front layer is {CNOT(0,4), CNOT(1,3)}. CNOT(0,2) is not included because it depends on the execution of CNOT(0,4).\n",
-    "\n",
-    "3. **Check gate executability**: For each gate in the front layer, the algorithm checks whether the mapped physical qubits are connected in the hardware graph:\n",
-    "If they are adjacent, the gate “is executed” and is withdrawn from the front layer.\n",
-    "If not, SWAP gate is required and step 4 is applied.\n",
-    "In our case, the CNOT(0,4) and CNOT(1,3) cannot be executed.\n",
-    "\n",
-    "4. **Compute routing cost**: The algorithm calculates the sum of the distance of the operations in the front layer:\n",
-    "$H = \\sum_{(a,b)} d(\\pi(a), \\pi(b))$\n",
-    "\n",
-    "5. **Evaluate SWAP candidates**: The algorithm considers SWAPs on edges of the hardware graph. For each candidate SWAP, they recompute the cost H and select the SWAP that minimizes the cost.\n",
-    "In our case, the SWAP(2,3) gives the distance $H=2+1=3$, which is better than other options.  So this SWAP is applied.\n",
-    "\n",
-    "6. **Iterate forward pass**: Steps 2–5 are repeated. The front layer is updated dynamically as gates become executable. Gates are executed whenever possible. SWAPs are inserted when necessary. At the end of this process, the algorithm produces a final mapping  for the forward pass.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "wo5jtgGJkC0A"
-   },
-   "source": [
-    "**Backward propagation**\n",
-    "\n",
-    "7. **Reverse circuit execution**: The algorithm then repeats the same procedure, but on the reversed circuit (executing gates in reverse order), starting from the final mapping obtained in the forward pass.\n",
-    "\n",
-    "8. **Improved initial mapping**: At the end of the backward pass, a new initial mapping $\\pi_0$ is obtained. This initial mapping is typically better than the original random initialization."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "fsVnXeFCkCx2"
-   },
-   "source": [
-    "In practice, SABRE may repeat multiple forward and backward passes (usually around 3 iterations), gradually improving the mapping until no significant improvement is observed.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "96vTzEyskCv4"
-   },
-   "source": [
-    "The SABRE algorithm has a polynomial computational complexity of approximately $\\mathcal{O}(Tm^2 n)$ where $T$ is the total number of two-qubit gates in the quantum circuit. This is significantly better than the exponential scaling of an exact solution. However, as the number of qubits in superconducting processors continues to increase, the runtime of the SABRE algorithm also grows accordingly.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "mOyw8ouzkCt6"
-   },
-   "source": [
-    "Machine learning models could offer better scalability while maintaining high accuracy.\n",
-    "In this demo, we consider supervised learning. Supervised learning requires a dataset with input data and corresponding labels to train the model. This is the purpose of QMapDataset, which provides datasets and tools to generate them for solving the qubit mapping problem using supervised machine learning models. QMapDataset is available in PennyLane. We will explore its features in the next section.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "uEh-QmbwkCsD"
-   },
-   "source": [
-    "## **3. QMapDataset**"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "C6PRla4WmADC"
-   },
-   "source": [
-    "QMapDataset is a dataset generator for the qubit mapping problem, designed for IBM quantum devices. The dataset is built from three main components: the hardware, the circuits, and the mapping.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "DQav8AHGmABJ"
-   },
-   "source": [
-    "**For the hardware**, First, we choose an IBM quantum processor. To make the dataset more diverse, the code can randomly permute the properties of the qubits and gates (data augmentation). We can control how often these permutations are applied.  "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "U7WANYQZl__0"
-   },
-   "source": [
-    "Then, the code collects all the relevant hardware information:\n",
-    "\n",
-    "* physical qubit properties ($T_1$, $T_2$, readout error)\n",
-    "* single-qubit gate errors for each qubit\n",
-    "* two-qubit gate errors for each connected pair of qubits\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "dySVIh7Pl_9Z"
-   },
-   "source": [
-    "**For the circuits**, These are sampled either from:\n",
-    "\n",
-    "* random quantum circuits\n",
-    "* well-known circuits (QPE, Grover, QFT, GHZ state preparation etc.)\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "sWa_g0APl_7y"
-   },
-   "source": [
-    "By default, the choice is split 50–50 between random and well-known circuits.\n",
-    "\n",
-    "For random quantum circuits, the code also sample:\n",
-    "* the number of logical qubits\n",
-    "* the circuit depth\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "vvrkGrV-l_5l"
-   },
-   "source": [
-    "The number of logical qubits is chosen between 3 and the number of physical qubits available on the hardware. The depth follows this distribution:  25% shallow (5–15 layers), 40% medium (16–50), 30% deep (51–150), and 5% very deep circuits (151–400)."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "ts3eQZQxl_3o"
-   },
-   "source": [
-    "To further enrich the dataset, we apply random permutations of qubits to the well-known circuits (since there are only a few of them). After that, the code collects:\n",
-    "\n",
-    "* Single-qubit gate counts for each logical qubit\n",
-    "* Two-qubit gate counts for each pair of logical qubits\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "7JvG3xxLl_1U"
-   },
-   "source": [
-    "Finally, **for the mapping**, the code uses the generated hardware and circuits to compute the qubit mapping with an advanced version of the SABRE algorithm (optimization_level = 3 in Qiskit)."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "kS5ZsjiLkCqF"
-   },
-   "source": [
-    "If you want more details, you can have a look at the GitHub repo: [rscadrien/QMapDataset: Generator of dataset for qubit mapping](https://github.com/rscadrien/QMapDataset)."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "CnOk-TREkCHV"
-   },
-   "source": [
-    "The good news is that you do not need to generate this dataset using QMapDataset yourself! We have already prepared datasets for three different types of IBM quantum processors: Eagle 3, Heron 1, and Heron 2. Each dataset contains 10,000 samples, where each sample consists of a circuit, hardware description, and mapping.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "xXjty4hDkCJv"
-   },
-   "source": [
-    "Let’s see how to use these datasets with PennyLane and extract their information.\n",
-    "\n",
-    "The first step is to load a dataset from the Xanadu server. For example, to load the dataset for the Eagle 3 architecture:\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "cFmt01_Om4QP"
-   },
-   "outputs": [],
-   "source": [
-    "import pennylane as qp\n",
-    "[ds] = qp.data.load(\"other\", name=\"qubit-mapping-eagle3\")"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "tH1A-toZm48N"
-   },
-   "source": [
-    "For the Heron 1 and Heron 2 architectures, you only need to replace the name argument with \"qubit-mapping-heron1\" or \"qubit-mapping-heron2\", respectively.\n",
-    "\n",
-    "Each dataset is divided into three attributes: circuits, hardware, and mappings, following the structure discussed earlier. For example, to access the first sample:\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "M90bkqSPm6sV"
-   },
-   "outputs": [],
-   "source": [
-    "sample0_circuit = ds.circuits[0]\n",
-    "sample0_hardware = ds.hardware[0]\n",
-    "sample0_mapping = ds.mappings[0]"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "J5wH6Bd1m4wz"
-   },
-   "source": [
-    "Each of these is a dictionary containing the features of the circuit, hardware, or mapping. For example, to access the single-qubit and two-qubit gate counts of the quantum circuit:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "kw_78wVvm9cD"
-   },
-   "outputs": [],
-   "source": [
-    "Single_qubit_count = sample0_circuit[\"single_qubit_counts\"]\n",
-    "Two_qubit_count = sample0_circuit[\"two_qubit_counts\"]"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "DXjNk5vNm-WP"
-   },
-   "source": [
-    "To retrieve $T_1$, $T_2$, and the two-qubit gate error from the hardware data:\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "Klm95l0LnBOw"
-   },
-   "outputs": [],
-   "source": [
-    "T_1 = sample0_hardware[\"T1\"]\n",
-    "T_2 = sample0_hardware[\"T2\"]\n",
-    "Two_qubit_error = sample0_hardware[\"ecr\"]"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "mrQ1aFYXnBrh"
-   },
-   "source": [
-    "Finally, to obtain the mapping (which can serve as the label in a supervised learning model):\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "nnj5S3fJnDFU"
-   },
-   "outputs": [],
-   "source": [
-    "mapping = sample0_mapping[\"final_mapping\"]"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "F6SduSotnJJ8"
-   },
-   "source": [
-    "Detailed information about all available features and how to access them can be found in the “Data” section of the Qubit Mapping Eagle 3 page on the PennyLane website. ([Qubit Mapping Eagle 3](https://pennylane.ai/datasets/qubit-mapping-eagle3) )."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "R16FjUi_nQQs"
-   },
-   "source": [
-    "## **4.Building a machine learning model to solve the qubit mapping problem**"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "RQLfsWGbnRFc"
-   },
-   "source": [
-    "In this session, we will show an example of how to preprocess data from QMapDataset and how to build a machine learning model that can solve the qubit mapping problem."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "HhmHF0bRnYq3"
-   },
-   "source": [
-    "### **4.1 Preprocessing**"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "LZ-DWZAKnZQp"
-   },
-   "source": [
-    "The qubit mapping problem strongly depends on the connectivity between the physical qubits in the given hardware, as well as the connectivity between the logical qubits in the circuits (represented by two-qubit gate counts). Therefore, a graph is a natural representation for solving the qubit mapping problem. We will see how to construct graphs for both the circuits and the hardware using the QMapDataset available in PennyLane.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "NabNn5wKnak5"
-   },
-   "source": [
-    "A graph is a way of showing relationships. It is made up of points, called nodes (or vertices), that are connected by lines, called edges. Both nodes and edges can also have extra information attached to them, called features.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "GXahu_-pnbgd"
-   },
-   "source": [
-    "For the **circuit**, the nodes represent the logical qubits, and edges connect pairs of logical qubits between which two-qubit gates are applied in the quantum circuit. The features of each node are the counts of different single-qubit gates applied to the corresponding logical qubit. The features of each edge are the counts of different two-qubit gates applied between the two logical qubits. Since the relative values of these features are what matter, we standardize them. This is the code to build the graph for the quantum circuit of the QMapDataset:\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "dUql_BFlneiy"
-   },
-   "outputs": [],
-   "source": [
-    "import torch\n",
-    "from torch_geometric.data import Data\n",
-    "def build_circuit_graph (circuit_data):\n",
-    "    # -----------------------------\n",
-    "    # STEP 1: Extract and scale single-qubit gate counts\n",
-    "    # Each function returns a vector of size (n_qubits,)\n",
-    "    # representing how often each gate was applied per qubit\n",
-    "    # -----------------------------\n",
-    "    rz_counts_scaled = get_single_qubit_counts(circuit_data, \"rz\")\n",
-    "    sx_counts_scaled = get_single_qubit_counts(circuit_data, \"sx\")\n",
-    "    x_counts_scaled = get_single_qubit_counts(circuit_data, \"x\")\n",
-    "\n",
-    "    # -----------------------------\n",
-    "    # STEP 2: Extract and scale two-qubit gate counts\n",
-    "    # These describe interactions between pairs of qubits\n",
-    "    # -----------------------------\n",
-    "    ecr_scaled = get_two_qubit_counts(circuit_data,'ecr')\n",
-    "\n",
-    "    # -----------------------------\n",
-    "    # STEP 3: Build node feature matrix\n",
-    "    # Each node = a logical qubit\n",
-    "    # Features = [rz count, sx count, x count]\n",
-    "    # Shape: (n_qubits, 3)\n",
-    "    # -----------------------------\n",
-    "    X =torch.tensor(\n",
-    "    list(zip(rz_counts_scaled, sx_counts_scaled, x_counts_scaled)),\n",
-    "    dtype=torch.float\n",
-    "    )\n",
-    "    # -----------------------------\n",
-    "    # STEP 4: Build graph connectivity (edges)\n",
-    "    # edge_index: which qubits are connected\n",
-    "    # edge_attr: weight/feature of each edge (scaled gate counts)\n",
-    "    # -----------------------------\n",
-    "    ecr_index, ecr_attr = edge_info(ecr_scaled)\n",
-    "    # -----------------------------\n",
-    "    # STEP 5: Create PyTorch Geometric graph object\n",
-    "    # x -> node features\n",
-    "    # edge_index -> connectivity (COO format)\n",
-    "    # edge_attr -> edge features\n",
-    "    # -----------------------------\n",
-    "    G_circuit = Data(\n",
-    "    x=X,\n",
-    "    edge_index=ecr_index,\n",
-    "    edge_attr=ecr_attr,\n",
-    "    )\n",
-    "    return G_circuit\n",
-    "\n",
-    "def get_single_qubit_counts(data, gate_name):\n",
-    "    # Number of logical qubits in the circuit\n",
-    "    n = data[\"n_logical_qubits\"]\n",
-    "    # Try to retrieve counts for the given gate type\n",
-    "    counts = data.get(\"single_qubit_counts\", {}).get(gate_name, None)\n",
-    "    # If no data exists, return a zero vector\n",
-    "    if counts is None:\n",
-    "        return np.zeros(n)\n",
-    "    else:\n",
-    "    # Otherwise normalize counts across qubits (z-score normalization)\n",
-    "        return standard_scaling_across_qubits(np.array(counts))\n",
-    "\n",
-    "def get_two_qubit_counts(data, gate_name):\n",
-    "    # Safely retrieve list of two-qubit gate interactions\n",
-    "    counts = data.get(\"two_qubit_counts\", {}).get(gate_name, [])\n",
-    "\n",
-    "    # If no interactions exist, return empty list\n",
-    "    if not counts:\n",
-    "        return counts\n",
-    "    else:\n",
-    "    # Otherwise scale the edge weights\n",
-    "      return scaled_two_qubit_list(counts)\n",
-    "\n",
-    "def standard_scaling_across_qubits(vector):\n",
-    "    # Compute mean and standard deviation for normalization\n",
-    "    mean = vector.mean()\n",
-    "    std = vector.std()\n",
-    "    # Avoid division by zero if all values are identical\n",
-    "    if std == 0 :\n",
-    "        vector_scaled = vector\n",
-    "    else:\n",
-    "        vector_scaled = (vector - mean)/std\n",
-    "    return vector_scaled\n",
-    "\n",
-    "def scaled_two_qubit_list(two_data):\n",
-    "    # Extract raw edge weights from dictionary list\n",
-    "    values = np.array([d['value']for d in two_data], dtype=float)\n",
-    "    # Normalize weights\n",
-    "    scaled_values = standard_scaling_across_qubits(values)\n",
-    "    # Reattach scaled values back into original edge dictionaries\n",
-    "    two_gate_scaled = [\n",
-    "    {**d, 'value':v_scaled}\n",
-    "    for d, v_scaled in zip(two_data, scaled_values)\n",
-    "    ]\n",
-    "    return two_gate_scaled\n",
-    "\n",
-    "def edge_info(gate_dict):\n",
-    "    # Lists to store graph connectivity\n",
-    "    edge_index = []\n",
-    "    edge_attr = []\n",
-    "    # Convert list of edge dictionaries into tensor format\n",
-    "    for gate in gate_dict:\n",
-    "        i,j,w = gate[\"row\"], gate[\"col\"], gate[\"value\"]\n",
-    "        edge_index.append([i,j])\n",
-    "        edge_attr.append([w])\n",
-    "    # Convert to PyTorch tensors\n",
-    "    # edge_index shape: [2, num_edges]\n",
-    "    edge_index = torch.tensor(edge_index, dtype=torch.long).T\n",
-    "    edge_attr = torch.tensor(edge_attr, dtype=torch.float)\n",
-    "    return edge_index, edge_attr"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "Q3NbrOeVnfQ5"
-   },
-   "source": [
-    "For example, if this code is applied for the circuit of the 27th sample, we obtain:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "TNXkhztEngJd"
-   },
-   "outputs": [],
-   "source": [
-    "Graph_circuit = build_circuit_graph(ds.circuits[26])"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "EZiBosOnnhGZ"
-   },
-   "source": [
-    "Using the networkx package, we can visualize the graph using the following graph:\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "vik1-T78nkY-"
-   },
-   "outputs": [],
-   "source": [
-    "import matplotlib.pyplot as plt\n",
-    "import networkx as nx\n",
-    "from torch_geometric.utils import to_networkx\n",
-    "# Convert PyG Data object to NetworkX graph\n",
-    "G = to_networkx(\n",
-    "    test,\n",
-    "    edge_attrs=[\"edge_attr\"],  # keep CX counts\n",
-    "    node_attrs=[\"x\"],          # keep node features\n",
-    "    to_undirected=False\n",
-    ")\n",
-    "edge_labels = {\n",
-    "    (u, v): f\"{d['edge_attr'][0]:.2f}\"\n",
-    "    for u, v, d in G.edges(data=True)\n",
-    "}\n",
-    "nx.draw(\n",
-    "    G,\n",
-    "    pos,\n",
-    "    with_labels=True,\n",
-    "    node_size=800,\n",
-    ")\n",
-    "\n",
-    "nx.draw_networkx_edge_labels(\n",
-    "    G,\n",
-    "    pos,\n",
-    "    edge_labels=edge_labels\n",
-    ")\n",
-    "\n",
-    "plt.show()"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "-74N13M3nmml"
-   },
-   "source": [
-    "We obtain the following graph:\n",
-    "![My graph](graph_circuit.png)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "DeAOr5biodV6"
-   },
-   "source": [
-    "For the **hardware**, the nodes represent the physical qubits, and edges connect pairs of physical qubits that are directly connected in the device architecture. The features of each node are the $T_1$ and $T_2$ coherence times, readout error, and single-qubit gate errors for each physical qubit. The features of each edge are the errors of the different two-qubit gates applied between connected physical qubits. As with the circuits, the feature values are standardized. The code to build the graph for the hardware is pretty similar to the one for the circuit:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "mAFplmqGn6tl"
-   },
-   "outputs": [],
-   "source": [
-    "def build_hardware_graph(hardware_data):\n",
-    "    # Scaling single qubit properties and gate\n",
-    "    T1_scaled = standard_scaling_across_qubits(np.array(hardware_data['T1']))\n",
-    "    T2_scaled = standard_scaling_across_qubits(np.array(hardware_data['T2']))\n",
-    "    readout_error_scaled = standard_scaling_across_qubits(np.array(hardware_data['readout_error']))\n",
-    "    sx_error_scaled = standard_scaling_across_qubits(np.array(hardware_data['sx_error']))\n",
-    "    x_error_scaled = standard_scaling_across_qubits(np.array(hardware_data['x_error']))\n",
-    "    #Scaling two qubit gate properties\n",
-    "    cx_error_scaled = scaled_two_qubit_list(hardware_data['cx'])\n",
-    "    #Build the node feature matrix\n",
-    "    X = torch.tensor(\n",
-    "    list(zip(T1_scaled, T2_scaled, readout_error_scaled, sx_error_scaled, x_error_scaled)),\n",
-    "    dtype=torch.float\n",
-    "    )\n",
-    "    #Build the edge index and attributes matrix\n",
-    "    cx_index, cx_attr = edge_info(cx_error_scaled)\n",
-    "    #Build the graph\n",
-    "    G_hardware = Data(\n",
-    "    x=X,\n",
-    "    edge_index=cx_index,\n",
-    "    edge_attr=cx_attr,\n",
-    "    )\n",
-    "    return G_hardware"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "gUAwb-arokXE"
-   },
-   "source": [
-    "We can again visualize the graph using networkx:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "# Draw graph\n",
-    "\n",
-    "plt.figure(figsize=(10, 10))\n",
-    "\n",
-    "pos = nx.spring_layout(\n",
-    "    G_hardware,\n",
-    "    seed=42,\n",
-    "    k=1 / np.sqrt(len(G_hardware)),\n",
-    "    iterations=200\n",
-    ")\n",
-    "\n",
-    "nx.draw_networkx_edges(G_hardware, pos, alpha=0.2, width=0.5)\n",
-    "nx.draw_networkx_nodes(G_hardware, pos, node_size=40)\n",
-    "\n",
-    "plt.axis(\"off\")\n",
-    "plt.savefig(\"graph_hardware.png\", dpi=300, bbox_inches=\"tight\")\n",
-    "plt.show()"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "![My graph](graph_hardware.png)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "gu9V-yu3o1Sg"
-   },
-   "source": [
-    "### **4.2 Machine learning model**"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "ub3VapYRo1PD"
-   },
-   "source": [
-    "We will show how to build a model that can learn qubit mapping. For this model, we will use an architecture adapted to the graph structure of the input: a Graph Neural Network (GNN). The key principle behind GNNs is that a node learns by aggregating information from its neighbors. A GNN typically works in layers, and each layer is mostly composed of three steps:\n",
-    "\n",
-    "* Message passing : Each node receives information from its neighbors\n",
-    "* Aggregation: Combine neighbor information (e.g., sum, mean, max)\n",
-    "* Update: Use a neural network to update the node’s embedding.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "bcHuT4jWo1NN"
-   },
-   "source": [
-    "In our case, we will use two separate GNNs: one for the quantum circuit and one for the hardware. After several GNN layers, we obtain updated embedding vectors for the nodes of both graphs, $h_L^{(i)}$ and $h_P^{(j)}$. ($h_L^{(i)}$ is the vector of the i th node of the quantum circuit graph while $h_P^{(j)}$ is the vector of the jth node of the hardware graph).\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "4cfX4ur8o1LG"
-   },
-   "source": [
-    " We then compare these embeddings using a dot product:\n",
-    "\n",
-    "$S_{ij}=h_L^{(i)}\\cdot h_P^{(j)}$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "LtUfHGxXo1JC"
-   },
-   "source": [
-    "The matrix  element $S_{ij}$ represents the logit that the logical qubit $lq_i$ is mapped to the physical qubit $pq_j$. The probability that logical qubit k maps to physical qubit j is calculated from the logit:\n",
-    "\n",
-    "$p_{ij}=\\frac{\\text{exp}(S_{ij})}{\\sum_{j^\\prime} \\text{exp}(S_{ij}^\\prime)}$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "G1KJt8P9o0le"
-   },
-   "source": [
-    " During training, the matrix $p_{ij}$ should ideally converge to the assignment matrix $A_{ij}$ presented in Section 2, where each row contains exactly one nonzero element equal to 1. After training, during inference, the strict assignment constraints are enforced by applying the Hungarian algorithm.  Unfortunately, this algorithm is not differentiable and therefore cannot be used during training.The code for this architecture is as follows :\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "NlJMmFyzpkdB"
-   },
-   "outputs": [],
-   "source": [
-    "class QubitMapping_dot_Model(nn.Module):\n",
-    "    def __init__(self, node_c_dim, node_h_dim, edge_c_dim, edge_h_dim, hidden_dim):\n",
-    "        super().__init__()\n",
-    "\n",
-    "        # === GNN for the LOGICAL CIRCUIT graph ===\n",
-    "        # These layers learn embeddings for logical qubits based on circuit structure\n",
-    "\n",
-    "        # First layer: edge-conditioned convolution\n",
-    "        # Edge features (e.g., gate types) generate weights dynamically\n",
-    "        self.nn_c1 = make_edge_mlp(edge_c_dim, node_c_dim * hidden_dim)\n",
-    "        self.gnn_c1 = NNConv(node_c_dim, hidden_dim, nn=self.nn_c1, aggr='add')\n",
-    "\n",
-    "        # Second layer: refines embeddings\n",
-    "        self.nn_c2 = make_edge_mlp(edge_c_dim, hidden_dim * hidden_dim)\n",
-    "        self.gnn_c2 = NNConv(hidden_dim, hidden_dim, nn=self.nn_c2, aggr='add')\n",
-    "\n",
-    "        # === GNN for the HARDWARE graph ===\n",
-    "        # These layers learn embeddings for physical qubits based on hardware connectivity\n",
-    "\n",
-    "        self.nn_h1 = make_edge_mlp(edge_h_dim, node_h_dim * hidden_dim)\n",
-    "        self.gnn_h1 = NNConv(node_h_dim, hidden_dim, nn=self.nn_h1, aggr='add')\n",
-    "\n",
-    "        self.nn_h2 = make_edge_mlp(edge_h_dim, hidden_dim * hidden_dim)\n",
-    "        self.gnn_h2 = NNConv(hidden_dim, hidden_dim, nn=self.nn_h2, aggr='add')\n",
-    "\n",
-    "\n",
-    "    def forward(self, G_circuit, G_hardware):\n",
-    "\n",
-    "        # === Step 1: Compute embeddings for LOGICAL qubits ===\n",
-    "        # Each node becomes a vector summarizing its role in the circuit\n",
-    "        x_c = F.relu(self.gnn_c1(\n",
-    "            G_circuit.x,              # node features\n",
-    "            G_circuit.edge_index,    # connectivity\n",
-    "            G_circuit.edge_attr      # edge features (e.g., gate info)\n",
-    "        ))\n",
-    "\n",
-    "        h_L = F.relu(self.gnn_c2(\n",
-    "            x_c,\n",
-    "            G_circuit.edge_index,\n",
-    "            G_circuit.edge_attr\n",
-    "        ))\n",
-    "\n",
-    "        # === Step 2: Compute embeddings for PHYSICAL qubits ===\n",
-    "        # Each node becomes a vector describing hardware properties\n",
-    "        x_h = F.relu(self.gnn_h1(\n",
-    "            G_hardware.x,\n",
-    "            G_hardware.edge_index,\n",
-    "            G_hardware.edge_attr\n",
-    "        ))\n",
-    "\n",
-    "        h_P = F.relu(self.gnn_h2(\n",
-    "            x_h,\n",
-    "            G_hardware.edge_index,\n",
-    "            G_hardware.edge_attr\n",
-    "        ))\n",
-    "\n",
-    "\n",
-    "        # === Step 3: Compute compatibility between logical and physical qubits ===\n",
-    "\n",
-    "        prob_list = []\n",
-    "\n",
-    "        # Number of graphs in the batch\n",
-    "        num_graphs = G_circuit.batch.max().item() + 1\n",
-    "\n",
-    "        # Loop over each graph in the batch\n",
-    "        for i in range(num_graphs):\n",
-    "\n",
-    "            # --- Select nodes belonging to graph i ---\n",
-    "            # (because batching mixes all graphs together)\n",
-    "            mask_c = (G_circuit.batch == i)\n",
-    "            mask_h = (G_hardware.batch == i)\n",
-    "\n",
-    "            # Extract embeddings for this specific graph\n",
-    "            h_L_i = h_L[mask_c]  # logical qubits\n",
-    "            h_P_i = h_P[mask_h]  # physical qubits\n",
-    "\n",
-    "            # --- Compute compatibility matrix ---\n",
-    "            # Dot product measures how well each pair matches\n",
-    "            # Shape: [num_logical_qubits, num_physical_qubits]\n",
-    "            S_i = torch.matmul(h_L_i, h_P_i.T)\n",
-    "\n",
-    "            if self.training:\n",
-    "                # During training:\n",
-    "                # return raw scores (used in loss function)\n",
-    "                prob_i = S_i\n",
-    "                prob_list.append(prob_i)\n",
-    "\n",
-    "            else:\n",
-    "                # During inference:\n",
-    "                # find the BEST one-to-one assignment\n",
-    "\n",
-    "                # Move to CPU + numpy for Hungarian algorithm\n",
-    "                S_cpu = S_i.detach().cpu().numpy()\n",
-    "\n",
-    "                # Hungarian algorithm finds assignment maximizing total score\n",
-    "                row_ind, col_ind = linear_sum_assignment(-S_cpu)\n",
-    "\n",
-    "                # Build mapping: logical qubit -> physical qubit\n",
-    "                mapping_i = torch.zeros(h_L_i.size(0), dtype=torch.long)\n",
-    "                mapping_i[row_ind] = torch.tensor(col_ind, dtype=torch.long)\n",
-    "\n",
-    "                prob_list.append(mapping_i)\n",
-    "\n",
-    "        # === Step 4: Merge results from all graphs ===\n",
-    "\n",
-    "        if self.training:\n",
-    "            # Concatenate score matrices\n",
-    "            # Shape: [total_logical_nodes_in_batch, num_physical_nodes_per_graph]\n",
-    "            prob = torch.cat(prob_list, dim=0)\n",
-    "            return prob\n",
-    "        else:\n",
-    "            # Concatenate mappings\n",
-    "            # Shape: [total_logical_nodes_in_batch]\n",
-    "            mapping = torch.cat(prob_list, dim=0)\n",
-    "            return mapping"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "tXekkRp2plFf"
-   },
-   "source": [
-    "### **4.3 Loss function and metric**"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "U_IriuM0pn7x"
-   },
-   "source": [
-    "We need to have a loss function to minimize for the training of the model."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "P7S0CBd8pn4j"
-   },
-   "source": [
-    "For each logical qubit k, we know the correct assignment (from the training data), which we denote as $k_c$. This is the physical qubit that the logical qubit should be mapped to."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "1YHatNH6pn2b"
-   },
-   "source": [
-    "The loss for the logical qubit k is then defined as:\n",
-    "\n",
-    "$L_k=\\log(p_{k,k_c})$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "3lmNMNC_pn0c"
-   },
-   "source": [
-    "Finally, the total loss is computed by averaging over all logical qubits :\n",
-    "\n",
-    "$L = \\frac{1}{n} \\sum_{k=1}^n -\\log(p_{k,k_c})$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "I0Ds5dOJpnHU"
-   },
-   "source": [
-    "This is known as the cross-entropy loss. With Pytorch, the cross entropy loss can be calculated by using nn.CrossEntropyLoss():\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "y8nGNCcKp7xZ"
-   },
-   "outputs": [],
-   "source": [
-    "loss_fn = nn.CrossEntropyLoss()"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "7Vb-3Td7p-L8"
-   },
-   "source": [
-    "This loss encourages the model to assign high probability to the correct qubit mapping and to assign low probability to incorrect mappings. However, it does not enforce one-to-one mapping  and does not prevent collisions (two logical qubits choosing the same physical qubit). It is the reason that it is better to use the Hungarian algorithm during inference.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "H1YqZkAip-IX"
-   },
-   "source": [
-    "In addition to the loss function, there are some metrics that help to evaluate the quality of the model and are easier to interpret than the loss function. We consider two different metrics:\n",
-    "* logical qubit-level accuracy\n",
-    "* full mapping accuracy\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "vSNLcnh5p8y7"
-   },
-   "source": [
-    "To understand the difference between them, let us consider an example with 5 logical qubits and 5 physical qubits. The correct mapping (from the training data) is [2,1,4,3,0] and the model predicts [2,0,4,3,1].\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "IkeLRv2hqDzh"
-   },
-   "source": [
-    "For the logical qubit-level accuracy, we simply count how many individual assignments are correct. In this example, 3 out of 5 qubits are correctly mapped. So the accuracy is 60%.     \n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "NBVjWv1DqE3Y"
-   },
-   "source": [
-    "For the full mapping accuracy, the prediction is considered correct only if the entire assignment is exactly correct. In this example, the predicted mapping is not identical to the ground truth. So the full mapping is incorrect. This metric is therefore much stricter and harder to improve.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "J8wtTMsaqFwP"
-   },
-   "source": [
-    "You could ask: Why use both metrics? These two metrics serve different goals. The logical qubit-level accuracy is useful to monitor progress during training. The full mapping accuracy reflects the actual goal of the problem: producing a correct global qubit mapping. During training, the logical-level accuracy improves first and helps track learning progress while the full-mapping accuracy becomes meaningful only later, when the model is already quite accurate. Note that the Hungarian algorithm significantly improves the full-mapping accuracy. As a result, the full-mapping accuracy is often higher on the validation set than on the training set.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "tShJDEofqG8H"
-   },
-   "source": [
-    "These metrics are calculated with the following code for the training and the validation dataset:\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "vuLpu-inqIdk"
-   },
-   "outputs": [],
-   "source": [
-    "def batch_graph_metrics_training(logit, qu_map, batch):\n",
-    "    \"\"\"\n",
-    "    Compute per-graph metrics in a single pass:\n",
-    "      - circuit-averaged node accuracy\n",
-    "      - exact-mapping accuracy\n",
-    "\n",
-    "    Args:\n",
-    "        logit: [N_total_nodes, N_P] predicted logit\n",
-    "        qu_map: [N_total_nodes] ground-truth physical qubits\n",
-    "        batch: [N_total_nodes] graph indices\n",
-    "\n",
-    "    Returns:\n",
-    "        node_acc_sum: sum of node-level accuracies per graph\n",
-    "        exact_sum: number of perfectly mapped graphs\n",
-    "        num_graphs: number of graphs in the batch\n",
-    "    \"\"\"\n",
-    "\n",
-    "    pred = logit.argmax(dim=1)\n",
-    "\n",
-    "    num_graphs = batch.max().item() + 1\n",
-    "    node_acc_sum = 0.0\n",
-    "    exact_sum = 0\n",
-    "\n",
-    "    for g in range(num_graphs):\n",
-    "        mask = (batch == g)\n",
-    "        correct_nodes = (pred[mask] == qu_map[mask]).sum().item()\n",
-    "        n_nodes = mask.sum().item()\n",
-    "\n",
-    "        node_acc_sum += correct_nodes / n_nodes\n",
-    "        if correct_nodes == n_nodes:\n",
-    "            exact_sum += 1\n",
-    "\n",
-    "    return node_acc_sum, exact_sum, num_graphs"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "P5qCPoLNqJnz"
-   },
-   "outputs": [],
-   "source": [
-    "def batch_graph_metrics_val(pred, qu_map, batch):\n",
-    "    \"\"\"\n",
-    "    Compute per-graph metrics in a single pass:\n",
-    "      - circuit-averaged node accuracy\n",
-    "      - exact-mapping accuracy\n",
-    "\n",
-    "    Args:\n",
-    "        pred: [N_total_nodes] predicted mapping\n",
-    "        qu_map: [N_total_nodes] ground-truth physical qubits\n",
-    "        batch: [N_total_nodes] graph indices\n",
-    "\n",
-    "    Returns:\n",
-    "        node_acc_sum: sum of node-level accuracies per graph\n",
-    "        exact_sum: number of perfectly mapped graphs\n",
-    "        num_graphs: number of graphs in the batch\n",
-    "    \"\"\"\n",
-    "\n",
-    "    num_graphs = batch.max().item() + 1\n",
-    "    node_acc_sum = 0.0\n",
-    "    exact_sum = 0\n",
-    "\n",
-    "    for g in range(num_graphs):\n",
-    "        mask = (batch == g)\n",
-    "        correct_nodes = (pred[mask] == qu_map[mask]).sum().item()\n",
-    "        n_nodes = mask.sum().item()\n",
-    "\n",
-    "        node_acc_sum += correct_nodes / n_nodes\n",
-    "        if correct_nodes == n_nodes:\n",
-    "            exact_sum += 1\n",
-    "\n",
-    "    return node_acc_sum, exact_sum, num_graphs"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "o-VijEzlqLKk"
-   },
-   "source": [
-    "### **4.3 Training Loop**"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "rxj6X-7LqP6H"
-   },
-   "source": [
-    "We now have all the elements to build our training loop! Here it is the code:\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "RwSgfSsNqRpL"
-   },
-   "outputs": [],
-   "source": [
-    "def train_loop(model, optimizer, loss_fn, loader_train, loader_valid, n_epoch, device):\n",
-    "    \"\"\"\n",
-    "    Training and validation loop for qubit mapping model.\n",
-    "    Returns:\n",
-    "        loss_hist_train, accuracy_hist_train, exact_accuracy_hist_train,\n",
-    "        accuracy_hist_valid, exact_accuracy_hist_valid\n",
-    "    \"\"\"\n",
-    "    loss_hist_train = [0.0] * n_epoch\n",
-    "    accuracy_hist_train = [0.0] * n_epoch\n",
-    "    exact_accuracy_hist_train = [0.0] * n_epoch\n",
-    "    accuracy_hist_valid = [0.0] * n_epoch\n",
-    "    exact_accuracy_hist_valid = [0.0] * n_epoch\n",
-    "\n",
-    "    for epoch in range(n_epoch):\n",
-    "        # ------------ TRAIN ------------------\n",
-    "        model.train()\n",
-    "        total_loss = 0.0\n",
-    "        total_acc = 0.0\n",
-    "        total_exact = 0\n",
-    "        total_graphs = 0\n",
-    "        for G_c, G_h,  qu_map in loader_train:\n",
-    "            G_c = G_c.to(device)\n",
-    "            G_h = G_h.to(device)\n",
-    "\n",
-    "            qu_map = qu_map.to(device)\n",
-    "\n",
-    "            optimizer.zero_grad()\n",
-    "            logit = model(G_c, G_h)\n",
-    "            loss = loss_fn(logit, qu_map)\n",
-    "            loss.backward()\n",
-    "            optimizer.step()\n",
-    "            #Metrics\n",
-    "            acc_sum, exact_sum, n_graphs = batch_graph_metrics_training(\n",
-    "                logit, qu_map, G_c.batch\n",
-    "            )\n",
-    "\n",
-    "            total_loss += loss.item() * n_graphs\n",
-    "            total_acc += acc_sum\n",
-    "            total_exact += exact_sum\n",
-    "            total_graphs += n_graphs\n",
-    "        #Store epoch metrics\n",
-    "        loss_hist_train[epoch] = total_loss  / total_graphs\n",
-    "        accuracy_hist_train[epoch] = total_acc / total_graphs\n",
-    "        exact_accuracy_hist_train[epoch] = total_exact / total_graphs\n",
-    "\n",
-    "        #VALIDATION\n",
-    "        model.eval()\n",
-    "        total_acc = 0.0\n",
-    "        total_exact = 0\n",
-    "        total_graphs = 0\n",
-    "\n",
-    "        with torch.no_grad():\n",
-    "            for G_c_val, G_h_val, qu_map_val in loader_valid:\n",
-    "                G_c_val = G_c_val.to(device)\n",
-    "                G_h_val = G_h_val.to(device)\n",
-    "                qu_map_val = qu_map_val.to(device)\n",
-    "\n",
-    "                mapping = model(G_c_val, G_h_val)\n",
-    "\n",
-    "                acc_sum, exact_sum, n_graphs = batch_graph_metrics_val(\n",
-    "                mapping, qu_map_val, G_c_val.batch\n",
-    "                )\n",
-    "\n",
-    "                total_acc += acc_sum\n",
-    "                total_exact += exact_sum\n",
-    "                total_graphs += n_graphs\n",
-    "\n",
-    "        accuracy_hist_valid[epoch] = total_acc / total_graphs\n",
-    "        exact_accuracy_hist_valid[epoch] = total_exact / total_graphs\n",
-    "        print(\n",
-    "            f\"Epoch {epoch+1:03d} | \"\n",
-    "            f\"Train Loss: {loss_hist_train[epoch]:.4f} | \"\n",
-    "            f\"Train Acc: {accuracy_hist_train[epoch]:.4f} | \"\n",
-    "            f\"Train Exact Acc: {exact_accuracy_hist_train[epoch]:.4f} | \"\n",
-    "            f\"Val Acc: {accuracy_hist_valid[epoch]:.4f} | \"\n",
-    "            f\"Val Exact Acc: {exact_accuracy_hist_valid[epoch]:.4f}\"\n",
-    "        )\n",
-    "\n",
-    "    return (loss_hist_train,\n",
-    "            accuracy_hist_train, exact_accuracy_hist_train,\n",
-    "            accuracy_hist_valid, exact_accuracy_hist_valid)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "W2_PN36iqStx"
-   },
-   "source": [
-    "The training can be run by defining the model, the optimizer, the loss function, the sample loaders, the number of epochs and the device.\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "rngz8XCLqTxs"
-   },
-   "outputs": [],
-   "source": [
-    "loss_hist_train, accuracy_hist_train, exact_accuracy_hist_train, \\\n",
-    "accuracy_hist_valid, exact_accuracy_hist_valid = train_loop(model, optimizer, loss_fn,\n",
-    "                                                            train_loader, val_loader,\n",
-    "                                                            n_epochs, device)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "Qr7-pybVqVSN"
-   },
-   "source": [
-    "For the optimizer, we can for example using an Adam optimizer which is an efficient optimizer which adapts the learning rate for each parameter based on estimates of the first and second moments (mean and variance) of gradients.\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "id": "e-s187KzqWkT"
-   },
-   "outputs": [],
-   "source": [
-    "optimizer = torch.optim.Adam(model.parameters(), lr=1e-3, weight_decay=1e-5)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "BarOe_RcqXDt"
-   },
-   "source": [
-    "With this architecture applied to a qubit mapping problem on hardware with 5 physical qubits (IBM Athens), a full-mapping accuracy of 90% can be obtained. This is not yet perfect, but it shows that a machine learning model can learn and solve the qubit mapping problem."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "mI5QzXpmqYrj"
-   },
-   "source": [
-    "## **5. Conclusion**"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "6Y8u-rT7qaSS"
-   },
-   "source": [
-    "In this demo, we introduced the qubit mapping problem. Qubit mapping is a fundamental problem for hardwares with limited connectivity, such as superconducting devices. Unfortunately, the qubit mapping problem is NP-hard. One way to find approximate solutions is to use heuristic methods, such as the SABRE algorithm presented in this demo.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "aQNULJR0qbV2"
-   },
-   "source": [
-    "It remains an open question whether a machine learning model can efficiently solve this problem and have better scaling than the SABRE algorithm. To explore this idea, we need a dataset, which is the purpose of QMapDataset. Three datasets for the IBM architectures Eagle R3, Heron r1, and Heron r2, respectively, are now available in PennyLane, and we showed how to access them.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "pI9wFDStqcOr"
-   },
-   "source": [
-    "Finally, we presented a Graph Neural Network model to solve the qubit mapping problem. This model shows encouraging results on small hardware. The next challenge is how to scale it to larger hardware, such as Eagle R3 (127 qubits), Heron r1 (133 qubits), and Heron r2 (156 qubits). Maybe the person who finds the solution will be you, using the dataset available in PennyLane!\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
-  }
- ],
- "metadata": {
-  "colab": {
-   "provenance": []
-  },
-  "kernelspec": {
-   "display_name": "Python 3 (ipykernel)",
-   "language": "python",
-   "name": "python3"
-  },
-  "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
-    "version": 3
-   },
-   "file_extension": ".py",
-   "mimetype": "text/x-python",
-   "name": "python",
-   "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython3",
-   "version": "3.13.10"
-  }
- },
- "nbformat": 4,
- "nbformat_minor": 4
-}
diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
new file mode 100644
index 0000000000..35d1ecddd3
--- /dev/null
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -0,0 +1,1066 @@
+r"""**Solving the qubit mapping problem with machine learning**

+===========================================================

+"""
+
+######################################################################
+# The demo introduces the qubit mapping problem, an NP-hard challenge in quantum computing systems

+# with limited hardware connectivity, and explores machine learning-based approaches for solving it.

+# It presents the QMapDataset available in PennyLane and shows how it can be used to train a Graph

+# Neural Network to tackle the qubit mapping problem.

+# 
+
+######################################################################
+# **1. What is the qubit mapping problem?**

+# -----------------------------------------

+# 
+
+######################################################################
+# When we write quantum circuits (with Pennylane, of course!), we usually assume that all the qubits

+# are connected to each other, or more precisely, that two-qubit gates can be applied between any

+# qubits. While some infrastructures have all-to-all connectivity (like the photonic devices developed

+# by Xanadu, the ion or the neutral atom platforms), superconducting quantum processors have fixed

+# connectivity between qubits. Only physically connected qubits can directly interact. For example,

+# the first generation of superconducting devices had 5 qubits with a simple linear connectivity:

+# 

+# :math:`pq_0-pq_1-pq_2-pq_3-pq_4`

+# 
+
+######################################################################
+# Physical qubit 0 and physical qubit 3 are not directly connected. If a two-qubit gate is required in

+# your code between qubits 0 and 3, it is necessary to apply SWAP gates. SWAP gates exchange the

+# states of two neighboring qubits for all basis states in the superposition (more details here `What

+# is a SWAP gate? \| PennyLane <https://pennylane.ai/qml/glossary/what-is-a-swap-gate>`__).In our

+# example, we would need to apply two SWAP gates: one between :math:`pq_0` and :math:`pq_1`, and one

+# between :math:`pq_1` and :math:`pq_2`. If the state was

+# :math:`\frac{1}{\sqrt{2}}(|10000\rangle + |00001\rangle)` before the SWAP gates, it becomes

+# :math:`\frac{1}{\sqrt{2}}(|00100\rangle + |00001\rangle)`. Then, we can apply the CNOT gate between

+# :math:`pq_2` and :math:`pq_3`.

+# 
+
+######################################################################
+# The lack of full connectivity between physical qubits does not alter the universality of quantum

+# computation in these devices but leads to an increase in the depth of the effective quantum circuit.

+# A SWAP gate corresponds to 3 CNOT gates. Therefore, instead of 1 CNOT gate, we have 7 CNOT gates.

+# This increase can be fatal for your quantum computation, especially on a NISQ device.

+# 
+
+######################################################################
+# Indeed, NISQ devices are sensitive to noise. In particular, two-qubit gates are noisy. Therefore,

+# when a large number of two-qubit gates is applied on a NISQ device, the quantum information is

+# progressively lost and randomized. Therefore, when running a quantum circuit on a NISQ device, we

+# try to minimize the number of two-qubit gates. Obviously, the increase by a factor of 7 in our

+# example is terrible for the noise robustness of our quantum computation. Even for fault-tolerant

+# quantum computers, reducing the number of CNOT gates would help to reduce the time of computation

+# and therefore the computational cost.

+# 
+
+######################################################################
+# It is here that qubit mapping can help mitigate the number of SWAP gates. There is no rule that

+# constrains the first qubit of your quantum circuit to be the first physical qubit, the second qubit

+# of your quantum circuit to be the second physical qubit, etc. We can separate the two sets of

+# qubits: on one side, we have the qubits of our quantum circuit, called logical qubits and denoted

+# :math:`lq_0`, :math:`lq_1`, etc., and on the other side, the physical qubits of the hardware denoted

+# :math:`pq_0`, :math:`pq_1` etc.. There does not necessarily have to be an identity mapping between

+# the two. Instead, we can define any mapping: :math:`\{lq\}\rightarrow \{pq\}`. The qubit mapping

+# problem corresponds to finding the initial mapping between the logical qubits and the physical

+# qubits that minimizes the number of SWAP gates during the execution of the quantum circuit by the

+# hardware. In our simple example, if we assign :math:`lq_0\rightarrow pq_0` and

+# :math:`lq_3\rightarrow pq_1`, a CNOT gate between physical qubits 0 and 1 corresponds to a CNOT gate

+# between logical qubits 0 and 3, as desired.

+# 
+
+######################################################################
+# Let�s consider a slightly more complex quantum circuit:

+# 
+
+import pennylane as qp
+dev = qp.device("default.qubit", wires=5)
+@qp.qnode(dev)
+def  circuit():
+    qp.CNOT(wires=[0,4])
+    qp.CNOT(wires=[1,3])
+    qp.CNOT(wires=[0,2])
+
+######################################################################
+# So, there are 3 CNOT gates: one between :math:`lq_0` and :math:`lq_4`, one between :math:`lq_1` and

+# :math:`lq_3`, and one between :math:`lq_0` and :math:`lq_2`. Let’s first examine the number of

+# SWAP gates with the identity mapping (:math:`lq_0\rightarrow pq_0`, :math:`lq_1\rightarrow pq_1`,

+# etc.):

+# 

+# :math:`pq_0(lq_0)-pq_1(lq_1)-pq_2(lq_2)-pq_3(lq_3)-pq_4(lq_4)`

+# 
+
+######################################################################
+# Here, in parentheses, the logical qubit mapped to the corresponding physical qubit is shown.

+# 
+
+######################################################################
+# For the first CNOT gate between :math:`lq_0` and :math:`lq_4`, we need to apply 3 SWAP gates

+# (SWAP(:math:`pq_0`, :math:`pq_1`), SWAP(:math:`pq_1`, :math:`pq_2`), and SWAP(:math:`pq_2`,

+# :math:`pq_3`)). The mapping becomes:

+# 

+# :math:`pq_0(lq_1)-pq_1(lq_2)-pq_2(lq_3)-pq_3(lq_0)-pq_4(lq_4)`

+# 
+
+######################################################################
+# Afterward, for the CNOT gate between :math:`lq_1` and :math:`lq_3`, we need a SWAP gate between

+# :math:`pq_1` and :math:`pq_2`:

+# 

+# :math:`pq_0(lq_1)-pq_1(lq_3)-pq_2(lq_2)-pq_4(lq_0)-pq_4(lq_4)`

+# 
+
+######################################################################
+# Finally, for the CNOT gate between :math:`lq_0` and :math:`lq_2`, we do not need any additional SWAP

+# gates. Therefore, in total, there are 4 SWAP gates.

+# 
+
+######################################################################
+# Now, let’s consider the following initial mapping:

+# 

+# :math:`pq_0(lq_1)-pq_1(lq_3)-pq_2(lq_2)-pq_3(lq_0)-pq_4(lq_4)`

+# 
+
+######################################################################
+# In this case, there is no need for SWAP gates. The three CNOT gates can be applied directly. We go

+# from 15 CNOT gates to 3 CNOT gates, a reduction by a factor of 5 just by changing the initial

+# mapping!

+# 
+
+######################################################################
+# Now that the qubit mapping problem is clear from these 5-qubit examples, let’s give a generalized

+# definition. We start with:

+# 

+# -  A quantum circuit C defined on n logical qubits :math:`lq_0, lq_1, \dots, lq_{n-1}`

+# -  A quantum device with m physical qubits :math:`pq_0, pq_1, \dots, pq_{m-1}` connected by some

+#    graph :math:`G_D`.

+# 
+
+######################################################################
+# The goal of the qubit mapping problem is to find the initial mapping

+# :math:`\pi_0: \{lq\}\rightarrow \{pq\}` in order to minimize the number of SWAP gates during the

+# execution of the quantum circuit C on the quantum device. Note that in general, the number of

+# logical qubits :math:`n` is smaller than the number of physical qubits :math:`m` (and cannot be

+# bigger).

+# 
+
+######################################################################
+# The initial qubit mapping problem can be represented as an assignment matrix A of dimension

+# :math:`n\times m`:

+# 

+# -  :math:`A_{ij}=1` if logical qubit :math:`lq_i` is mapped to the physical qubit :math:`pq_j`.

+# -  :math:`A_{ij}=0` otherwise

+# 
+
+######################################################################
+# There are two constraints on the assignment matrix A : \* Each logical qubit is assigned exactly one

+# physical qubit :math:`\sum_{j=1}^m A_{ij} = 1, ~\forall i`

+# 

+# -  Each physical qubit holds at most one logical qubit :math:`\sum_{i=1}^n A_{ij}\leq 1, ~\forall j`

+# 
+
+######################################################################
+# An important quantity is the distance between two physical qubits :math:`D_{ij}=d(pq_i, pq_j)`. The

+# distance is the shortest-path length between two physical qubits in the hardware connectivity graph,

+# and it directly estimates how many SWAPs are needed to make them interact. Of course, the distance

+# would depend on the hardware connectivity graph.

+# 
+
+######################################################################
+# We will explore in the next section the actual methods used to find the optimal assignment matrix A

+# and therefore to solve the qubit mapping problem.

+# 
+
+######################################################################
+# **2.How to solve the qubit mapping problem?**

+# ---------------------------------------------

+# 
+
+######################################################################
+# The qubit mapping problem is unfortunately a NP problem and is exponentially hard to find the

+# optimal solution when the number of logical and physical qubits increases. However, there are some

+# heuristic methods that allow us to solve it with a reasonable computational time. The most famous

+# heuristic algorithm is the SABRE algorithm, used for example by Qiskit in its compilation.

+# 
+
+######################################################################
+# The SABRE algorithm works by iteratively updating the qubit mapping guided by a cost function based

+# on the total distance.

+# 
+
+######################################################################
+# **Forward propagation** 1. **Initialize the mapping**: The algorithm starts by initializing a single

+# mapping between logical and physical qubits (typically random or identity). Example (5-qubit

+# device): :math:`pq_0(lq_0)-pq_1(lq_2)-pq_2(lq_3)-pq_3(lq_4)-pq_4(lq_1)`

+# 

+# 2. **Define the front layer**: The front layer is the set of gates that are ready to be executed,

+#    meaning all their dependencies in the circuit have been satisfied. For the quantum circuit

+#    (CNOT(0,4), CNOT(1,3) and CNOT(0,2), the front layer is {CNOT(0,4), CNOT(1,3)}. CNOT(0,2) is not

+#    included because it depends on the execution of CNOT(0,4).

+# 

+# 3. **Check gate executability**: For each gate in the front layer, the algorithm checks whether the

+#    mapped physical qubits are connected in the hardware graph: If they are adjacent, the gate �is

+#    executed� and is withdrawn from the front layer. If not, SWAP gate is required and step 4 is

+#    applied. In our case, the CNOT(0,4) and CNOT(1,3) cannot be executed.

+# 

+# 4. **Compute routing cost**: The algorithm calculates the sum of the distance of the operations in

+#    the front layer: :math:`H = \sum_{(a,b)} d(\pi(a), \pi(b))`

+# 

+# 5. **Evaluate SWAP candidates**: The algorithm considers SWAPs on edges of the hardware graph. For

+#    each candidate SWAP, they recompute the cost H and select the SWAP that minimizes the cost. In

+#    our case, the SWAP(2,3) gives the distance :math:`H=2+1=3`, which is better than other options.

+#    So this SWAP is applied.

+# 

+# 6. **Iterate forward pass**: Steps 2–5 are repeated. The front layer is updated dynamically as

+#    gates become executable. Gates are executed whenever possible. SWAPs are inserted when necessary.

+#    At the end of this process, the algorithm produces a final mapping for the forward pass.

+# 
+
+######################################################################
+# **Backward propagation**

+# 

+# 7. **Reverse circuit execution**: The algorithm then repeats the same procedure, but on the reversed

+#    circuit (executing gates in reverse order), starting from the final mapping obtained in the

+#    forward pass.

+# 

+# 8. **Improved initial mapping**: At the end of the backward pass, a new initial mapping

+#    :math:`\pi_0` is obtained. This initial mapping is typically better than the original random

+#    initialization.

+# 
+
+######################################################################
+# In practice, SABRE may repeat multiple forward and backward passes (usually around 3 iterations),

+# gradually improving the mapping until no significant improvement is observed.

+# 
+
+######################################################################
+# The SABRE algorithm has a polynomial computational complexity of approximately

+# :math:`\mathcal{O}(Tm^2 n)` where :math:`T` is the total number of two-qubit gates in the quantum

+# circuit. This is significantly better than the exponential scaling of an exact solution. However, as

+# the number of qubits in superconducting processors continues to increase, the runtime of the SABRE

+# algorithm also grows accordingly.

+# 
+
+######################################################################
+# Machine learning models could offer better scalability while maintaining high accuracy. In this

+# demo, we consider supervised learning. Supervised learning requires a dataset with input data and

+# corresponding labels to train the model. This is the purpose of QMapDataset, which provides datasets

+# and tools to generate them for solving the qubit mapping problem using supervised machine learning

+# models. QMapDataset is available in PennyLane. We will explore its features in the next section.

+# 
+
+######################################################################
+# **3. QMapDataset**

+# ------------------

+# 
+
+######################################################################
+# QMapDataset is a dataset generator for the qubit mapping problem, designed for IBM quantum devices.

+# The dataset is built from three main components: the hardware, the circuits, and the mapping.

+# 
+
+######################################################################
+# **For the hardware**, First, we choose an IBM quantum processor. To make the dataset more diverse,

+# the code can randomly permute the properties of the qubits and gates (data augmentation). We can

+# control how often these permutations are applied.

+# 
+
+######################################################################
+# Then, the code collects all the relevant hardware information:

+# 

+# -  physical qubit properties (:math:`T_1`, :math:`T_2`, readout error)

+# -  single-qubit gate errors for each qubit

+# -  two-qubit gate errors for each connected pair of qubits

+# 
+
+######################################################################
+# **For the circuits**, These are sampled either from:

+# 

+# -  random quantum circuits

+# -  well-known circuits (QPE, Grover, QFT, GHZ state preparation etc.)

+# 
+
+######################################################################
+# By default, the choice is split 50–50 between random and well-known circuits.

+# 

+# For random quantum circuits, the code also sample: \* the number of logical qubits \* the circuit

+# depth

+# 
+
+######################################################################
+# The number of logical qubits is chosen between 3 and the number of physical qubits available on the

+# hardware. The depth follows this distribution: 25% shallow (5–15 layers), 40% medium (16–50),

+# 30% deep (51–150), and 5% very deep circuits (151–400).

+# 
+
+######################################################################
+# To further enrich the dataset, we apply random permutations of qubits to the well-known circuits

+# (since there are only a few of them). After that, the code collects:

+# 

+# -  Single-qubit gate counts for each logical qubit

+# -  Two-qubit gate counts for each pair of logical qubits

+# 
+
+######################################################################
+# Finally, **for the mapping**, the code uses the generated hardware and circuits to compute the qubit

+# mapping with an advanced version of the SABRE algorithm (optimization_level = 3 in Qiskit).

+# 
+
+######################################################################
+# If you want more details, you can have a look at the GitHub repo: `rscadrien/QMapDataset: Generator

+# of dataset for qubit mapping <https://github.com/rscadrien/QMapDataset>`__.

+# 
+
+######################################################################
+# The good news is that you do not need to generate this dataset using QMapDataset yourself! We have

+# already prepared datasets for three different types of IBM quantum processors: Eagle 3, Heron 1, and

+# Heron 2. Each dataset contains 10,000 samples, where each sample consists of a circuit, hardware

+# description, and mapping.

+# 
+
+######################################################################
+# Let’s see how to use these datasets with PennyLane and extract their information.

+# 

+# The first step is to load a dataset from the Xanadu server. For example, to load the dataset for the

+# Eagle 3 architecture:

+# 
+
+import pennylane as qp
+[ds] = qp.data.load("other", name="qubit-mapping-eagle3")
+
+######################################################################
+# For the Heron 1 and Heron 2 architectures, you only need to replace the name argument with

+# �qubit-mapping-heron1� or �qubit-mapping-heron2�, respectively.

+# 

+# Each dataset is divided into three attributes: circuits, hardware, and mappings, following the

+# structure discussed earlier. For example, to access the first sample:

+# 
+
+sample0_circuit = ds.circuits[0]
+sample0_hardware = ds.hardware[0]
+sample0_mapping = ds.mappings[0]
+
+######################################################################
+# Each of these is a dictionary containing the features of the circuit, hardware, or mapping. For

+# example, to access the single-qubit and two-qubit gate counts of the quantum circuit:

+# 
+
+Single_qubit_count = sample0_circuit["single_qubit_counts"]
+Two_qubit_count = sample0_circuit["two_qubit_counts"]
+
+######################################################################
+# To retrieve :math:`T_1`, :math:`T_2`, and the two-qubit gate error from the hardware data:

+# 
+
+T_1 = sample0_hardware["T1"]
+T_2 = sample0_hardware["T2"]
+Two_qubit_error = sample0_hardware["ecr"]
+
+######################################################################
+# Finally, to obtain the mapping (which can serve as the label in a supervised learning model):

+# 
+
+mapping = sample0_mapping["final_mapping"]
+
+######################################################################
+# Detailed information about all available features and how to access them can be found in the �Data�

+# section of the Qubit Mapping Eagle 3 page on the PennyLane website. (`Qubit Mapping Eagle

+# 3 <https://pennylane.ai/datasets/qubit-mapping-eagle3>`__ ).

+# 
+
+######################################################################
+# **4.Building a machine learning model to solve the qubit mapping problem**

+# --------------------------------------------------------------------------

+# 
+
+######################################################################
+# In this session, we will show an example of how to preprocess data from QMapDataset and how to build

+# a machine learning model that can solve the qubit mapping problem.

+# 
+
+######################################################################
+# **4.1 Preprocessing**

+# ~~~~~~~~~~~~~~~~~~~~~

+# 
+
+######################################################################
+# The qubit mapping problem strongly depends on the connectivity between the physical qubits in the

+# given hardware, as well as the connectivity between the logical qubits in the circuits (represented

+# by two-qubit gate counts). Therefore, a graph is a natural representation for solving the qubit

+# mapping problem. We will see how to construct graphs for both the circuits and the hardware using

+# the QMapDataset available in PennyLane.

+# 
+
+######################################################################
+# A graph is a way of showing relationships. It is made up of points, called nodes (or vertices), that

+# are connected by lines, called edges. Both nodes and edges can also have extra information attached

+# to them, called features.

+# 
+
+######################################################################
+# For the **circuit**, the nodes represent the logical qubits, and edges connect pairs of logical

+# qubits between which two-qubit gates are applied in the quantum circuit. The features of each node

+# are the counts of different single-qubit gates applied to the corresponding logical qubit. The

+# features of each edge are the counts of different two-qubit gates applied between the two logical

+# qubits. Since the relative values of these features are what matter, we standardize them. This is

+# the code to build the graph for the quantum circuit of the QMapDataset:

+# 
+
+import torch
+from torch_geometric.data import Data
+def build_circuit_graph (circuit_data):
+    # -----------------------------
+    # STEP 1: Extract and scale single-qubit gate counts
+    # Each function returns a vector of size (n_qubits,)
+    # representing how often each gate was applied per qubit
+    # -----------------------------
+    rz_counts_scaled = get_single_qubit_counts(circuit_data, "rz")
+    sx_counts_scaled = get_single_qubit_counts(circuit_data, "sx")
+    x_counts_scaled = get_single_qubit_counts(circuit_data, "x")
+
+    # -----------------------------
+    # STEP 2: Extract and scale two-qubit gate counts
+    # These describe interactions between pairs of qubits
+    # -----------------------------
+    ecr_scaled = get_two_qubit_counts(circuit_data,'ecr')
+
+    # -----------------------------
+    # STEP 3: Build node feature matrix
+    # Each node = a logical qubit
+    # Features = [rz count, sx count, x count]
+    # Shape: (n_qubits, 3)
+    # -----------------------------
+    X =torch.tensor(
+    list(zip(rz_counts_scaled, sx_counts_scaled, x_counts_scaled)),
+    dtype=torch.float
+    )
+    # -----------------------------
+    # STEP 4: Build graph connectivity (edges)
+    # edge_index: which qubits are connected
+    # edge_attr: weight/feature of each edge (scaled gate counts)
+    # -----------------------------
+    ecr_index, ecr_attr = edge_info(ecr_scaled)
+    # -----------------------------
+    # STEP 5: Create PyTorch Geometric graph object
+    # x -> node features
+    # edge_index -> connectivity (COO format)
+    # edge_attr -> edge features
+    # -----------------------------
+    G_circuit = Data(
+    x=X,
+    edge_index=ecr_index,
+    edge_attr=ecr_attr,
+    )
+    return G_circuit
+
+def get_single_qubit_counts(data, gate_name):
+    # Number of logical qubits in the circuit
+    n = data["n_logical_qubits"]
+    # Try to retrieve counts for the given gate type
+    counts = data.get("single_qubit_counts", {}).get(gate_name, None)
+    # If no data exists, return a zero vector
+    if counts is None:
+        return np.zeros(n)
+    else:
+    # Otherwise normalize counts across qubits (z-score normalization)
+        return standard_scaling_across_qubits(np.array(counts))
+
+def get_two_qubit_counts(data, gate_name):
+    # Safely retrieve list of two-qubit gate interactions
+    counts = data.get("two_qubit_counts", {}).get(gate_name, [])
+
+    # If no interactions exist, return empty list
+    if not counts:
+        return counts
+    else:
+    # Otherwise scale the edge weights
+      return scaled_two_qubit_list(counts)
+
+def standard_scaling_across_qubits(vector):
+    # Compute mean and standard deviation for normalization
+    mean = vector.mean()
+    std = vector.std()
+    # Avoid division by zero if all values are identical
+    if std == 0 :
+        vector_scaled = vector
+    else:
+        vector_scaled = (vector - mean)/std
+    return vector_scaled
+
+def scaled_two_qubit_list(two_data):
+    # Extract raw edge weights from dictionary list
+    values = np.array([d['value']for d in two_data], dtype=float)
+    # Normalize weights
+    scaled_values = standard_scaling_across_qubits(values)
+    # Reattach scaled values back into original edge dictionaries
+    two_gate_scaled = [
+    {**d, 'value':v_scaled}
+    for d, v_scaled in zip(two_data, scaled_values)
+    ]
+    return two_gate_scaled
+
+def edge_info(gate_dict):
+    # Lists to store graph connectivity
+    edge_index = []
+    edge_attr = []
+    # Convert list of edge dictionaries into tensor format
+    for gate in gate_dict:
+        i,j,w = gate["row"], gate["col"], gate["value"]
+        edge_index.append([i,j])
+        edge_attr.append([w])
+    # Convert to PyTorch tensors
+    # edge_index shape: [2, num_edges]
+    edge_index = torch.tensor(edge_index, dtype=torch.long).T
+    edge_attr = torch.tensor(edge_attr, dtype=torch.float)
+    return edge_index, edge_attr
+
+######################################################################
+# For example, if this code is applied for the circuit of the 27th sample, we obtain:

+# 
+
+Graph_circuit = build_circuit_graph(ds.circuits[26])
+
+######################################################################
+# Using the networkx package, we can visualize the graph using the following graph:

+# 
+
+import matplotlib.pyplot as plt
+import networkx as nx
+from torch_geometric.utils import to_networkx
+# Convert PyG Data object to NetworkX graph
+G = to_networkx(
+    test,
+    edge_attrs=["edge_attr"],  # keep CX counts
+    node_attrs=["x"],          # keep node features
+    to_undirected=False
+)
+edge_labels = {
+    (u, v): f"{d['edge_attr'][0]:.2f}"
+    for u, v, d in G.edges(data=True)
+}
+nx.draw(
+    G,
+    pos,
+    with_labels=True,
+    node_size=800,
+)
+
+nx.draw_networkx_edge_labels(
+    G,
+    pos,
+    edge_labels=edge_labels
+)
+
+plt.show()
+
+######################################################################
+# We obtain the following graph: |My graph|

+# 

+# .. |My graph| image:: graph_circuit.png

+# 
+
+######################################################################
+# For the **hardware**, the nodes represent the physical qubits, and edges connect pairs of physical

+# qubits that are directly connected in the device architecture. The features of each node are the

+# :math:`T_1` and :math:`T_2` coherence times, readout error, and single-qubit gate errors for each

+# physical qubit. The features of each edge are the errors of the different two-qubit gates applied

+# between connected physical qubits. As with the circuits, the feature values are standardized. The

+# code to build the graph for the hardware is pretty similar to the one for the circuit:

+# 
+
+def build_hardware_graph(hardware_data):
+    # Scaling single qubit properties and gate
+    T1_scaled = standard_scaling_across_qubits(np.array(hardware_data['T1']))
+    T2_scaled = standard_scaling_across_qubits(np.array(hardware_data['T2']))
+    readout_error_scaled = standard_scaling_across_qubits(np.array(hardware_data['readout_error']))
+    sx_error_scaled = standard_scaling_across_qubits(np.array(hardware_data['sx_error']))
+    x_error_scaled = standard_scaling_across_qubits(np.array(hardware_data['x_error']))
+    #Scaling two qubit gate properties
+    cx_error_scaled = scaled_two_qubit_list(hardware_data['cx'])
+    #Build the node feature matrix
+    X = torch.tensor(
+    list(zip(T1_scaled, T2_scaled, readout_error_scaled, sx_error_scaled, x_error_scaled)),
+    dtype=torch.float
+    )
+    #Build the edge index and attributes matrix
+    cx_index, cx_attr = edge_info(cx_error_scaled)
+    #Build the graph
+    G_hardware = Data(
+    x=X,
+    edge_index=cx_index,
+    edge_attr=cx_attr,
+    )
+    return G_hardware
+
+######################################################################
+# We can again visualize the graph using networkx:

+# 
+
+# Draw graph
+
+plt.figure(figsize=(10, 10))
+
+pos = nx.spring_layout(
+    G_hardware,
+    seed=42,
+    k=1 / np.sqrt(len(G_hardware)),
+    iterations=200
+)
+
+nx.draw_networkx_edges(G_hardware, pos, alpha=0.2, width=0.5)
+nx.draw_networkx_nodes(G_hardware, pos, node_size=40)
+
+plt.axis("off")
+plt.savefig("graph_hardware.png", dpi=300, bbox_inches="tight")
+plt.show()
+
+######################################################################
+# .. figure:: graph_hardware.png

+#    :alt: My graph

+# 

+#    My graph

+# 
+
+######################################################################
+# **4.2 Machine learning model**

+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

+# 
+
+######################################################################
+# We will show how to build a model that can learn qubit mapping. For this model, we will use an

+# architecture adapted to the graph structure of the input: a Graph Neural Network (GNN). The key

+# principle behind GNNs is that a node learns by aggregating information from its neighbors. A GNN

+# typically works in layers, and each layer is mostly composed of three steps:

+# 

+# -  Message passing : Each node receives information from its neighbors

+# -  Aggregation: Combine neighbor information (e.g., sum, mean, max)

+# -  Update: Use a neural network to update the node’s embedding.

+# 
+
+######################################################################
+# In our case, we will use two separate GNNs: one for the quantum circuit and one for the hardware.

+# After several GNN layers, we obtain updated embedding vectors for the nodes of both graphs,

+# :math:`h_L^{(i)}` and :math:`h_P^{(j)}`. (:math:`h_L^{(i)}` is the vector of the i th node of the

+# quantum circuit graph while :math:`h_P^{(j)}` is the vector of the jth node of the hardware graph).

+# 
+
+######################################################################
+# We then compare these embeddings using a dot product:

+# 

+# :math:`S_{ij}=h_L^{(i)}\cdot h_P^{(j)}`

+# 
+
+######################################################################
+# The matrix element :math:`S_{ij}` represents the logit that the logical qubit :math:`lq_i` is mapped

+# to the physical qubit :math:`pq_j`. The probability that logical qubit k maps to physical qubit j is

+# calculated from the logit:

+# 

+# :math:`p_{ij}=\frac{\text{exp}(S_{ij})}{\sum_{j^\prime} \text{exp}(S_{ij}^\prime)}`

+# 
+
+######################################################################
+# During training, the matrix :math:`p_{ij}` should ideally converge to the assignment matrix

+# :math:`A_{ij}` presented in Section 2, where each row contains exactly one nonzero element equal to

+# 1. After training, during inference, the strict assignment constraints are enforced by applying the

+# Hungarian algorithm. Unfortunately, this algorithm is not differentiable and therefore cannot be

+# used during training.The code for this architecture is as follows :

+# 
+
+class QubitMapping_dot_Model(nn.Module):
+    def __init__(self, node_c_dim, node_h_dim, edge_c_dim, edge_h_dim, hidden_dim):
+        super().__init__()
+
+        # === GNN for the LOGICAL CIRCUIT graph ===
+        # These layers learn embeddings for logical qubits based on circuit structure
+
+        # First layer: edge-conditioned convolution
+        # Edge features (e.g., gate types) generate weights dynamically
+        self.nn_c1 = make_edge_mlp(edge_c_dim, node_c_dim * hidden_dim)
+        self.gnn_c1 = NNConv(node_c_dim, hidden_dim, nn=self.nn_c1, aggr='add')
+
+        # Second layer: refines embeddings
+        self.nn_c2 = make_edge_mlp(edge_c_dim, hidden_dim * hidden_dim)
+        self.gnn_c2 = NNConv(hidden_dim, hidden_dim, nn=self.nn_c2, aggr='add')
+
+        # === GNN for the HARDWARE graph ===
+        # These layers learn embeddings for physical qubits based on hardware connectivity
+
+        self.nn_h1 = make_edge_mlp(edge_h_dim, node_h_dim * hidden_dim)
+        self.gnn_h1 = NNConv(node_h_dim, hidden_dim, nn=self.nn_h1, aggr='add')
+
+        self.nn_h2 = make_edge_mlp(edge_h_dim, hidden_dim * hidden_dim)
+        self.gnn_h2 = NNConv(hidden_dim, hidden_dim, nn=self.nn_h2, aggr='add')
+
+
+    def forward(self, G_circuit, G_hardware):
+
+        # === Step 1: Compute embeddings for LOGICAL qubits ===
+        # Each node becomes a vector summarizing its role in the circuit
+        x_c = F.relu(self.gnn_c1(
+            G_circuit.x,              # node features
+            G_circuit.edge_index,    # connectivity
+            G_circuit.edge_attr      # edge features (e.g., gate info)
+        ))
+
+        h_L = F.relu(self.gnn_c2(
+            x_c,
+            G_circuit.edge_index,
+            G_circuit.edge_attr
+        ))
+
+        # === Step 2: Compute embeddings for PHYSICAL qubits ===
+        # Each node becomes a vector describing hardware properties
+        x_h = F.relu(self.gnn_h1(
+            G_hardware.x,
+            G_hardware.edge_index,
+            G_hardware.edge_attr
+        ))
+
+        h_P = F.relu(self.gnn_h2(
+            x_h,
+            G_hardware.edge_index,
+            G_hardware.edge_attr
+        ))
+
+
+        # === Step 3: Compute compatibility between logical and physical qubits ===
+
+        prob_list = []
+
+        # Number of graphs in the batch
+        num_graphs = G_circuit.batch.max().item() + 1
+
+        # Loop over each graph in the batch
+        for i in range(num_graphs):
+
+            # --- Select nodes belonging to graph i ---
+            # (because batching mixes all graphs together)
+            mask_c = (G_circuit.batch == i)
+            mask_h = (G_hardware.batch == i)
+
+            # Extract embeddings for this specific graph
+            h_L_i = h_L[mask_c]  # logical qubits
+            h_P_i = h_P[mask_h]  # physical qubits
+
+            # --- Compute compatibility matrix ---
+            # Dot product measures how well each pair matches
+            # Shape: [num_logical_qubits, num_physical_qubits]
+            S_i = torch.matmul(h_L_i, h_P_i.T)
+
+            if self.training:
+                # During training:
+                # return raw scores (used in loss function)
+                prob_i = S_i
+                prob_list.append(prob_i)
+
+            else:
+                # During inference:
+                # find the BEST one-to-one assignment
+
+                # Move to CPU + numpy for Hungarian algorithm
+                S_cpu = S_i.detach().cpu().numpy()
+
+                # Hungarian algorithm finds assignment maximizing total score
+                row_ind, col_ind = linear_sum_assignment(-S_cpu)
+
+                # Build mapping: logical qubit -> physical qubit
+                mapping_i = torch.zeros(h_L_i.size(0), dtype=torch.long)
+                mapping_i[row_ind] = torch.tensor(col_ind, dtype=torch.long)
+
+                prob_list.append(mapping_i)
+
+        # === Step 4: Merge results from all graphs ===
+
+        if self.training:
+            # Concatenate score matrices
+            # Shape: [total_logical_nodes_in_batch, num_physical_nodes_per_graph]
+            prob = torch.cat(prob_list, dim=0)
+            return prob
+        else:
+            # Concatenate mappings
+            # Shape: [total_logical_nodes_in_batch]
+            mapping = torch.cat(prob_list, dim=0)
+            return mapping
+
+######################################################################
+# **4.3 Loss function and metric**

+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

+# 
+
+######################################################################
+# We need to have a loss function to minimize for the training of the model.

+# 
+
+######################################################################
+# For each logical qubit k, we know the correct assignment (from the training data), which we denote

+# as :math:`k_c`. This is the physical qubit that the logical qubit should be mapped to.

+# 
+
+######################################################################
+# The loss for the logical qubit k is then defined as:

+# 

+# :math:`L_k=\log(p_{k,k_c})`

+# 
+
+######################################################################
+# Finally, the total loss is computed by averaging over all logical qubits :

+# 

+# :math:`L = \frac{1}{n} \sum_{k=1}^n -\log(p_{k,k_c})`

+# 
+
+######################################################################
+# This is known as the cross-entropy loss. With Pytorch, the cross entropy loss can be calculated by

+# using nn.CrossEntropyLoss():

+# 
+
+loss_fn = nn.CrossEntropyLoss()
+
+######################################################################
+# This loss encourages the model to assign high probability to the correct qubit mapping and to assign

+# low probability to incorrect mappings. However, it does not enforce one-to-one mapping and does not

+# prevent collisions (two logical qubits choosing the same physical qubit). It is the reason that it

+# is better to use the Hungarian algorithm during inference.

+# 
+
+######################################################################
+# In addition to the loss function, there are some metrics that help to evaluate the quality of the

+# model and are easier to interpret than the loss function. We consider two different metrics: \*

+# logical qubit-level accuracy \* full mapping accuracy

+# 
+
+######################################################################
+# To understand the difference between them, let us consider an example with 5 logical qubits and 5

+# physical qubits. The correct mapping (from the training data) is [2,1,4,3,0] and the model predicts

+# [2,0,4,3,1].

+# 
+
+######################################################################
+# For the logical qubit-level accuracy, we simply count how many individual assignments are correct.

+# In this example, 3 out of 5 qubits are correctly mapped. So the accuracy is 60%.

+# 
+
+######################################################################
+# For the full mapping accuracy, the prediction is considered correct only if the entire assignment is

+# exactly correct. In this example, the predicted mapping is not identical to the ground truth. So the

+# full mapping is incorrect. This metric is therefore much stricter and harder to improve.

+# 
+
+######################################################################
+# You could ask: Why use both metrics? These two metrics serve different goals. The logical

+# qubit-level accuracy is useful to monitor progress during training. The full mapping accuracy

+# reflects the actual goal of the problem: producing a correct global qubit mapping. During training,

+# the logical-level accuracy improves first and helps track learning progress while the full-mapping

+# accuracy becomes meaningful only later, when the model is already quite accurate. Note that the

+# Hungarian algorithm significantly improves the full-mapping accuracy. As a result, the full-mapping

+# accuracy is often higher on the validation set than on the training set.

+# 
+
+######################################################################
+# These metrics are calculated with the following code for the training and the validation dataset:

+# 
+
+def batch_graph_metrics_training(logit, qu_map, batch):
+    """
+    Compute per-graph metrics in a single pass:
+      - circuit-averaged node accuracy
+      - exact-mapping accuracy
+
+    Args:
+        logit: [N_total_nodes, N_P] predicted logit
+        qu_map: [N_total_nodes] ground-truth physical qubits
+        batch: [N_total_nodes] graph indices
+
+    Returns:
+        node_acc_sum: sum of node-level accuracies per graph
+        exact_sum: number of perfectly mapped graphs
+        num_graphs: number of graphs in the batch
+    """
+
+    pred = logit.argmax(dim=1)
+
+    num_graphs = batch.max().item() + 1
+    node_acc_sum = 0.0
+    exact_sum = 0
+
+    for g in range(num_graphs):
+        mask = (batch == g)
+        correct_nodes = (pred[mask] == qu_map[mask]).sum().item()
+        n_nodes = mask.sum().item()
+
+        node_acc_sum += correct_nodes / n_nodes
+        if correct_nodes == n_nodes:
+            exact_sum += 1
+
+    return node_acc_sum, exact_sum, num_graphs
+
+def batch_graph_metrics_val(pred, qu_map, batch):
+    """
+    Compute per-graph metrics in a single pass:
+      - circuit-averaged node accuracy
+      - exact-mapping accuracy
+
+    Args:
+        pred: [N_total_nodes] predicted mapping
+        qu_map: [N_total_nodes] ground-truth physical qubits
+        batch: [N_total_nodes] graph indices
+
+    Returns:
+        node_acc_sum: sum of node-level accuracies per graph
+        exact_sum: number of perfectly mapped graphs
+        num_graphs: number of graphs in the batch
+    """
+
+    num_graphs = batch.max().item() + 1
+    node_acc_sum = 0.0
+    exact_sum = 0
+
+    for g in range(num_graphs):
+        mask = (batch == g)
+        correct_nodes = (pred[mask] == qu_map[mask]).sum().item()
+        n_nodes = mask.sum().item()
+
+        node_acc_sum += correct_nodes / n_nodes
+        if correct_nodes == n_nodes:
+            exact_sum += 1
+
+    return node_acc_sum, exact_sum, num_graphs
+
+######################################################################
+# **4.3 Training Loop**

+# ~~~~~~~~~~~~~~~~~~~~~

+# 
+
+######################################################################
+# We now have all the elements to build our training loop! Here it is the code:

+# 
+
+def train_loop(model, optimizer, loss_fn, loader_train, loader_valid, n_epoch, device):
+    """
+    Training and validation loop for qubit mapping model.
+    Returns:
+        loss_hist_train, accuracy_hist_train, exact_accuracy_hist_train,
+        accuracy_hist_valid, exact_accuracy_hist_valid
+    """
+    loss_hist_train = [0.0] * n_epoch
+    accuracy_hist_train = [0.0] * n_epoch
+    exact_accuracy_hist_train = [0.0] * n_epoch
+    accuracy_hist_valid = [0.0] * n_epoch
+    exact_accuracy_hist_valid = [0.0] * n_epoch
+
+    for epoch in range(n_epoch):
+        # ------------ TRAIN ------------------
+        model.train()
+        total_loss = 0.0
+        total_acc = 0.0
+        total_exact = 0
+        total_graphs = 0
+        for G_c, G_h,  qu_map in loader_train:
+            G_c = G_c.to(device)
+            G_h = G_h.to(device)
+
+            qu_map = qu_map.to(device)
+
+            optimizer.zero_grad()
+            logit = model(G_c, G_h)
+            loss = loss_fn(logit, qu_map)
+            loss.backward()
+            optimizer.step()
+            #Metrics
+            acc_sum, exact_sum, n_graphs = batch_graph_metrics_training(
+                logit, qu_map, G_c.batch
+            )
+
+            total_loss += loss.item() * n_graphs
+            total_acc += acc_sum
+            total_exact += exact_sum
+            total_graphs += n_graphs
+        #Store epoch metrics
+        loss_hist_train[epoch] = total_loss  / total_graphs
+        accuracy_hist_train[epoch] = total_acc / total_graphs
+        exact_accuracy_hist_train[epoch] = total_exact / total_graphs
+
+        #VALIDATION
+        model.eval()
+        total_acc = 0.0
+        total_exact = 0
+        total_graphs = 0
+
+        with torch.no_grad():
+            for G_c_val, G_h_val, qu_map_val in loader_valid:
+                G_c_val = G_c_val.to(device)
+                G_h_val = G_h_val.to(device)
+                qu_map_val = qu_map_val.to(device)
+
+                mapping = model(G_c_val, G_h_val)
+
+                acc_sum, exact_sum, n_graphs = batch_graph_metrics_val(
+                mapping, qu_map_val, G_c_val.batch
+                )
+
+                total_acc += acc_sum
+                total_exact += exact_sum
+                total_graphs += n_graphs
+
+        accuracy_hist_valid[epoch] = total_acc / total_graphs
+        exact_accuracy_hist_valid[epoch] = total_exact / total_graphs
+        print(
+            f"Epoch {epoch+1:03d} | "
+            f"Train Loss: {loss_hist_train[epoch]:.4f} | "
+            f"Train Acc: {accuracy_hist_train[epoch]:.4f} | "
+            f"Train Exact Acc: {exact_accuracy_hist_train[epoch]:.4f} | "
+            f"Val Acc: {accuracy_hist_valid[epoch]:.4f} | "
+            f"Val Exact Acc: {exact_accuracy_hist_valid[epoch]:.4f}"
+        )
+
+    return (loss_hist_train,
+            accuracy_hist_train, exact_accuracy_hist_train,
+            accuracy_hist_valid, exact_accuracy_hist_valid)
+
+######################################################################
+# The training can be run by defining the model, the optimizer, the loss function, the sample loaders,

+# the number of epochs and the device.

+# 
+
+loss_hist_train, accuracy_hist_train, exact_accuracy_hist_train, \
+accuracy_hist_valid, exact_accuracy_hist_valid = train_loop(model, optimizer, loss_fn,
+                                                            train_loader, val_loader,
+                                                            n_epochs, device)
+
+######################################################################
+# For the optimizer, we can for example using an Adam optimizer which is an efficient optimizer which

+# adapts the learning rate for each parameter based on estimates of the first and second moments (mean

+# and variance) of gradients.

+# 
+
+optimizer = torch.optim.Adam(model.parameters(), lr=1e-3, weight_decay=1e-5)
+
+######################################################################
+# With this architecture applied to a qubit mapping problem on hardware with 5 physical qubits (IBM

+# Athens), a full-mapping accuracy of 90% can be obtained. This is not yet perfect, but it shows that

+# a machine learning model can learn and solve the qubit mapping problem.

+# 
+
+######################################################################
+# **5. Conclusion**

+# -----------------

+# 
+
+######################################################################
+# In this demo, we introduced the qubit mapping problem. Qubit mapping is a fundamental problem for

+# hardwares with limited connectivity, such as superconducting devices. Unfortunately, the qubit

+# mapping problem is NP-hard. One way to find approximate solutions is to use heuristic methods, such

+# as the SABRE algorithm presented in this demo.

+# 
+
+######################################################################
+# It remains an open question whether a machine learning model can efficiently solve this problem and

+# have better scaling than the SABRE algorithm. To explore this idea, we need a dataset, which is the

+# purpose of QMapDataset. Three datasets for the IBM architectures Eagle R3, Heron r1, and Heron r2,

+# respectively, are now available in PennyLane, and we showed how to access them.

+# 
+
+######################################################################
+# Finally, we presented a Graph Neural Network model to solve the qubit mapping problem. This model

+# shows encouraging results on small hardware. The next challenge is how to scale it to larger

+# hardware, such as Eagle R3 (127 qubits), Heron r1 (133 qubits), and Heron r2 (156 qubits). Maybe the

+# person who finds the solution will be you, using the dataset available in PennyLane!

+# 
\ No newline at end of file

From 8c2312c2742880ca2fa16bab59b3142b8e24abbf Mon Sep 17 00:00:00 2001
From: drdren <104710745+drdren@users.noreply.github.com>
Date: Thu, 2 Jul 2026 09:33:07 -0400
Subject: [PATCH 06/21] Remove blank lines

---
 demonstrations_v2/QMapDataset_demo/demo.py | 794 +++++++++------------
 1 file changed, 329 insertions(+), 465 deletions(-)

diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index 35d1ecddd3..037e2c0738 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -1,76 +1,68 @@
-r"""**Solving the qubit mapping problem with machine learning**

-===========================================================

+r"""Solving the qubit mapping problem with machine learning
+=============================================================
 """
-
 ######################################################################
-# The demo introduces the qubit mapping problem, an NP-hard challenge in quantum computing systems

-# with limited hardware connectivity, and explores machine learning-based approaches for solving it.

-# It presents the QMapDataset available in PennyLane and shows how it can be used to train a Graph

-# Neural Network to tackle the qubit mapping problem.

+# The demo introduces the qubit mapping problem, an NP-hard challenge in quantum computing systems
+# with limited hardware connectivity, and explores machine learning-based approaches for solving it.
+# It presents the QMapDataset available in PennyLane and shows how it can be used to train a Graph
+# Neural Network to tackle the qubit mapping problem.
 # 
-
 ######################################################################
-# **1. What is the qubit mapping problem?**

-# -----------------------------------------

+# **1. What is the qubit mapping problem?**
+# -----------------------------------------
 # 
-
 ######################################################################
-# When we write quantum circuits (with Pennylane, of course!), we usually assume that all the qubits

-# are connected to each other, or more precisely, that two-qubit gates can be applied between any

-# qubits. While some infrastructures have all-to-all connectivity (like the photonic devices developed

-# by Xanadu, the ion or the neutral atom platforms), superconducting quantum processors have fixed

-# connectivity between qubits. Only physically connected qubits can directly interact. For example,

-# the first generation of superconducting devices had 5 qubits with a simple linear connectivity:

-# 

-# :math:`pq_0-pq_1-pq_2-pq_3-pq_4`

+# When we write quantum circuits (with Pennylane, of course!), we usually assume that all the qubits
+# are connected to each other, or more precisely, that two-qubit gates can be applied between any
+# qubits. While some infrastructures have all-to-all connectivity (like the photonic devices developed
+# by Xanadu, the ion or the neutral atom platforms), superconducting quantum processors have fixed
+# connectivity between qubits. Only physically connected qubits can directly interact. For example,
+# the first generation of superconducting devices had 5 qubits with a simple linear connectivity:
+# 
+# :math:`pq_0-pq_1-pq_2-pq_3-pq_4`
 # 
-
 ######################################################################
-# Physical qubit 0 and physical qubit 3 are not directly connected. If a two-qubit gate is required in

-# your code between qubits 0 and 3, it is necessary to apply SWAP gates. SWAP gates exchange the

-# states of two neighboring qubits for all basis states in the superposition (more details here `What

-# is a SWAP gate? \| PennyLane <https://pennylane.ai/qml/glossary/what-is-a-swap-gate>`__).In our

-# example, we would need to apply two SWAP gates: one between :math:`pq_0` and :math:`pq_1`, and one

-# between :math:`pq_1` and :math:`pq_2`. If the state was

-# :math:`\frac{1}{\sqrt{2}}(|10000\rangle + |00001\rangle)` before the SWAP gates, it becomes

-# :math:`\frac{1}{\sqrt{2}}(|00100\rangle + |00001\rangle)`. Then, we can apply the CNOT gate between

-# :math:`pq_2` and :math:`pq_3`.

+# Physical qubit 0 and physical qubit 3 are not directly connected. If a two-qubit gate is required in
+# your code between qubits 0 and 3, it is necessary to apply SWAP gates. SWAP gates exchange the
+# states of two neighboring qubits for all basis states in the superposition (more details here `What
+# is a SWAP gate? \| PennyLane <https://pennylane.ai/qml/glossary/what-is-a-swap-gate>`__).In our
+# example, we would need to apply two SWAP gates: one between :math:`pq_0` and :math:`pq_1`, and one
+# between :math:`pq_1` and :math:`pq_2`. If the state was
+# :math:`\frac{1}{\sqrt{2}}(|10000\rangle + |00001\rangle)` before the SWAP gates, it becomes
+# :math:`\frac{1}{\sqrt{2}}(|00100\rangle + |00001\rangle)`. Then, we can apply the CNOT gate between
+# :math:`pq_2` and :math:`pq_3`.
 # 
-
 ######################################################################
-# The lack of full connectivity between physical qubits does not alter the universality of quantum

-# computation in these devices but leads to an increase in the depth of the effective quantum circuit.

-# A SWAP gate corresponds to 3 CNOT gates. Therefore, instead of 1 CNOT gate, we have 7 CNOT gates.

-# This increase can be fatal for your quantum computation, especially on a NISQ device.

+# The lack of full connectivity between physical qubits does not alter the universality of quantum
+# computation in these devices but leads to an increase in the depth of the effective quantum circuit.
+# A SWAP gate corresponds to 3 CNOT gates. Therefore, instead of 1 CNOT gate, we have 7 CNOT gates.
+# This increase can be fatal for your quantum computation, especially on a NISQ device.
 # 
-
 ######################################################################
-# Indeed, NISQ devices are sensitive to noise. In particular, two-qubit gates are noisy. Therefore,

-# when a large number of two-qubit gates is applied on a NISQ device, the quantum information is

-# progressively lost and randomized. Therefore, when running a quantum circuit on a NISQ device, we

-# try to minimize the number of two-qubit gates. Obviously, the increase by a factor of 7 in our

-# example is terrible for the noise robustness of our quantum computation. Even for fault-tolerant

-# quantum computers, reducing the number of CNOT gates would help to reduce the time of computation

-# and therefore the computational cost.

+# Indeed, NISQ devices are sensitive to noise. In particular, two-qubit gates are noisy. Therefore,
+# when a large number of two-qubit gates is applied on a NISQ device, the quantum information is
+# progressively lost and randomized. Therefore, when running a quantum circuit on a NISQ device, we
+# try to minimize the number of two-qubit gates. Obviously, the increase by a factor of 7 in our
+# example is terrible for the noise robustness of our quantum computation. Even for fault-tolerant
+# quantum computers, reducing the number of CNOT gates would help to reduce the time of computation
+# and therefore the computational cost.
 # 
-
 ######################################################################
-# It is here that qubit mapping can help mitigate the number of SWAP gates. There is no rule that

-# constrains the first qubit of your quantum circuit to be the first physical qubit, the second qubit

-# of your quantum circuit to be the second physical qubit, etc. We can separate the two sets of

-# qubits: on one side, we have the qubits of our quantum circuit, called logical qubits and denoted

-# :math:`lq_0`, :math:`lq_1`, etc., and on the other side, the physical qubits of the hardware denoted

-# :math:`pq_0`, :math:`pq_1` etc.. There does not necessarily have to be an identity mapping between

-# the two. Instead, we can define any mapping: :math:`\{lq\}\rightarrow \{pq\}`. The qubit mapping

-# problem corresponds to finding the initial mapping between the logical qubits and the physical

-# qubits that minimizes the number of SWAP gates during the execution of the quantum circuit by the

-# hardware. In our simple example, if we assign :math:`lq_0\rightarrow pq_0` and

-# :math:`lq_3\rightarrow pq_1`, a CNOT gate between physical qubits 0 and 1 corresponds to a CNOT gate

-# between logical qubits 0 and 3, as desired.

+# It is here that qubit mapping can help mitigate the number of SWAP gates. There is no rule that
+# constrains the first qubit of your quantum circuit to be the first physical qubit, the second qubit
+# of your quantum circuit to be the second physical qubit, etc. We can separate the two sets of
+# qubits: on one side, we have the qubits of our quantum circuit, called logical qubits and denoted
+# :math:`lq_0`, :math:`lq_1`, etc., and on the other side, the physical qubits of the hardware denoted
+# :math:`pq_0`, :math:`pq_1` etc.. There does not necessarily have to be an identity mapping between
+# the two. Instead, we can define any mapping: :math:`\{lq\}\rightarrow \{pq\}`. The qubit mapping
+# problem corresponds to finding the initial mapping between the logical qubits and the physical
+# qubits that minimizes the number of SWAP gates during the execution of the quantum circuit by the
+# hardware. In our simple example, if we assign :math:`lq_0\rightarrow pq_0` and
+# :math:`lq_3\rightarrow pq_1`, a CNOT gate between physical qubits 0 and 1 corresponds to a CNOT gate
+# between logical qubits 0 and 3, as desired.
 # 
-
 ######################################################################
-# Let�s consider a slightly more complex quantum circuit:

+# Let's consider a slightly more complex quantum circuit:
 # 
 
 import pennylane as qp
@@ -82,257 +74,225 @@ def  circuit():
     qp.CNOT(wires=[0,2])
 
 ######################################################################
-# So, there are 3 CNOT gates: one between :math:`lq_0` and :math:`lq_4`, one between :math:`lq_1` and

-# :math:`lq_3`, and one between :math:`lq_0` and :math:`lq_2`. Let’s first examine the number of

-# SWAP gates with the identity mapping (:math:`lq_0\rightarrow pq_0`, :math:`lq_1\rightarrow pq_1`,

-# etc.):

-# 

-# :math:`pq_0(lq_0)-pq_1(lq_1)-pq_2(lq_2)-pq_3(lq_3)-pq_4(lq_4)`

+# So, there are 3 CNOT gates: one between :math:`lq_0` and :math:`lq_4`, one between :math:`lq_1` and
+# :math:`lq_3`, and one between :math:`lq_0` and :math:`lq_2`. Let’s first examine the number of
+# SWAP gates with the identity mapping (:math:`lq_0\rightarrow pq_0`, :math:`lq_1\rightarrow pq_1`,
+# etc.):
+# 
+# :math:`pq_0(lq_0)-pq_1(lq_1)-pq_2(lq_2)-pq_3(lq_3)-pq_4(lq_4)`
 # 
-
 ######################################################################
-# Here, in parentheses, the logical qubit mapped to the corresponding physical qubit is shown.

+# Here, in parentheses, the logical qubit mapped to the corresponding physical qubit is shown.
 # 
-
 ######################################################################
-# For the first CNOT gate between :math:`lq_0` and :math:`lq_4`, we need to apply 3 SWAP gates

-# (SWAP(:math:`pq_0`, :math:`pq_1`), SWAP(:math:`pq_1`, :math:`pq_2`), and SWAP(:math:`pq_2`,

-# :math:`pq_3`)). The mapping becomes:

-# 

-# :math:`pq_0(lq_1)-pq_1(lq_2)-pq_2(lq_3)-pq_3(lq_0)-pq_4(lq_4)`

+# For the first CNOT gate between :math:`lq_0` and :math:`lq_4`, we need to apply 3 SWAP gates
+# (SWAP(:math:`pq_0`, :math:`pq_1`), SWAP(:math:`pq_1`, :math:`pq_2`), and SWAP(:math:`pq_2`,
+# :math:`pq_3`)). The mapping becomes:
+# 
+# :math:`pq_0(lq_1)-pq_1(lq_2)-pq_2(lq_3)-pq_3(lq_0)-pq_4(lq_4)`
 # 
-
 ######################################################################
-# Afterward, for the CNOT gate between :math:`lq_1` and :math:`lq_3`, we need a SWAP gate between

-# :math:`pq_1` and :math:`pq_2`:

-# 

-# :math:`pq_0(lq_1)-pq_1(lq_3)-pq_2(lq_2)-pq_4(lq_0)-pq_4(lq_4)`

+# Afterward, for the CNOT gate between :math:`lq_1` and :math:`lq_3`, we need a SWAP gate between
+# :math:`pq_1` and :math:`pq_2`:
+# 
+# :math:`pq_0(lq_1)-pq_1(lq_3)-pq_2(lq_2)-pq_4(lq_0)-pq_4(lq_4)`
 # 
-
 ######################################################################
-# Finally, for the CNOT gate between :math:`lq_0` and :math:`lq_2`, we do not need any additional SWAP

-# gates. Therefore, in total, there are 4 SWAP gates.

+# Finally, for the CNOT gate between :math:`lq_0` and :math:`lq_2`, we do not need any additional SWAP
+# gates. Therefore, in total, there are 4 SWAP gates.
 # 
-
 ######################################################################
-# Now, let’s consider the following initial mapping:

-# 

-# :math:`pq_0(lq_1)-pq_1(lq_3)-pq_2(lq_2)-pq_3(lq_0)-pq_4(lq_4)`

+# Now, let’s consider the following initial mapping:
+# 
+# :math:`pq_0(lq_1)-pq_1(lq_3)-pq_2(lq_2)-pq_3(lq_0)-pq_4(lq_4)`
 # 
-
 ######################################################################
-# In this case, there is no need for SWAP gates. The three CNOT gates can be applied directly. We go

-# from 15 CNOT gates to 3 CNOT gates, a reduction by a factor of 5 just by changing the initial

-# mapping!

+# In this case, there is no need for SWAP gates. The three CNOT gates can be applied directly. We go
+# from 15 CNOT gates to 3 CNOT gates, a reduction by a factor of 5 just by changing the initial
+# mapping!
 # 
-
 ######################################################################
-# Now that the qubit mapping problem is clear from these 5-qubit examples, let’s give a generalized

-# definition. We start with:

-# 

-# -  A quantum circuit C defined on n logical qubits :math:`lq_0, lq_1, \dots, lq_{n-1}`

-# -  A quantum device with m physical qubits :math:`pq_0, pq_1, \dots, pq_{m-1}` connected by some

-#    graph :math:`G_D`.

+# Now that the qubit mapping problem is clear from these 5-qubit examples, let’s give a generalized
+# definition. We start with:
+# 
+# -  A quantum circuit C defined on n logical qubits :math:`lq_0, lq_1, \dots, lq_{n-1}`
+# -  A quantum device with m physical qubits :math:`pq_0, pq_1, \dots, pq_{m-1}` connected by some
+#    graph :math:`G_D`.
 # 
-
 ######################################################################
-# The goal of the qubit mapping problem is to find the initial mapping

-# :math:`\pi_0: \{lq\}\rightarrow \{pq\}` in order to minimize the number of SWAP gates during the

-# execution of the quantum circuit C on the quantum device. Note that in general, the number of

-# logical qubits :math:`n` is smaller than the number of physical qubits :math:`m` (and cannot be

-# bigger).

+# The goal of the qubit mapping problem is to find the initial mapping
+# :math:`\pi_0: \{lq\}\rightarrow \{pq\}` in order to minimize the number of SWAP gates during the
+# execution of the quantum circuit C on the quantum device. Note that in general, the number of
+# logical qubits :math:`n` is smaller than the number of physical qubits :math:`m` (and cannot be
+# bigger).
 # 
-
 ######################################################################
-# The initial qubit mapping problem can be represented as an assignment matrix A of dimension

-# :math:`n\times m`:

-# 

-# -  :math:`A_{ij}=1` if logical qubit :math:`lq_i` is mapped to the physical qubit :math:`pq_j`.

-# -  :math:`A_{ij}=0` otherwise

+# The initial qubit mapping problem can be represented as an assignment matrix A of dimension
+# :math:`n\times m`:
+# 
+# -  :math:`A_{ij}=1` if logical qubit :math:`lq_i` is mapped to the physical qubit :math:`pq_j`.
+# -  :math:`A_{ij}=0` otherwise
 # 
-
 ######################################################################
-# There are two constraints on the assignment matrix A : \* Each logical qubit is assigned exactly one

-# physical qubit :math:`\sum_{j=1}^m A_{ij} = 1, ~\forall i`

-# 

-# -  Each physical qubit holds at most one logical qubit :math:`\sum_{i=1}^n A_{ij}\leq 1, ~\forall j`

+# There are two constraints on the assignment matrix A : \* Each logical qubit is assigned exactly one
+# physical qubit :math:`\sum_{j=1}^m A_{ij} = 1, ~\forall i`
+# 
+# -  Each physical qubit holds at most one logical qubit :math:`\sum_{i=1}^n A_{ij}\leq 1, ~\forall j`
 # 
-
 ######################################################################
-# An important quantity is the distance between two physical qubits :math:`D_{ij}=d(pq_i, pq_j)`. The

-# distance is the shortest-path length between two physical qubits in the hardware connectivity graph,

-# and it directly estimates how many SWAPs are needed to make them interact. Of course, the distance

-# would depend on the hardware connectivity graph.

+# An important quantity is the distance between two physical qubits :math:`D_{ij}=d(pq_i, pq_j)`. The
+# distance is the shortest-path length between two physical qubits in the hardware connectivity graph,
+# and it directly estimates how many SWAPs are needed to make them interact. Of course, the distance
+# would depend on the hardware connectivity graph.
 # 
-
 ######################################################################
-# We will explore in the next section the actual methods used to find the optimal assignment matrix A

-# and therefore to solve the qubit mapping problem.

+# We will explore in the next section the actual methods used to find the optimal assignment matrix A
+# and therefore to solve the qubit mapping problem.
 # 
-
 ######################################################################
-# **2.How to solve the qubit mapping problem?**

-# ---------------------------------------------

+# **2.How to solve the qubit mapping problem?**
+# ---------------------------------------------
 # 
-
 ######################################################################
-# The qubit mapping problem is unfortunately a NP problem and is exponentially hard to find the

-# optimal solution when the number of logical and physical qubits increases. However, there are some

-# heuristic methods that allow us to solve it with a reasonable computational time. The most famous

-# heuristic algorithm is the SABRE algorithm, used for example by Qiskit in its compilation.

+# The qubit mapping problem is unfortunately a NP problem and is exponentially hard to find the
+# optimal solution when the number of logical and physical qubits increases. However, there are some
+# heuristic methods that allow us to solve it with a reasonable computational time. The most famous
+# heuristic algorithm is the SABRE algorithm, used for example by Qiskit in its compilation.
 # 
-
 ######################################################################
-# The SABRE algorithm works by iteratively updating the qubit mapping guided by a cost function based

-# on the total distance.

+# The SABRE algorithm works by iteratively updating the qubit mapping guided by a cost function based
+# on the total distance.
 # 
-
 ######################################################################
-# **Forward propagation** 1. **Initialize the mapping**: The algorithm starts by initializing a single

-# mapping between logical and physical qubits (typically random or identity). Example (5-qubit

-# device): :math:`pq_0(lq_0)-pq_1(lq_2)-pq_2(lq_3)-pq_3(lq_4)-pq_4(lq_1)`

-# 

-# 2. **Define the front layer**: The front layer is the set of gates that are ready to be executed,

-#    meaning all their dependencies in the circuit have been satisfied. For the quantum circuit

-#    (CNOT(0,4), CNOT(1,3) and CNOT(0,2), the front layer is {CNOT(0,4), CNOT(1,3)}. CNOT(0,2) is not

-#    included because it depends on the execution of CNOT(0,4).

-# 

-# 3. **Check gate executability**: For each gate in the front layer, the algorithm checks whether the

-#    mapped physical qubits are connected in the hardware graph: If they are adjacent, the gate �is

-#    executed� and is withdrawn from the front layer. If not, SWAP gate is required and step 4 is

-#    applied. In our case, the CNOT(0,4) and CNOT(1,3) cannot be executed.

-# 

-# 4. **Compute routing cost**: The algorithm calculates the sum of the distance of the operations in

-#    the front layer: :math:`H = \sum_{(a,b)} d(\pi(a), \pi(b))`

-# 

-# 5. **Evaluate SWAP candidates**: The algorithm considers SWAPs on edges of the hardware graph. For

-#    each candidate SWAP, they recompute the cost H and select the SWAP that minimizes the cost. In

-#    our case, the SWAP(2,3) gives the distance :math:`H=2+1=3`, which is better than other options.

-#    So this SWAP is applied.

-# 

-# 6. **Iterate forward pass**: Steps 2–5 are repeated. The front layer is updated dynamically as

-#    gates become executable. Gates are executed whenever possible. SWAPs are inserted when necessary.

-#    At the end of this process, the algorithm produces a final mapping for the forward pass.

+# **Forward propagation** 1. **Initialize the mapping**: The algorithm starts by initializing a single
+# mapping between logical and physical qubits (typically random or identity). Example (5-qubit
+# device): :math:`pq_0(lq_0)-pq_1(lq_2)-pq_2(lq_3)-pq_3(lq_4)-pq_4(lq_1)`
+# 
+# 2. **Define the front layer**: The front layer is the set of gates that are ready to be executed,
+#    meaning all their dependencies in the circuit have been satisfied. For the quantum circuit
+#    (CNOT(0,4), CNOT(1,3) and CNOT(0,2), the front layer is {CNOT(0,4), CNOT(1,3)}. CNOT(0,2) is not
+#    included because it depends on the execution of CNOT(0,4).
+# 
+# 3. **Check gate executability**: For each gate in the front layer, the algorithm checks whether the
+#    mapped physical qubits are connected in the hardware graph: If they are adjacent, the gate �is
+#    executed� and is withdrawn from the front layer. If not, SWAP gate is required and step 4 is
+#    applied. In our case, the CNOT(0,4) and CNOT(1,3) cannot be executed.
+# 
+# 4. **Compute routing cost**: The algorithm calculates the sum of the distance of the operations in
+#    the front layer: :math:`H = \sum_{(a,b)} d(\pi(a), \pi(b))`
+# 
+# 5. **Evaluate SWAP candidates**: The algorithm considers SWAPs on edges of the hardware graph. For
+#    each candidate SWAP, they recompute the cost H and select the SWAP that minimizes the cost. In
+#    our case, the SWAP(2,3) gives the distance :math:`H=2+1=3`, which is better than other options.
+#    So this SWAP is applied.
+# 
+# 6. **Iterate forward pass**: Steps 2–5 are repeated. The front layer is updated dynamically as
+#    gates become executable. Gates are executed whenever possible. SWAPs are inserted when necessary.
+#    At the end of this process, the algorithm produces a final mapping for the forward pass.
 # 
-
 ######################################################################
-# **Backward propagation**

-# 

-# 7. **Reverse circuit execution**: The algorithm then repeats the same procedure, but on the reversed

-#    circuit (executing gates in reverse order), starting from the final mapping obtained in the

-#    forward pass.

-# 

-# 8. **Improved initial mapping**: At the end of the backward pass, a new initial mapping

-#    :math:`\pi_0` is obtained. This initial mapping is typically better than the original random

-#    initialization.

+# **Backward propagation**
+# 
+# 7. **Reverse circuit execution**: The algorithm then repeats the same procedure, but on the reversed
+#    circuit (executing gates in reverse order), starting from the final mapping obtained in the
+#    forward pass.
+# 
+# 8. **Improved initial mapping**: At the end of the backward pass, a new initial mapping
+#    :math:`\pi_0` is obtained. This initial mapping is typically better than the original random
+#    initialization.
 # 
-
 ######################################################################
-# In practice, SABRE may repeat multiple forward and backward passes (usually around 3 iterations),

-# gradually improving the mapping until no significant improvement is observed.

+# In practice, SABRE may repeat multiple forward and backward passes (usually around 3 iterations),
+# gradually improving the mapping until no significant improvement is observed.
 # 
-
 ######################################################################
-# The SABRE algorithm has a polynomial computational complexity of approximately

-# :math:`\mathcal{O}(Tm^2 n)` where :math:`T` is the total number of two-qubit gates in the quantum

-# circuit. This is significantly better than the exponential scaling of an exact solution. However, as

-# the number of qubits in superconducting processors continues to increase, the runtime of the SABRE

-# algorithm also grows accordingly.

+# The SABRE algorithm has a polynomial computational complexity of approximately
+# :math:`\mathcal{O}(Tm^2 n)` where :math:`T` is the total number of two-qubit gates in the quantum
+# circuit. This is significantly better than the exponential scaling of an exact solution. However, as
+# the number of qubits in superconducting processors continues to increase, the runtime of the SABRE
+# algorithm also grows accordingly.
 # 
-
 ######################################################################
-# Machine learning models could offer better scalability while maintaining high accuracy. In this

-# demo, we consider supervised learning. Supervised learning requires a dataset with input data and

-# corresponding labels to train the model. This is the purpose of QMapDataset, which provides datasets

-# and tools to generate them for solving the qubit mapping problem using supervised machine learning

-# models. QMapDataset is available in PennyLane. We will explore its features in the next section.

+# Machine learning models could offer better scalability while maintaining high accuracy. In this
+# demo, we consider supervised learning. Supervised learning requires a dataset with input data and
+# corresponding labels to train the model. This is the purpose of QMapDataset, which provides datasets
+# and tools to generate them for solving the qubit mapping problem using supervised machine learning
+# models. QMapDataset is available in PennyLane. We will explore its features in the next section.
 # 
-
 ######################################################################
-# **3. QMapDataset**

-# ------------------

+# **3. QMapDataset**
+# ------------------
 # 
-
 ######################################################################
-# QMapDataset is a dataset generator for the qubit mapping problem, designed for IBM quantum devices.

-# The dataset is built from three main components: the hardware, the circuits, and the mapping.

+# QMapDataset is a dataset generator for the qubit mapping problem, designed for IBM quantum devices.
+# The dataset is built from three main components: the hardware, the circuits, and the mapping.
 # 
-
 ######################################################################
-# **For the hardware**, First, we choose an IBM quantum processor. To make the dataset more diverse,

-# the code can randomly permute the properties of the qubits and gates (data augmentation). We can

-# control how often these permutations are applied.

+# **For the hardware**, First, we choose an IBM quantum processor. To make the dataset more diverse,
+# the code can randomly permute the properties of the qubits and gates (data augmentation). We can
+# control how often these permutations are applied.
 # 
-
 ######################################################################
-# Then, the code collects all the relevant hardware information:

-# 

-# -  physical qubit properties (:math:`T_1`, :math:`T_2`, readout error)

-# -  single-qubit gate errors for each qubit

-# -  two-qubit gate errors for each connected pair of qubits

+# Then, the code collects all the relevant hardware information:
+# 
+# -  physical qubit properties (:math:`T_1`, :math:`T_2`, readout error)
+# -  single-qubit gate errors for each qubit
+# -  two-qubit gate errors for each connected pair of qubits
 # 
-
 ######################################################################
-# **For the circuits**, These are sampled either from:

-# 

-# -  random quantum circuits

-# -  well-known circuits (QPE, Grover, QFT, GHZ state preparation etc.)

+# **For the circuits**, These are sampled either from:
+# 
+# -  random quantum circuits
+# -  well-known circuits (QPE, Grover, QFT, GHZ state preparation etc.)
 # 
-
 ######################################################################
-# By default, the choice is split 50–50 between random and well-known circuits.

-# 

-# For random quantum circuits, the code also sample: \* the number of logical qubits \* the circuit

-# depth

+# By default, the choice is split 50–50 between random and well-known circuits.
+# 
+# For random quantum circuits, the code also sample: \* the number of logical qubits \* the circuit
+# depth
 # 
-
 ######################################################################
-# The number of logical qubits is chosen between 3 and the number of physical qubits available on the

-# hardware. The depth follows this distribution: 25% shallow (5–15 layers), 40% medium (16–50),

-# 30% deep (51–150), and 5% very deep circuits (151–400).

+# The number of logical qubits is chosen between 3 and the number of physical qubits available on the
+# hardware. The depth follows this distribution: 25% shallow (5–15 layers), 40% medium (16–50),
+# 30% deep (51–150), and 5% very deep circuits (151–400).
 # 
-
 ######################################################################
-# To further enrich the dataset, we apply random permutations of qubits to the well-known circuits

-# (since there are only a few of them). After that, the code collects:

-# 

-# -  Single-qubit gate counts for each logical qubit

-# -  Two-qubit gate counts for each pair of logical qubits

+# To further enrich the dataset, we apply random permutations of qubits to the well-known circuits
+# (since there are only a few of them). After that, the code collects:
+# 
+# -  Single-qubit gate counts for each logical qubit
+# -  Two-qubit gate counts for each pair of logical qubits
 # 
-
 ######################################################################
-# Finally, **for the mapping**, the code uses the generated hardware and circuits to compute the qubit

-# mapping with an advanced version of the SABRE algorithm (optimization_level = 3 in Qiskit).

+# Finally, **for the mapping**, the code uses the generated hardware and circuits to compute the qubit
+# mapping with an advanced version of the SABRE algorithm (optimization_level = 3 in Qiskit).
 # 
-
 ######################################################################
-# If you want more details, you can have a look at the GitHub repo: `rscadrien/QMapDataset: Generator

-# of dataset for qubit mapping <https://github.com/rscadrien/QMapDataset>`__.

+# If you want more details, you can have a look at the GitHub repo: `rscadrien/QMapDataset: Generator
+# of dataset for qubit mapping <https://github.com/rscadrien/QMapDataset>`__.
 # 
-
 ######################################################################
-# The good news is that you do not need to generate this dataset using QMapDataset yourself! We have

-# already prepared datasets for three different types of IBM quantum processors: Eagle 3, Heron 1, and

-# Heron 2. Each dataset contains 10,000 samples, where each sample consists of a circuit, hardware

-# description, and mapping.

+# The good news is that you do not need to generate this dataset using QMapDataset yourself! We have
+# already prepared datasets for three different types of IBM quantum processors: Eagle 3, Heron 1, and
+# Heron 2. Each dataset contains 10,000 samples, where each sample consists of a circuit, hardware
+# description, and mapping.
 # 
-
 ######################################################################
-# Let’s see how to use these datasets with PennyLane and extract their information.

-# 

-# The first step is to load a dataset from the Xanadu server. For example, to load the dataset for the

-# Eagle 3 architecture:

+# Let’s see how to use these datasets with PennyLane and extract their information.
+# 
+# The first step is to load a dataset from the Xanadu server. For example, to load the dataset for the
+# Eagle 3 architecture:
 # 
 
 import pennylane as qp
 [ds] = qp.data.load("other", name="qubit-mapping-eagle3")
 
 ######################################################################
-# For the Heron 1 and Heron 2 architectures, you only need to replace the name argument with

-# �qubit-mapping-heron1� or �qubit-mapping-heron2�, respectively.

-# 

-# Each dataset is divided into three attributes: circuits, hardware, and mappings, following the

-# structure discussed earlier. For example, to access the first sample:

+# For the Heron 1 and Heron 2 architectures, you only need to replace the name argument with
+# �qubit-mapping-heron1� or �qubit-mapping-heron2�, respectively.
+# 
+# Each dataset is divided into three attributes: circuits, hardware, and mappings, following the
+# structure discussed earlier. For example, to access the first sample:
 # 
 
 sample0_circuit = ds.circuits[0]
@@ -340,15 +300,15 @@ def  circuit():
 sample0_mapping = ds.mappings[0]
 
 ######################################################################
-# Each of these is a dictionary containing the features of the circuit, hardware, or mapping. For

-# example, to access the single-qubit and two-qubit gate counts of the quantum circuit:

+# Each of these is a dictionary containing the features of the circuit, hardware, or mapping. For
+# example, to access the single-qubit and two-qubit gate counts of the quantum circuit:
 # 
 
 Single_qubit_count = sample0_circuit["single_qubit_counts"]
 Two_qubit_count = sample0_circuit["two_qubit_counts"]
 
 ######################################################################
-# To retrieve :math:`T_1`, :math:`T_2`, and the two-qubit gate error from the hardware data:

+# To retrieve :math:`T_1`, :math:`T_2`, and the two-qubit gate error from the hardware data:
 # 
 
 T_1 = sample0_hardware["T1"]
@@ -356,53 +316,47 @@ def  circuit():
 Two_qubit_error = sample0_hardware["ecr"]
 
 ######################################################################
-# Finally, to obtain the mapping (which can serve as the label in a supervised learning model):

+# Finally, to obtain the mapping (which can serve as the label in a supervised learning model):
 # 
 
 mapping = sample0_mapping["final_mapping"]
 
 ######################################################################
-# Detailed information about all available features and how to access them can be found in the �Data�

-# section of the Qubit Mapping Eagle 3 page on the PennyLane website. (`Qubit Mapping Eagle

-# 3 <https://pennylane.ai/datasets/qubit-mapping-eagle3>`__ ).

+# Detailed information about all available features and how to access them can be found in the "Data"
+# section of the Qubit Mapping Eagle 3 page on the PennyLane website. (`Qubit Mapping Eagle
+# 3 <https://pennylane.ai/datasets/qubit-mapping-eagle3>`__ ).
 # 
-
 ######################################################################
-# **4.Building a machine learning model to solve the qubit mapping problem**

-# --------------------------------------------------------------------------

+# **4.Building a machine learning model to solve the qubit mapping problem**
+# --------------------------------------------------------------------------
 # 
-
 ######################################################################
-# In this session, we will show an example of how to preprocess data from QMapDataset and how to build

-# a machine learning model that can solve the qubit mapping problem.

+# In this session, we will show an example of how to preprocess data from QMapDataset and how to build
+# a machine learning model that can solve the qubit mapping problem.
 # 
-
 ######################################################################
-# **4.1 Preprocessing**

-# ~~~~~~~~~~~~~~~~~~~~~

+# **4.1 Preprocessing**
+# ~~~~~~~~~~~~~~~~~~~~~
 # 
-
 ######################################################################
-# The qubit mapping problem strongly depends on the connectivity between the physical qubits in the

-# given hardware, as well as the connectivity between the logical qubits in the circuits (represented

-# by two-qubit gate counts). Therefore, a graph is a natural representation for solving the qubit

-# mapping problem. We will see how to construct graphs for both the circuits and the hardware using

-# the QMapDataset available in PennyLane.

+# The qubit mapping problem strongly depends on the connectivity between the physical qubits in the
+# given hardware, as well as the connectivity between the logical qubits in the circuits (represented
+# by two-qubit gate counts). Therefore, a graph is a natural representation for solving the qubit
+# mapping problem. We will see how to construct graphs for both the circuits and the hardware using
+# the QMapDataset available in PennyLane.
 # 
-
 ######################################################################
-# A graph is a way of showing relationships. It is made up of points, called nodes (or vertices), that

-# are connected by lines, called edges. Both nodes and edges can also have extra information attached

-# to them, called features.

+# A graph is a way of showing relationships. It is made up of points, called nodes (or vertices), that
+# are connected by lines, called edges. Both nodes and edges can also have extra information attached
+# to them, called features.
 # 
-
 ######################################################################
-# For the **circuit**, the nodes represent the logical qubits, and edges connect pairs of logical

-# qubits between which two-qubit gates are applied in the quantum circuit. The features of each node

-# are the counts of different single-qubit gates applied to the corresponding logical qubit. The

-# features of each edge are the counts of different two-qubit gates applied between the two logical

-# qubits. Since the relative values of these features are what matter, we standardize them. This is

-# the code to build the graph for the quantum circuit of the QMapDataset:

+# For the **circuit**, the nodes represent the logical qubits, and edges connect pairs of logical
+# qubits between which two-qubit gates are applied in the quantum circuit. The features of each node
+# are the counts of different single-qubit gates applied to the corresponding logical qubit. The
+# features of each edge are the counts of different two-qubit gates applied between the two logical
+# qubits. Since the relative values of these features are what matter, we standardize them. This is
+# the code to build the graph for the quantum circuit of the QMapDataset:
 # 
 
 import torch
@@ -416,13 +370,11 @@ def build_circuit_graph (circuit_data):
     rz_counts_scaled = get_single_qubit_counts(circuit_data, "rz")
     sx_counts_scaled = get_single_qubit_counts(circuit_data, "sx")
     x_counts_scaled = get_single_qubit_counts(circuit_data, "x")
-
     # -----------------------------
     # STEP 2: Extract and scale two-qubit gate counts
     # These describe interactions between pairs of qubits
     # -----------------------------
     ecr_scaled = get_two_qubit_counts(circuit_data,'ecr')
-
     # -----------------------------
     # STEP 3: Build node feature matrix
     # Each node = a logical qubit
@@ -451,7 +403,6 @@ def build_circuit_graph (circuit_data):
     edge_attr=ecr_attr,
     )
     return G_circuit
-
 def get_single_qubit_counts(data, gate_name):
     # Number of logical qubits in the circuit
     n = data["n_logical_qubits"]
@@ -463,18 +414,15 @@ def get_single_qubit_counts(data, gate_name):
     else:
     # Otherwise normalize counts across qubits (z-score normalization)
         return standard_scaling_across_qubits(np.array(counts))
-
 def get_two_qubit_counts(data, gate_name):
     # Safely retrieve list of two-qubit gate interactions
     counts = data.get("two_qubit_counts", {}).get(gate_name, [])
-
     # If no interactions exist, return empty list
     if not counts:
         return counts
     else:
     # Otherwise scale the edge weights
       return scaled_two_qubit_list(counts)
-
 def standard_scaling_across_qubits(vector):
     # Compute mean and standard deviation for normalization
     mean = vector.mean()
@@ -485,7 +433,6 @@ def standard_scaling_across_qubits(vector):
     else:
         vector_scaled = (vector - mean)/std
     return vector_scaled
-
 def scaled_two_qubit_list(two_data):
     # Extract raw edge weights from dictionary list
     values = np.array([d['value']for d in two_data], dtype=float)
@@ -497,7 +444,6 @@ def scaled_two_qubit_list(two_data):
     for d, v_scaled in zip(two_data, scaled_values)
     ]
     return two_gate_scaled
-
 def edge_info(gate_dict):
     # Lists to store graph connectivity
     edge_index = []
@@ -514,13 +460,13 @@ def edge_info(gate_dict):
     return edge_index, edge_attr
 
 ######################################################################
-# For example, if this code is applied for the circuit of the 27th sample, we obtain:

+# For example, if this code is applied for the circuit of the 27th sample, we obtain:
 # 
 
 Graph_circuit = build_circuit_graph(ds.circuits[26])
 
 ######################################################################
-# Using the networkx package, we can visualize the graph using the following graph:

+# Using the networkx package, we can visualize the graph using the following graph:
 # 
 
 import matplotlib.pyplot as plt
@@ -543,28 +489,25 @@ def edge_info(gate_dict):
     with_labels=True,
     node_size=800,
 )
-
 nx.draw_networkx_edge_labels(
     G,
     pos,
     edge_labels=edge_labels
 )
-
 plt.show()
 
 ######################################################################
-# We obtain the following graph: |My graph|

-# 

-# .. |My graph| image:: graph_circuit.png

+# We obtain the following graph: |My graph|
+# 
+# .. |My graph| image:: graph_circuit.png
 # 
-
 ######################################################################
-# For the **hardware**, the nodes represent the physical qubits, and edges connect pairs of physical

-# qubits that are directly connected in the device architecture. The features of each node are the

-# :math:`T_1` and :math:`T_2` coherence times, readout error, and single-qubit gate errors for each

-# physical qubit. The features of each edge are the errors of the different two-qubit gates applied

-# between connected physical qubits. As with the circuits, the feature values are standardized. The

-# code to build the graph for the hardware is pretty similar to the one for the circuit:

+# For the **hardware**, the nodes represent the physical qubits, and edges connect pairs of physical
+# qubits that are directly connected in the device architecture. The features of each node are the
+# :math:`T_1` and :math:`T_2` coherence times, readout error, and single-qubit gate errors for each
+# physical qubit. The features of each edge are the errors of the different two-qubit gates applied
+# between connected physical qubits. As with the circuits, the feature values are standardized. The
+# code to build the graph for the hardware is pretty similar to the one for the circuit:
 # 
 
 def build_hardware_graph(hardware_data):
@@ -592,107 +535,88 @@ def build_hardware_graph(hardware_data):
     return G_hardware
 
 ######################################################################
-# We can again visualize the graph using networkx:

+# We can again visualize the graph using networkx:
 # 
-
 # Draw graph
 
 plt.figure(figsize=(10, 10))
-
 pos = nx.spring_layout(
     G_hardware,
     seed=42,
     k=1 / np.sqrt(len(G_hardware)),
     iterations=200
 )
-
 nx.draw_networkx_edges(G_hardware, pos, alpha=0.2, width=0.5)
 nx.draw_networkx_nodes(G_hardware, pos, node_size=40)
-
 plt.axis("off")
 plt.savefig("graph_hardware.png", dpi=300, bbox_inches="tight")
 plt.show()
 
 ######################################################################
-# .. figure:: graph_hardware.png

-#    :alt: My graph

-# 

-#    My graph

+# .. figure:: graph_hardware.png
+#    :alt: My graph
+# 
+#    My graph
 # 
-
 ######################################################################
-# **4.2 Machine learning model**

-# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

+# **4.2 Machine learning model**
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 # 
-
 ######################################################################
-# We will show how to build a model that can learn qubit mapping. For this model, we will use an

-# architecture adapted to the graph structure of the input: a Graph Neural Network (GNN). The key

-# principle behind GNNs is that a node learns by aggregating information from its neighbors. A GNN

-# typically works in layers, and each layer is mostly composed of three steps:

-# 

-# -  Message passing : Each node receives information from its neighbors

-# -  Aggregation: Combine neighbor information (e.g., sum, mean, max)

-# -  Update: Use a neural network to update the node’s embedding.

+# We will show how to build a model that can learn qubit mapping. For this model, we will use an
+# architecture adapted to the graph structure of the input: a Graph Neural Network (GNN). The key
+# principle behind GNNs is that a node learns by aggregating information from its neighbors. A GNN
+# typically works in layers, and each layer is mostly composed of three steps:
+# 
+# -  Message passing : Each node receives information from its neighbors
+# -  Aggregation: Combine neighbor information (e.g., sum, mean, max)
+# -  Update: Use a neural network to update the node’s embedding.
 # 
-
 ######################################################################
-# In our case, we will use two separate GNNs: one for the quantum circuit and one for the hardware.

-# After several GNN layers, we obtain updated embedding vectors for the nodes of both graphs,

-# :math:`h_L^{(i)}` and :math:`h_P^{(j)}`. (:math:`h_L^{(i)}` is the vector of the i th node of the

-# quantum circuit graph while :math:`h_P^{(j)}` is the vector of the jth node of the hardware graph).

+# In our case, we will use two separate GNNs: one for the quantum circuit and one for the hardware.
+# After several GNN layers, we obtain updated embedding vectors for the nodes of both graphs,
+# :math:`h_L^{(i)}` and :math:`h_P^{(j)}`. (:math:`h_L^{(i)}` is the vector of the i th node of the
+# quantum circuit graph while :math:`h_P^{(j)}` is the vector of the jth node of the hardware graph).
 # 
-
 ######################################################################
-# We then compare these embeddings using a dot product:

-# 

-# :math:`S_{ij}=h_L^{(i)}\cdot h_P^{(j)}`

+# We then compare these embeddings using a dot product:
+# 
+# :math:`S_{ij}=h_L^{(i)}\cdot h_P^{(j)}`
 # 
-
 ######################################################################
-# The matrix element :math:`S_{ij}` represents the logit that the logical qubit :math:`lq_i` is mapped

-# to the physical qubit :math:`pq_j`. The probability that logical qubit k maps to physical qubit j is

-# calculated from the logit:

-# 

-# :math:`p_{ij}=\frac{\text{exp}(S_{ij})}{\sum_{j^\prime} \text{exp}(S_{ij}^\prime)}`

+# The matrix element :math:`S_{ij}` represents the logit that the logical qubit :math:`lq_i` is mapped
+# to the physical qubit :math:`pq_j`. The probability that logical qubit k maps to physical qubit j is
+# calculated from the logit:
+# 
+# :math:`p_{ij}=\frac{\text{exp}(S_{ij})}{\sum_{j^\prime} \text{exp}(S_{ij}^\prime)}`
 # 
-
 ######################################################################
-# During training, the matrix :math:`p_{ij}` should ideally converge to the assignment matrix

-# :math:`A_{ij}` presented in Section 2, where each row contains exactly one nonzero element equal to

-# 1. After training, during inference, the strict assignment constraints are enforced by applying the

-# Hungarian algorithm. Unfortunately, this algorithm is not differentiable and therefore cannot be

-# used during training.The code for this architecture is as follows :

+# During training, the matrix :math:`p_{ij}` should ideally converge to the assignment matrix
+# :math:`A_{ij}` presented in Section 2, where each row contains exactly one nonzero element equal to
+# 1. After training, during inference, the strict assignment constraints are enforced by applying the
+# Hungarian algorithm. Unfortunately, this algorithm is not differentiable and therefore cannot be
+# used during training.The code for this architecture is as follows :
 # 
 
 class QubitMapping_dot_Model(nn.Module):
     def __init__(self, node_c_dim, node_h_dim, edge_c_dim, edge_h_dim, hidden_dim):
         super().__init__()
-
         # === GNN for the LOGICAL CIRCUIT graph ===
         # These layers learn embeddings for logical qubits based on circuit structure
-
         # First layer: edge-conditioned convolution
         # Edge features (e.g., gate types) generate weights dynamically
         self.nn_c1 = make_edge_mlp(edge_c_dim, node_c_dim * hidden_dim)
         self.gnn_c1 = NNConv(node_c_dim, hidden_dim, nn=self.nn_c1, aggr='add')
-
         # Second layer: refines embeddings
         self.nn_c2 = make_edge_mlp(edge_c_dim, hidden_dim * hidden_dim)
         self.gnn_c2 = NNConv(hidden_dim, hidden_dim, nn=self.nn_c2, aggr='add')
-
         # === GNN for the HARDWARE graph ===
         # These layers learn embeddings for physical qubits based on hardware connectivity
-
         self.nn_h1 = make_edge_mlp(edge_h_dim, node_h_dim * hidden_dim)
         self.gnn_h1 = NNConv(node_h_dim, hidden_dim, nn=self.nn_h1, aggr='add')
-
         self.nn_h2 = make_edge_mlp(edge_h_dim, hidden_dim * hidden_dim)
         self.gnn_h2 = NNConv(hidden_dim, hidden_dim, nn=self.nn_h2, aggr='add')
-
-
     def forward(self, G_circuit, G_hardware):
-
         # === Step 1: Compute embeddings for LOGICAL qubits ===
         # Each node becomes a vector summarizing its role in the circuit
         x_c = F.relu(self.gnn_c1(
@@ -700,13 +624,11 @@ def forward(self, G_circuit, G_hardware):
             G_circuit.edge_index,    # connectivity
             G_circuit.edge_attr      # edge features (e.g., gate info)
         ))
-
         h_L = F.relu(self.gnn_c2(
             x_c,
             G_circuit.edge_index,
             G_circuit.edge_attr
         ))
-
         # === Step 2: Compute embeddings for PHYSICAL qubits ===
         # Each node becomes a vector describing hardware properties
         x_h = F.relu(self.gnn_h1(
@@ -714,62 +636,45 @@ def forward(self, G_circuit, G_hardware):
             G_hardware.edge_index,
             G_hardware.edge_attr
         ))
-
         h_P = F.relu(self.gnn_h2(
             x_h,
             G_hardware.edge_index,
             G_hardware.edge_attr
         ))
-
-
         # === Step 3: Compute compatibility between logical and physical qubits ===
-
         prob_list = []
-
         # Number of graphs in the batch
         num_graphs = G_circuit.batch.max().item() + 1
-
         # Loop over each graph in the batch
         for i in range(num_graphs):
-
             # --- Select nodes belonging to graph i ---
             # (because batching mixes all graphs together)
             mask_c = (G_circuit.batch == i)
             mask_h = (G_hardware.batch == i)
-
             # Extract embeddings for this specific graph
             h_L_i = h_L[mask_c]  # logical qubits
             h_P_i = h_P[mask_h]  # physical qubits
-
             # --- Compute compatibility matrix ---
             # Dot product measures how well each pair matches
             # Shape: [num_logical_qubits, num_physical_qubits]
             S_i = torch.matmul(h_L_i, h_P_i.T)
-
             if self.training:
                 # During training:
                 # return raw scores (used in loss function)
                 prob_i = S_i
                 prob_list.append(prob_i)
-
             else:
                 # During inference:
                 # find the BEST one-to-one assignment
-
                 # Move to CPU + numpy for Hungarian algorithm
                 S_cpu = S_i.detach().cpu().numpy()
-
                 # Hungarian algorithm finds assignment maximizing total score
                 row_ind, col_ind = linear_sum_assignment(-S_cpu)
-
                 # Build mapping: logical qubit -> physical qubit
                 mapping_i = torch.zeros(h_L_i.size(0), dtype=torch.long)
                 mapping_i[row_ind] = torch.tensor(col_ind, dtype=torch.long)
-
                 prob_list.append(mapping_i)
-
         # === Step 4: Merge results from all graphs ===
-
         if self.training:
             # Concatenate score matrices
             # Shape: [total_logical_nodes_in_batch, num_physical_nodes_per_graph]
@@ -782,80 +687,69 @@ def forward(self, G_circuit, G_hardware):
             return mapping
 
 ######################################################################
-# **4.3 Loss function and metric**

-# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

+# **4.3 Loss function and metric**
+# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 # 
-
 ######################################################################
-# We need to have a loss function to minimize for the training of the model.

+# We need to have a loss function to minimize for the training of the model.
 # 
-
 ######################################################################
-# For each logical qubit k, we know the correct assignment (from the training data), which we denote

-# as :math:`k_c`. This is the physical qubit that the logical qubit should be mapped to.

+# For each logical qubit k, we know the correct assignment (from the training data), which we denote
+# as :math:`k_c`. This is the physical qubit that the logical qubit should be mapped to.
 # 
-
 ######################################################################
-# The loss for the logical qubit k is then defined as:

-# 

-# :math:`L_k=\log(p_{k,k_c})`

+# The loss for the logical qubit k is then defined as:
+# 
+# :math:`L_k=\log(p_{k,k_c})`
 # 
-
 ######################################################################
-# Finally, the total loss is computed by averaging over all logical qubits :

-# 

-# :math:`L = \frac{1}{n} \sum_{k=1}^n -\log(p_{k,k_c})`

+# Finally, the total loss is computed by averaging over all logical qubits :
+# 
+# :math:`L = \frac{1}{n} \sum_{k=1}^n -\log(p_{k,k_c})`
 # 
-
 ######################################################################
-# This is known as the cross-entropy loss. With Pytorch, the cross entropy loss can be calculated by

-# using nn.CrossEntropyLoss():

+# This is known as the cross-entropy loss. With Pytorch, the cross entropy loss can be calculated by
+# using nn.CrossEntropyLoss():
 # 
 
 loss_fn = nn.CrossEntropyLoss()
 
 ######################################################################
-# This loss encourages the model to assign high probability to the correct qubit mapping and to assign

-# low probability to incorrect mappings. However, it does not enforce one-to-one mapping and does not

-# prevent collisions (two logical qubits choosing the same physical qubit). It is the reason that it

-# is better to use the Hungarian algorithm during inference.

+# This loss encourages the model to assign high probability to the correct qubit mapping and to assign
+# low probability to incorrect mappings. However, it does not enforce one-to-one mapping and does not
+# prevent collisions (two logical qubits choosing the same physical qubit). It is the reason that it
+# is better to use the Hungarian algorithm during inference.
 # 
-
 ######################################################################
-# In addition to the loss function, there are some metrics that help to evaluate the quality of the

-# model and are easier to interpret than the loss function. We consider two different metrics: \*

-# logical qubit-level accuracy \* full mapping accuracy

+# In addition to the loss function, there are some metrics that help to evaluate the quality of the
+# model and are easier to interpret than the loss function. We consider two different metrics: \*
+# logical qubit-level accuracy \* full mapping accuracy
 # 
-
 ######################################################################
-# To understand the difference between them, let us consider an example with 5 logical qubits and 5

-# physical qubits. The correct mapping (from the training data) is [2,1,4,3,0] and the model predicts

-# [2,0,4,3,1].

+# To understand the difference between them, let us consider an example with 5 logical qubits and 5
+# physical qubits. The correct mapping (from the training data) is [2,1,4,3,0] and the model predicts
+# [2,0,4,3,1].
 # 
-
 ######################################################################
-# For the logical qubit-level accuracy, we simply count how many individual assignments are correct.

-# In this example, 3 out of 5 qubits are correctly mapped. So the accuracy is 60%.

+# For the logical qubit-level accuracy, we simply count how many individual assignments are correct.
+# In this example, 3 out of 5 qubits are correctly mapped. So the accuracy is 60%.
 # 
-
 ######################################################################
-# For the full mapping accuracy, the prediction is considered correct only if the entire assignment is

-# exactly correct. In this example, the predicted mapping is not identical to the ground truth. So the

-# full mapping is incorrect. This metric is therefore much stricter and harder to improve.

+# For the full mapping accuracy, the prediction is considered correct only if the entire assignment is
+# exactly correct. In this example, the predicted mapping is not identical to the ground truth. So the
+# full mapping is incorrect. This metric is therefore much stricter and harder to improve.
 # 
-
 ######################################################################
-# You could ask: Why use both metrics? These two metrics serve different goals. The logical

-# qubit-level accuracy is useful to monitor progress during training. The full mapping accuracy

-# reflects the actual goal of the problem: producing a correct global qubit mapping. During training,

-# the logical-level accuracy improves first and helps track learning progress while the full-mapping

-# accuracy becomes meaningful only later, when the model is already quite accurate. Note that the

-# Hungarian algorithm significantly improves the full-mapping accuracy. As a result, the full-mapping

-# accuracy is often higher on the validation set than on the training set.

+# You could ask: Why use both metrics? These two metrics serve different goals. The logical
+# qubit-level accuracy is useful to monitor progress during training. The full mapping accuracy
+# reflects the actual goal of the problem: producing a correct global qubit mapping. During training,
+# the logical-level accuracy improves first and helps track learning progress while the full-mapping
+# accuracy becomes meaningful only later, when the model is already quite accurate. Note that the
+# Hungarian algorithm significantly improves the full-mapping accuracy. As a result, the full-mapping
+# accuracy is often higher on the validation set than on the training set.
 # 
-
 ######################################################################
-# These metrics are calculated with the following code for the training and the validation dataset:

+# These metrics are calculated with the following code for the training and the validation dataset:
 # 
 
 def batch_graph_metrics_training(logit, qu_map, batch):
@@ -863,74 +757,59 @@ def batch_graph_metrics_training(logit, qu_map, batch):
     Compute per-graph metrics in a single pass:
       - circuit-averaged node accuracy
       - exact-mapping accuracy
-
     Args:
         logit: [N_total_nodes, N_P] predicted logit
         qu_map: [N_total_nodes] ground-truth physical qubits
         batch: [N_total_nodes] graph indices
-
     Returns:
         node_acc_sum: sum of node-level accuracies per graph
         exact_sum: number of perfectly mapped graphs
         num_graphs: number of graphs in the batch
     """
-
     pred = logit.argmax(dim=1)
-
     num_graphs = batch.max().item() + 1
     node_acc_sum = 0.0
     exact_sum = 0
-
     for g in range(num_graphs):
         mask = (batch == g)
         correct_nodes = (pred[mask] == qu_map[mask]).sum().item()
         n_nodes = mask.sum().item()
-
         node_acc_sum += correct_nodes / n_nodes
         if correct_nodes == n_nodes:
             exact_sum += 1
-
     return node_acc_sum, exact_sum, num_graphs
-
 def batch_graph_metrics_val(pred, qu_map, batch):
     """
     Compute per-graph metrics in a single pass:
       - circuit-averaged node accuracy
       - exact-mapping accuracy
-
     Args:
         pred: [N_total_nodes] predicted mapping
         qu_map: [N_total_nodes] ground-truth physical qubits
         batch: [N_total_nodes] graph indices
-
     Returns:
         node_acc_sum: sum of node-level accuracies per graph
         exact_sum: number of perfectly mapped graphs
         num_graphs: number of graphs in the batch
     """
-
     num_graphs = batch.max().item() + 1
     node_acc_sum = 0.0
     exact_sum = 0
-
     for g in range(num_graphs):
         mask = (batch == g)
         correct_nodes = (pred[mask] == qu_map[mask]).sum().item()
         n_nodes = mask.sum().item()
-
         node_acc_sum += correct_nodes / n_nodes
         if correct_nodes == n_nodes:
             exact_sum += 1
-
     return node_acc_sum, exact_sum, num_graphs
 
 ######################################################################
-# **4.3 Training Loop**

-# ~~~~~~~~~~~~~~~~~~~~~

+# **4.3 Training Loop**
+# ~~~~~~~~~~~~~~~~~~~~~
 # 
-
 ######################################################################
-# We now have all the elements to build our training loop! Here it is the code:

+# We now have all the elements to build our training loop! Here it is the code:
 # 
 
 def train_loop(model, optimizer, loss_fn, loader_train, loader_valid, n_epoch, device):
@@ -945,7 +824,6 @@ def train_loop(model, optimizer, loss_fn, loader_train, loader_valid, n_epoch, d
     exact_accuracy_hist_train = [0.0] * n_epoch
     accuracy_hist_valid = [0.0] * n_epoch
     exact_accuracy_hist_valid = [0.0] * n_epoch
-
     for epoch in range(n_epoch):
         # ------------ TRAIN ------------------
         model.train()
@@ -956,9 +834,7 @@ def train_loop(model, optimizer, loss_fn, loader_train, loader_valid, n_epoch, d
         for G_c, G_h,  qu_map in loader_train:
             G_c = G_c.to(device)
             G_h = G_h.to(device)
-
             qu_map = qu_map.to(device)
-
             optimizer.zero_grad()
             logit = model(G_c, G_h)
             loss = loss_fn(logit, qu_map)
@@ -968,7 +844,6 @@ def train_loop(model, optimizer, loss_fn, loader_train, loader_valid, n_epoch, d
             acc_sum, exact_sum, n_graphs = batch_graph_metrics_training(
                 logit, qu_map, G_c.batch
             )
-
             total_loss += loss.item() * n_graphs
             total_acc += acc_sum
             total_exact += exact_sum
@@ -977,29 +852,23 @@ def train_loop(model, optimizer, loss_fn, loader_train, loader_valid, n_epoch, d
         loss_hist_train[epoch] = total_loss  / total_graphs
         accuracy_hist_train[epoch] = total_acc / total_graphs
         exact_accuracy_hist_train[epoch] = total_exact / total_graphs
-
         #VALIDATION
         model.eval()
         total_acc = 0.0
         total_exact = 0
         total_graphs = 0
-
         with torch.no_grad():
             for G_c_val, G_h_val, qu_map_val in loader_valid:
                 G_c_val = G_c_val.to(device)
                 G_h_val = G_h_val.to(device)
                 qu_map_val = qu_map_val.to(device)
-
                 mapping = model(G_c_val, G_h_val)
-
                 acc_sum, exact_sum, n_graphs = batch_graph_metrics_val(
                 mapping, qu_map_val, G_c_val.batch
                 )
-
                 total_acc += acc_sum
                 total_exact += exact_sum
                 total_graphs += n_graphs
-
         accuracy_hist_valid[epoch] = total_acc / total_graphs
         exact_accuracy_hist_valid[epoch] = total_exact / total_graphs
         print(
@@ -1010,14 +879,13 @@ def train_loop(model, optimizer, loss_fn, loader_train, loader_valid, n_epoch, d
             f"Val Acc: {accuracy_hist_valid[epoch]:.4f} | "
             f"Val Exact Acc: {exact_accuracy_hist_valid[epoch]:.4f}"
         )
-
     return (loss_hist_train,
             accuracy_hist_train, exact_accuracy_hist_train,
             accuracy_hist_valid, exact_accuracy_hist_valid)
 
 ######################################################################
-# The training can be run by defining the model, the optimizer, the loss function, the sample loaders,

-# the number of epochs and the device.

+# The training can be run by defining the model, the optimizer, the loss function, the sample loaders,
+# the number of epochs and the device.
 # 
 
 loss_hist_train, accuracy_hist_train, exact_accuracy_hist_train, \
@@ -1026,41 +894,37 @@ def train_loop(model, optimizer, loss_fn, loader_train, loader_valid, n_epoch, d
                                                             n_epochs, device)
 
 ######################################################################
-# For the optimizer, we can for example using an Adam optimizer which is an efficient optimizer which

-# adapts the learning rate for each parameter based on estimates of the first and second moments (mean

-# and variance) of gradients.

+# For the optimizer, we can for example using an Adam optimizer which is an efficient optimizer which
+# adapts the learning rate for each parameter based on estimates of the first and second moments (mean
+# and variance) of gradients.
 # 
 
 optimizer = torch.optim.Adam(model.parameters(), lr=1e-3, weight_decay=1e-5)
 
 ######################################################################
-# With this architecture applied to a qubit mapping problem on hardware with 5 physical qubits (IBM

-# Athens), a full-mapping accuracy of 90% can be obtained. This is not yet perfect, but it shows that

-# a machine learning model can learn and solve the qubit mapping problem.

+# With this architecture applied to a qubit mapping problem on hardware with 5 physical qubits (IBM
+# Athens), a full-mapping accuracy of 90% can be obtained. This is not yet perfect, but it shows that
+# a machine learning model can learn and solve the qubit mapping problem.
 # 
-
 ######################################################################
-# **5. Conclusion**

-# -----------------

+# **5. Conclusion**
+# -----------------
 # 
-
 ######################################################################
-# In this demo, we introduced the qubit mapping problem. Qubit mapping is a fundamental problem for

-# hardwares with limited connectivity, such as superconducting devices. Unfortunately, the qubit

-# mapping problem is NP-hard. One way to find approximate solutions is to use heuristic methods, such

-# as the SABRE algorithm presented in this demo.

+# In this demo, we introduced the qubit mapping problem. Qubit mapping is a fundamental problem for
+# hardwares with limited connectivity, such as superconducting devices. Unfortunately, the qubit
+# mapping problem is NP-hard. One way to find approximate solutions is to use heuristic methods, such
+# as the SABRE algorithm presented in this demo.
 # 
-
 ######################################################################
-# It remains an open question whether a machine learning model can efficiently solve this problem and

-# have better scaling than the SABRE algorithm. To explore this idea, we need a dataset, which is the

-# purpose of QMapDataset. Three datasets for the IBM architectures Eagle R3, Heron r1, and Heron r2,

-# respectively, are now available in PennyLane, and we showed how to access them.

+# It remains an open question whether a machine learning model can efficiently solve this problem and
+# have better scaling than the SABRE algorithm. To explore this idea, we need a dataset, which is the
+# purpose of QMapDataset. Three datasets for the IBM architectures Eagle R3, Heron r1, and Heron r2,
+# respectively, are now available in PennyLane, and we showed how to access them.
 # 
-
 ######################################################################
-# Finally, we presented a Graph Neural Network model to solve the qubit mapping problem. This model

-# shows encouraging results on small hardware. The next challenge is how to scale it to larger

-# hardware, such as Eagle R3 (127 qubits), Heron r1 (133 qubits), and Heron r2 (156 qubits). Maybe the

-# person who finds the solution will be you, using the dataset available in PennyLane!

+# Finally, we presented a Graph Neural Network model to solve the qubit mapping problem. This model
+# shows encouraging results on small hardware. The next challenge is how to scale it to larger
+# hardware, such as Eagle R3 (127 qubits), Heron r1 (133 qubits), and Heron r2 (156 qubits). Maybe the
+# person who finds the solution will be you, using the dataset available in PennyLane!
 # 
\ No newline at end of file

From f80b56609804a91b264ca5806f33641e7bba0bc5 Mon Sep 17 00:00:00 2001
From: drdren <104710745+drdren@users.noreply.github.com>
Date: Thu, 2 Jul 2026 09:37:35 -0400
Subject: [PATCH 07/21] Fix the erroneous symbols

---
 demonstrations_v2/QMapDataset_demo/demo.py | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index 037e2c0738..7d4662403e 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -174,8 +174,8 @@ def  circuit():
 #    included because it depends on the execution of CNOT(0,4).
 # 
 # 3. **Check gate executability**: For each gate in the front layer, the algorithm checks whether the
-#    mapped physical qubits are connected in the hardware graph: If they are adjacent, the gate �is
-#    executed� and is withdrawn from the front layer. If not, SWAP gate is required and step 4 is
+#    mapped physical qubits are connected in the hardware graph: If they are adjacent, the gate "is
+#    executed" and is withdrawn from the front layer. If not, SWAP gate is required and step 4 is
 #    applied. In our case, the CNOT(0,4) and CNOT(1,3) cannot be executed.
 # 
 # 4. **Compute routing cost**: The algorithm calculates the sum of the distance of the operations in
@@ -289,7 +289,7 @@ def  circuit():
 
 ######################################################################
 # For the Heron 1 and Heron 2 architectures, you only need to replace the name argument with
-# �qubit-mapping-heron1� or �qubit-mapping-heron2�, respectively.
+# "qubit-mapping-heron1" or "qubit-mapping-heron2", respectively.
 # 
 # Each dataset is divided into three attributes: circuits, hardware, and mappings, following the
 # structure discussed earlier. For example, to access the first sample:

From d778712f922047c388d6b3d3954b4ecb4e61cd22 Mon Sep 17 00:00:00 2001
From: drdren <104710745+drdren@users.noreply.github.com>
Date: Thu, 2 Jul 2026 09:39:54 -0400
Subject: [PATCH 08/21] Update format

---
 demonstrations_v2/QMapDataset_demo/demo.py | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index 7d4662403e..66ede09c8c 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -1,13 +1,13 @@
 r"""Solving the qubit mapping problem with machine learning
 =============================================================
+
+The demo introduces the qubit mapping problem, an NP-hard challenge in quantum computing systems
+with limited hardware connectivity, and explores machine learning-based approaches for solving it.
+It presents the QMapDataset available in PennyLane and shows how it can be used to train a Graph
+Neural Network to tackle the qubit mapping problem.
+ 
 """
 ######################################################################
-# The demo introduces the qubit mapping problem, an NP-hard challenge in quantum computing systems
-# with limited hardware connectivity, and explores machine learning-based approaches for solving it.
-# It presents the QMapDataset available in PennyLane and shows how it can be used to train a Graph
-# Neural Network to tackle the qubit mapping problem.
-# 
-######################################################################
 # **1. What is the qubit mapping problem?**
 # -----------------------------------------
 # 

From 6a217db5dd5fa38666c88852990fce45c9f30687 Mon Sep 17 00:00:00 2001
From: drdren <104710745+drdren@users.noreply.github.com>
Date: Thu, 2 Jul 2026 09:40:47 -0400
Subject: [PATCH 09/21] Update format again

---
 demonstrations_v2/QMapDataset_demo/demo.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index 66ede09c8c..f5347d5e0b 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -5,8 +5,8 @@
 with limited hardware connectivity, and explores machine learning-based approaches for solving it.
 It presents the QMapDataset available in PennyLane and shows how it can be used to train a Graph
 Neural Network to tackle the qubit mapping problem.
- 
 """
+
 ######################################################################
 # **1. What is the qubit mapping problem?**
 # -----------------------------------------

From 6c187c450b4db919dcca0625026b1b57494a693e Mon Sep 17 00:00:00 2001
From: drdren <104710745+drdren@users.noreply.github.com>
Date: Thu, 2 Jul 2026 10:00:09 -0400
Subject: [PATCH 10/21] Import numpy

---
 demonstrations_v2/QMapDataset_demo/demo.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index f5347d5e0b..d336b61b6b 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -358,7 +358,7 @@ def  circuit():
 # qubits. Since the relative values of these features are what matter, we standardize them. This is
 # the code to build the graph for the quantum circuit of the QMapDataset:
 # 
-
+import numpy as np
 import torch
 from torch_geometric.data import Data
 def build_circuit_graph (circuit_data):

From d3999a9774699ea55861395b103439f228e847c1 Mon Sep 17 00:00:00 2001
From: drdren <104710745+drdren@users.noreply.github.com>
Date: Thu, 2 Jul 2026 11:42:12 -0400
Subject: [PATCH 11/21] Remove extraneous sections

---
 demonstrations_v2/QMapDataset_demo/demo.py | 71 +---------------------
 1 file changed, 1 insertion(+), 70 deletions(-)

diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index d336b61b6b..ec515e1631 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -11,7 +11,6 @@
 # **1. What is the qubit mapping problem?**
 # -----------------------------------------
 # 
-######################################################################
 # When we write quantum circuits (with Pennylane, of course!), we usually assume that all the qubits
 # are connected to each other, or more precisely, that two-qubit gates can be applied between any
 # qubits. While some infrastructures have all-to-all connectivity (like the photonic devices developed
@@ -21,7 +20,6 @@
 # 
 # :math:`pq_0-pq_1-pq_2-pq_3-pq_4`
 # 
-######################################################################
 # Physical qubit 0 and physical qubit 3 are not directly connected. If a two-qubit gate is required in
 # your code between qubits 0 and 3, it is necessary to apply SWAP gates. SWAP gates exchange the
 # states of two neighboring qubits for all basis states in the superposition (more details here `What
@@ -32,13 +30,11 @@
 # :math:`\frac{1}{\sqrt{2}}(|00100\rangle + |00001\rangle)`. Then, we can apply the CNOT gate between
 # :math:`pq_2` and :math:`pq_3`.
 # 
-######################################################################
 # The lack of full connectivity between physical qubits does not alter the universality of quantum
 # computation in these devices but leads to an increase in the depth of the effective quantum circuit.
 # A SWAP gate corresponds to 3 CNOT gates. Therefore, instead of 1 CNOT gate, we have 7 CNOT gates.
 # This increase can be fatal for your quantum computation, especially on a NISQ device.
 # 
-######################################################################
 # Indeed, NISQ devices are sensitive to noise. In particular, two-qubit gates are noisy. Therefore,
 # when a large number of two-qubit gates is applied on a NISQ device, the quantum information is
 # progressively lost and randomized. Therefore, when running a quantum circuit on a NISQ device, we
@@ -47,7 +43,6 @@
 # quantum computers, reducing the number of CNOT gates would help to reduce the time of computation
 # and therefore the computational cost.
 # 
-######################################################################
 # It is here that qubit mapping can help mitigate the number of SWAP gates. There is no rule that
 # constrains the first qubit of your quantum circuit to be the first physical qubit, the second qubit
 # of your quantum circuit to be the second physical qubit, etc. We can separate the two sets of
@@ -61,9 +56,7 @@
 # :math:`lq_3\rightarrow pq_1`, a CNOT gate between physical qubits 0 and 1 corresponds to a CNOT gate
 # between logical qubits 0 and 3, as desired.
 # 
-######################################################################
 # Let's consider a slightly more complex quantum circuit:
-# 
 
 import pennylane as qp
 dev = qp.device("default.qubit", wires=5)
@@ -81,37 +74,30 @@ def  circuit():
 # 
 # :math:`pq_0(lq_0)-pq_1(lq_1)-pq_2(lq_2)-pq_3(lq_3)-pq_4(lq_4)`
 # 
-######################################################################
 # Here, in parentheses, the logical qubit mapped to the corresponding physical qubit is shown.
 # 
-######################################################################
 # For the first CNOT gate between :math:`lq_0` and :math:`lq_4`, we need to apply 3 SWAP gates
 # (SWAP(:math:`pq_0`, :math:`pq_1`), SWAP(:math:`pq_1`, :math:`pq_2`), and SWAP(:math:`pq_2`,
 # :math:`pq_3`)). The mapping becomes:
 # 
 # :math:`pq_0(lq_1)-pq_1(lq_2)-pq_2(lq_3)-pq_3(lq_0)-pq_4(lq_4)`
 # 
-######################################################################
 # Afterward, for the CNOT gate between :math:`lq_1` and :math:`lq_3`, we need a SWAP gate between
 # :math:`pq_1` and :math:`pq_2`:
 # 
 # :math:`pq_0(lq_1)-pq_1(lq_3)-pq_2(lq_2)-pq_4(lq_0)-pq_4(lq_4)`
 # 
-######################################################################
 # Finally, for the CNOT gate between :math:`lq_0` and :math:`lq_2`, we do not need any additional SWAP
 # gates. Therefore, in total, there are 4 SWAP gates.
 # 
-######################################################################
 # Now, let’s consider the following initial mapping:
 # 
 # :math:`pq_0(lq_1)-pq_1(lq_3)-pq_2(lq_2)-pq_3(lq_0)-pq_4(lq_4)`
 # 
-######################################################################
 # In this case, there is no need for SWAP gates. The three CNOT gates can be applied directly. We go
 # from 15 CNOT gates to 3 CNOT gates, a reduction by a factor of 5 just by changing the initial
 # mapping!
 # 
-######################################################################
 # Now that the qubit mapping problem is clear from these 5-qubit examples, let’s give a generalized
 # definition. We start with:
 # 
@@ -119,37 +105,31 @@ def  circuit():
 # -  A quantum device with m physical qubits :math:`pq_0, pq_1, \dots, pq_{m-1}` connected by some
 #    graph :math:`G_D`.
 # 
-######################################################################
 # The goal of the qubit mapping problem is to find the initial mapping
 # :math:`\pi_0: \{lq\}\rightarrow \{pq\}` in order to minimize the number of SWAP gates during the
 # execution of the quantum circuit C on the quantum device. Note that in general, the number of
 # logical qubits :math:`n` is smaller than the number of physical qubits :math:`m` (and cannot be
 # bigger).
 # 
-######################################################################
 # The initial qubit mapping problem can be represented as an assignment matrix A of dimension
 # :math:`n\times m`:
 # 
 # -  :math:`A_{ij}=1` if logical qubit :math:`lq_i` is mapped to the physical qubit :math:`pq_j`.
 # -  :math:`A_{ij}=0` otherwise
 # 
-######################################################################
 # There are two constraints on the assignment matrix A : \* Each logical qubit is assigned exactly one
 # physical qubit :math:`\sum_{j=1}^m A_{ij} = 1, ~\forall i`
 # 
 # -  Each physical qubit holds at most one logical qubit :math:`\sum_{i=1}^n A_{ij}\leq 1, ~\forall j`
 # 
-######################################################################
 # An important quantity is the distance between two physical qubits :math:`D_{ij}=d(pq_i, pq_j)`. The
 # distance is the shortest-path length between two physical qubits in the hardware connectivity graph,
 # and it directly estimates how many SWAPs are needed to make them interact. Of course, the distance
 # would depend on the hardware connectivity graph.
 # 
-######################################################################
 # We will explore in the next section the actual methods used to find the optimal assignment matrix A
 # and therefore to solve the qubit mapping problem.
 # 
-######################################################################
 # **2.How to solve the qubit mapping problem?**
 # ---------------------------------------------
 # 
@@ -159,11 +139,9 @@ def  circuit():
 # heuristic methods that allow us to solve it with a reasonable computational time. The most famous
 # heuristic algorithm is the SABRE algorithm, used for example by Qiskit in its compilation.
 # 
-######################################################################
 # The SABRE algorithm works by iteratively updating the qubit mapping guided by a cost function based
 # on the total distance.
 # 
-######################################################################
 # **Forward propagation** 1. **Initialize the mapping**: The algorithm starts by initializing a single
 # mapping between logical and physical qubits (typically random or identity). Example (5-qubit
 # device): :math:`pq_0(lq_0)-pq_1(lq_2)-pq_2(lq_3)-pq_3(lq_4)-pq_4(lq_1)`
@@ -190,7 +168,6 @@ def  circuit():
 #    gates become executable. Gates are executed whenever possible. SWAPs are inserted when necessary.
 #    At the end of this process, the algorithm produces a final mapping for the forward pass.
 # 
-######################################################################
 # **Backward propagation**
 # 
 # 7. **Reverse circuit execution**: The algorithm then repeats the same procedure, but on the reversed
@@ -201,88 +178,72 @@ def  circuit():
 #    :math:`\pi_0` is obtained. This initial mapping is typically better than the original random
 #    initialization.
 # 
-######################################################################
 # In practice, SABRE may repeat multiple forward and backward passes (usually around 3 iterations),
 # gradually improving the mapping until no significant improvement is observed.
 # 
-######################################################################
 # The SABRE algorithm has a polynomial computational complexity of approximately
 # :math:`\mathcal{O}(Tm^2 n)` where :math:`T` is the total number of two-qubit gates in the quantum
 # circuit. This is significantly better than the exponential scaling of an exact solution. However, as
 # the number of qubits in superconducting processors continues to increase, the runtime of the SABRE
 # algorithm also grows accordingly.
 # 
-######################################################################
 # Machine learning models could offer better scalability while maintaining high accuracy. In this
 # demo, we consider supervised learning. Supervised learning requires a dataset with input data and
 # corresponding labels to train the model. This is the purpose of QMapDataset, which provides datasets
 # and tools to generate them for solving the qubit mapping problem using supervised machine learning
 # models. QMapDataset is available in PennyLane. We will explore its features in the next section.
 # 
-######################################################################
 # **3. QMapDataset**
 # ------------------
 # 
-######################################################################
 # QMapDataset is a dataset generator for the qubit mapping problem, designed for IBM quantum devices.
 # The dataset is built from three main components: the hardware, the circuits, and the mapping.
 # 
-######################################################################
 # **For the hardware**, First, we choose an IBM quantum processor. To make the dataset more diverse,
 # the code can randomly permute the properties of the qubits and gates (data augmentation). We can
 # control how often these permutations are applied.
 # 
-######################################################################
 # Then, the code collects all the relevant hardware information:
 # 
 # -  physical qubit properties (:math:`T_1`, :math:`T_2`, readout error)
 # -  single-qubit gate errors for each qubit
 # -  two-qubit gate errors for each connected pair of qubits
 # 
-######################################################################
 # **For the circuits**, These are sampled either from:
 # 
 # -  random quantum circuits
 # -  well-known circuits (QPE, Grover, QFT, GHZ state preparation etc.)
 # 
-######################################################################
 # By default, the choice is split 50–50 between random and well-known circuits.
 # 
 # For random quantum circuits, the code also sample: \* the number of logical qubits \* the circuit
 # depth
 # 
-######################################################################
 # The number of logical qubits is chosen between 3 and the number of physical qubits available on the
 # hardware. The depth follows this distribution: 25% shallow (5–15 layers), 40% medium (16–50),
 # 30% deep (51–150), and 5% very deep circuits (151–400).
 # 
-######################################################################
 # To further enrich the dataset, we apply random permutations of qubits to the well-known circuits
 # (since there are only a few of them). After that, the code collects:
 # 
 # -  Single-qubit gate counts for each logical qubit
 # -  Two-qubit gate counts for each pair of logical qubits
 # 
-######################################################################
 # Finally, **for the mapping**, the code uses the generated hardware and circuits to compute the qubit
 # mapping with an advanced version of the SABRE algorithm (optimization_level = 3 in Qiskit).
 # 
-######################################################################
 # If you want more details, you can have a look at the GitHub repo: `rscadrien/QMapDataset: Generator
 # of dataset for qubit mapping <https://github.com/rscadrien/QMapDataset>`__.
 # 
-######################################################################
 # The good news is that you do not need to generate this dataset using QMapDataset yourself! We have
 # already prepared datasets for three different types of IBM quantum processors: Eagle 3, Heron 1, and
 # Heron 2. Each dataset contains 10,000 samples, where each sample consists of a circuit, hardware
 # description, and mapping.
 # 
-######################################################################
 # Let’s see how to use these datasets with PennyLane and extract their information.
 # 
 # The first step is to load a dataset from the Xanadu server. For example, to load the dataset for the
 # Eagle 3 architecture:
-# 
 
 import pennylane as qp
 [ds] = qp.data.load("other", name="qubit-mapping-eagle3")
@@ -326,31 +287,25 @@ def  circuit():
 # section of the Qubit Mapping Eagle 3 page on the PennyLane website. (`Qubit Mapping Eagle
 # 3 <https://pennylane.ai/datasets/qubit-mapping-eagle3>`__ ).
 # 
-######################################################################
 # **4.Building a machine learning model to solve the qubit mapping problem**
 # --------------------------------------------------------------------------
 # 
-######################################################################
 # In this session, we will show an example of how to preprocess data from QMapDataset and how to build
 # a machine learning model that can solve the qubit mapping problem.
 # 
-######################################################################
 # **4.1 Preprocessing**
 # ~~~~~~~~~~~~~~~~~~~~~
 # 
-######################################################################
 # The qubit mapping problem strongly depends on the connectivity between the physical qubits in the
 # given hardware, as well as the connectivity between the logical qubits in the circuits (represented
 # by two-qubit gate counts). Therefore, a graph is a natural representation for solving the qubit
 # mapping problem. We will see how to construct graphs for both the circuits and the hardware using
 # the QMapDataset available in PennyLane.
 # 
-######################################################################
 # A graph is a way of showing relationships. It is made up of points, called nodes (or vertices), that
 # are connected by lines, called edges. Both nodes and edges can also have extra information attached
 # to them, called features.
 # 
-######################################################################
 # For the **circuit**, the nodes represent the logical qubits, and edges connect pairs of logical
 # qubits between which two-qubit gates are applied in the quantum circuit. The features of each node
 # are the counts of different single-qubit gates applied to the corresponding logical qubit. The
@@ -501,14 +456,12 @@ def edge_info(gate_dict):
 # 
 # .. |My graph| image:: graph_circuit.png
 # 
-######################################################################
 # For the **hardware**, the nodes represent the physical qubits, and edges connect pairs of physical
 # qubits that are directly connected in the device architecture. The features of each node are the
 # :math:`T_1` and :math:`T_2` coherence times, readout error, and single-qubit gate errors for each
 # physical qubit. The features of each edge are the errors of the different two-qubit gates applied
 # between connected physical qubits. As with the circuits, the feature values are standardized. The
-# code to build the graph for the hardware is pretty similar to the one for the circuit:
-# 
+# code to build the graph for the hardware is pretty similar to the one for the circuit: 
 
 def build_hardware_graph(hardware_data):
     # Scaling single qubit properties and gate
@@ -558,11 +511,9 @@ def build_hardware_graph(hardware_data):
 # 
 #    My graph
 # 
-######################################################################
 # **4.2 Machine learning model**
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 # 
-######################################################################
 # We will show how to build a model that can learn qubit mapping. For this model, we will use an
 # architecture adapted to the graph structure of the input: a Graph Neural Network (GNN). The key
 # principle behind GNNs is that a node learns by aggregating information from its neighbors. A GNN
@@ -572,25 +523,21 @@ def build_hardware_graph(hardware_data):
 # -  Aggregation: Combine neighbor information (e.g., sum, mean, max)
 # -  Update: Use a neural network to update the node’s embedding.
 # 
-######################################################################
 # In our case, we will use two separate GNNs: one for the quantum circuit and one for the hardware.
 # After several GNN layers, we obtain updated embedding vectors for the nodes of both graphs,
 # :math:`h_L^{(i)}` and :math:`h_P^{(j)}`. (:math:`h_L^{(i)}` is the vector of the i th node of the
 # quantum circuit graph while :math:`h_P^{(j)}` is the vector of the jth node of the hardware graph).
 # 
-######################################################################
 # We then compare these embeddings using a dot product:
 # 
 # :math:`S_{ij}=h_L^{(i)}\cdot h_P^{(j)}`
 # 
-######################################################################
 # The matrix element :math:`S_{ij}` represents the logit that the logical qubit :math:`lq_i` is mapped
 # to the physical qubit :math:`pq_j`. The probability that logical qubit k maps to physical qubit j is
 # calculated from the logit:
 # 
 # :math:`p_{ij}=\frac{\text{exp}(S_{ij})}{\sum_{j^\prime} \text{exp}(S_{ij}^\prime)}`
 # 
-######################################################################
 # During training, the matrix :math:`p_{ij}` should ideally converge to the assignment matrix
 # :math:`A_{ij}` presented in Section 2, where each row contains exactly one nonzero element equal to
 # 1. After training, during inference, the strict assignment constraints are enforced by applying the
@@ -690,24 +637,19 @@ def forward(self, G_circuit, G_hardware):
 # **4.3 Loss function and metric**
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 # 
-######################################################################
 # We need to have a loss function to minimize for the training of the model.
 # 
-######################################################################
 # For each logical qubit k, we know the correct assignment (from the training data), which we denote
 # as :math:`k_c`. This is the physical qubit that the logical qubit should be mapped to.
 # 
-######################################################################
 # The loss for the logical qubit k is then defined as:
 # 
 # :math:`L_k=\log(p_{k,k_c})`
 # 
-######################################################################
 # Finally, the total loss is computed by averaging over all logical qubits :
 # 
 # :math:`L = \frac{1}{n} \sum_{k=1}^n -\log(p_{k,k_c})`
 # 
-######################################################################
 # This is known as the cross-entropy loss. With Pytorch, the cross entropy loss can be calculated by
 # using nn.CrossEntropyLoss():
 # 
@@ -720,26 +662,21 @@ def forward(self, G_circuit, G_hardware):
 # prevent collisions (two logical qubits choosing the same physical qubit). It is the reason that it
 # is better to use the Hungarian algorithm during inference.
 # 
-######################################################################
 # In addition to the loss function, there are some metrics that help to evaluate the quality of the
 # model and are easier to interpret than the loss function. We consider two different metrics: \*
 # logical qubit-level accuracy \* full mapping accuracy
 # 
-######################################################################
 # To understand the difference between them, let us consider an example with 5 logical qubits and 5
 # physical qubits. The correct mapping (from the training data) is [2,1,4,3,0] and the model predicts
 # [2,0,4,3,1].
 # 
-######################################################################
 # For the logical qubit-level accuracy, we simply count how many individual assignments are correct.
 # In this example, 3 out of 5 qubits are correctly mapped. So the accuracy is 60%.
 # 
-######################################################################
 # For the full mapping accuracy, the prediction is considered correct only if the entire assignment is
 # exactly correct. In this example, the predicted mapping is not identical to the ground truth. So the
 # full mapping is incorrect. This metric is therefore much stricter and harder to improve.
 # 
-######################################################################
 # You could ask: Why use both metrics? These two metrics serve different goals. The logical
 # qubit-level accuracy is useful to monitor progress during training. The full mapping accuracy
 # reflects the actual goal of the problem: producing a correct global qubit mapping. During training,
@@ -748,7 +685,6 @@ def forward(self, G_circuit, G_hardware):
 # Hungarian algorithm significantly improves the full-mapping accuracy. As a result, the full-mapping
 # accuracy is often higher on the validation set than on the training set.
 # 
-######################################################################
 # These metrics are calculated with the following code for the training and the validation dataset:
 # 
 
@@ -808,7 +744,6 @@ def batch_graph_metrics_val(pred, qu_map, batch):
 # **4.3 Training Loop**
 # ~~~~~~~~~~~~~~~~~~~~~
 # 
-######################################################################
 # We now have all the elements to build our training loop! Here it is the code:
 # 
 
@@ -906,23 +841,19 @@ def train_loop(model, optimizer, loss_fn, loader_train, loader_valid, n_epoch, d
 # Athens), a full-mapping accuracy of 90% can be obtained. This is not yet perfect, but it shows that
 # a machine learning model can learn and solve the qubit mapping problem.
 # 
-######################################################################
 # **5. Conclusion**
 # -----------------
 # 
-######################################################################
 # In this demo, we introduced the qubit mapping problem. Qubit mapping is a fundamental problem for
 # hardwares with limited connectivity, such as superconducting devices. Unfortunately, the qubit
 # mapping problem is NP-hard. One way to find approximate solutions is to use heuristic methods, such
 # as the SABRE algorithm presented in this demo.
 # 
-######################################################################
 # It remains an open question whether a machine learning model can efficiently solve this problem and
 # have better scaling than the SABRE algorithm. To explore this idea, we need a dataset, which is the
 # purpose of QMapDataset. Three datasets for the IBM architectures Eagle R3, Heron r1, and Heron r2,
 # respectively, are now available in PennyLane, and we showed how to access them.
 # 
-######################################################################
 # Finally, we presented a Graph Neural Network model to solve the qubit mapping problem. This model
 # shows encouraging results on small hardware. The next challenge is how to scale it to larger
 # hardware, such as Eagle R3 (127 qubits), Heron r1 (133 qubits), and Heron r2 (156 qubits). Maybe the

From 27c11b894b51ba2787810fce3110ba6642bc4c75 Mon Sep 17 00:00:00 2001
From: drdren <104710745+drdren@users.noreply.github.com>
Date: Thu, 2 Jul 2026 12:16:50 -0400
Subject: [PATCH 12/21] Fix the "test" and "pos" variables error

---
 demonstrations_v2/QMapDataset_demo/demo.py | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index ec515e1631..f8f77d2196 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -429,7 +429,7 @@ def edge_info(gate_dict):
 from torch_geometric.utils import to_networkx
 # Convert PyG Data object to NetworkX graph
 G = to_networkx(
-    test,
+    Graph_circuit,
     edge_attrs=["edge_attr"],  # keep CX counts
     node_attrs=["x"],          # keep node features
     to_undirected=False
@@ -440,13 +440,13 @@ def edge_info(gate_dict):
 }
 nx.draw(
     G,
-    pos,
+    nx.spring_layout(G, seed=42),
     with_labels=True,
     node_size=800,
 )
 nx.draw_networkx_edge_labels(
     G,
-    pos,
+    nx.spring_layout(G, seed=42),
     edge_labels=edge_labels
 )
 plt.show()

From 2c0c34903275439e71a2cea6c7c4df368c3339bb Mon Sep 17 00:00:00 2001
From: drdren <104710745+drdren@users.noreply.github.com>
Date: Thu, 2 Jul 2026 12:26:34 -0400
Subject: [PATCH 13/21] Add imports to nn etc

---
 demonstrations_v2/QMapDataset_demo/demo.py | 6 ++++++
 1 file changed, 6 insertions(+)

diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index f8f77d2196..3a969f4c44 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -545,6 +545,12 @@ def build_hardware_graph(hardware_data):
 # used during training.The code for this architecture is as follows :
 # 
 
+import torch, torch.nn as nn, torch.nn.functional as F
+from torch_geometric.data import Batch
+from torch_geometric.nn import NNConv
+from torch.utils.data import Dataset, DataLoader, random_split
+from scipy.optimize import linear_sum_assignment
+
 class QubitMapping_dot_Model(nn.Module):
     def __init__(self, node_c_dim, node_h_dim, edge_c_dim, edge_h_dim, hidden_dim):
         super().__init__()

From 46a2901ad903bb6bc7a47f58d413a194823792b2 Mon Sep 17 00:00:00 2001
From: Adrien Devolder <adridevolder@hotmail.com>
Date: Thu, 2 Jul 2026 13:58:59 -0400
Subject: [PATCH 14/21] Debug demo code

---
 demonstrations_v2/QMapDataset_demo/demo.py | 51 ++++++++++++++++++----
 1 file changed, 43 insertions(+), 8 deletions(-)

diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index 3a969f4c44..a5ed1cb8c6 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -487,10 +487,18 @@ def build_hardware_graph(hardware_data):
     )
     return G_hardware
 
+Graph_hardware = build_hardware_graph(ds.hardware[26])
+
 ######################################################################
 # We can again visualize the graph using networkx:
 # 
 # Draw graph
+G_hardware = to_networkx(
+    Graph_hardware,
+    edge_attrs=["edge_attr"],  # keep CX counts
+    node_attrs=["x"],          # keep node features
+    to_undirected=False
+)
 
 plt.figure(figsize=(10, 10))
 pos = nx.spring_layout(
@@ -639,6 +647,41 @@ def forward(self, G_circuit, G_hardware):
             mapping = torch.cat(prob_list, dim=0)
             return mapping
 
+def make_edge_mlp(in_dim, out_dim, hid_dim=32):
+    """
+    Create a small multilayer perceptron (MLP) used to process edge features.
+
+    Parameters
+    ----------
+    in_dim : int
+        Dimension of the input edge feature vector.
+    out_dim : int
+        Dimension of the output representation (e.g., message, weight,
+        or embedding associated with an edge).
+    hid_dim : int, optional
+        Number of neurons in the hidden layer (default: 32).
+
+    Returns
+    -------
+    torch.nn.Sequential
+        Two-layer feed-forward neural network:
+            Linear(in_dim -> hid_dim)
+            ReLU activation
+            Linear(hid_dim -> out_dim)
+
+    Notes
+    -----
+    In graph neural networks, this MLP is typically applied to edge
+    attributes to generate edge embeddings or edge-dependent messages
+    that are used during the message-passing/convolution step.
+    """
+    return nn.Sequential(
+        nn.Linear(in_dim, hid_dim),
+        nn.ReLU(),
+        nn.Linear(hid_dim, out_dim)
+    )
+
+
 ######################################################################
 # **4.3 Loss function and metric**
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -828,20 +871,12 @@ def train_loop(model, optimizer, loss_fn, loader_train, loader_valid, n_epoch, d
 # The training can be run by defining the model, the optimizer, the loss function, the sample loaders,
 # the number of epochs and the device.
 # 
-
-loss_hist_train, accuracy_hist_train, exact_accuracy_hist_train, \
-accuracy_hist_valid, exact_accuracy_hist_valid = train_loop(model, optimizer, loss_fn,
-                                                            train_loader, val_loader,
-                                                            n_epochs, device)
-
 ######################################################################
 # For the optimizer, we can for example using an Adam optimizer which is an efficient optimizer which
 # adapts the learning rate for each parameter based on estimates of the first and second moments (mean
 # and variance) of gradients.
 # 
 
-optimizer = torch.optim.Adam(model.parameters(), lr=1e-3, weight_decay=1e-5)
-
 ######################################################################
 # With this architecture applied to a qubit mapping problem on hardware with 5 physical qubits (IBM
 # Athens), a full-mapping accuracy of 90% can be obtained. This is not yet perfect, but it shows that

From 21b30a5dc967d40a8031bb1114d4ca80f59c9a7d Mon Sep 17 00:00:00 2001
From: Adrien Devolder <adridevolder@hotmail.com>
Date: Thu, 2 Jul 2026 14:16:42 -0400
Subject: [PATCH 15/21] Debug the demo code part2

---
 demonstrations_v2/QMapDataset_demo/demo.py | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index a5ed1cb8c6..1f62949b4e 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -471,14 +471,14 @@ def build_hardware_graph(hardware_data):
     sx_error_scaled = standard_scaling_across_qubits(np.array(hardware_data['sx_error']))
     x_error_scaled = standard_scaling_across_qubits(np.array(hardware_data['x_error']))
     #Scaling two qubit gate properties
-    cx_error_scaled = scaled_two_qubit_list(hardware_data['cx'])
+    ecr_error_scaled = scaled_two_qubit_list(hardware_data['ecr'])
     #Build the node feature matrix
     X = torch.tensor(
     list(zip(T1_scaled, T2_scaled, readout_error_scaled, sx_error_scaled, x_error_scaled)),
     dtype=torch.float
     )
     #Build the edge index and attributes matrix
-    cx_index, cx_attr = edge_info(cx_error_scaled)
+    cx_index, cx_attr = edge_info(ecr_error_scaled)
     #Build the graph
     G_hardware = Data(
     x=X,

From 36c15845f9170cf659077e90269146accd946881 Mon Sep 17 00:00:00 2001
From: Adrien Devolder <adridevolder@hotmail.com>
Date: Thu, 2 Jul 2026 14:28:16 -0400
Subject: [PATCH 16/21] Debug the demo code part 3

---
 demonstrations_v2/QMapDataset_demo/demo.py | 1 -
 1 file changed, 1 deletion(-)

diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index 1f62949b4e..aab499dd0a 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -871,7 +871,6 @@ def train_loop(model, optimizer, loss_fn, loader_train, loader_valid, n_epoch, d
 # The training can be run by defining the model, the optimizer, the loss function, the sample loaders,
 # the number of epochs and the device.
 # 
-######################################################################
 # For the optimizer, we can for example using an Adam optimizer which is an efficient optimizer which
 # adapts the learning rate for each parameter based on estimates of the first and second moments (mean
 # and variance) of gradients.

From e9a322d0419db4a0ba6419041014a330c3dd2c48 Mon Sep 17 00:00:00 2001
From: Adrien Devolder <130473693+rscadrien@users.noreply.github.com>
Date: Thu, 2 Jul 2026 14:47:53 -0400
Subject: [PATCH 17/21] Update demonstrations_v2/QMapDataset_demo/demo.py

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>
---
 demonstrations_v2/QMapDataset_demo/demo.py | 1 -
 1 file changed, 1 deletion(-)

diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index aab499dd0a..b604b6777b 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -133,7 +133,6 @@ def  circuit():
 # **2.How to solve the qubit mapping problem?**
 # ---------------------------------------------
 # 
-######################################################################
 # The qubit mapping problem is unfortunately a NP problem and is exponentially hard to find the
 # optimal solution when the number of logical and physical qubits increases. However, there are some
 # heuristic methods that allow us to solve it with a reasonable computational time. The most famous

From 796e2a2d07e9b85fbd1c1f1b63c2ded0cf44b44f Mon Sep 17 00:00:00 2001
From: drdren <104710745+drdren@users.noreply.github.com>
Date: Thu, 2 Jul 2026 14:54:31 -0400
Subject: [PATCH 18/21] Update graphs

---
 .../qubit-mapping}/Thumbnail_example.png          | Bin
 .../qubit-mapping}/graph_circuit.png              | Bin
 .../qubit-mapping}/graph_hardware.png             | Bin
 demonstrations_v2/QMapDataset_demo/demo.py        |  14 ++++++++++----
 4 files changed, 10 insertions(+), 4 deletions(-)
 rename {demonstrations_v2/QMapDataset_demo => _static/demonstration_assets/qubit-mapping}/Thumbnail_example.png (100%)
 rename {demonstrations_v2/QMapDataset_demo => _static/demonstration_assets/qubit-mapping}/graph_circuit.png (100%)
 rename {demonstrations_v2/QMapDataset_demo => _static/demonstration_assets/qubit-mapping}/graph_hardware.png (100%)

diff --git a/demonstrations_v2/QMapDataset_demo/Thumbnail_example.png b/_static/demonstration_assets/qubit-mapping/Thumbnail_example.png
similarity index 100%
rename from demonstrations_v2/QMapDataset_demo/Thumbnail_example.png
rename to _static/demonstration_assets/qubit-mapping/Thumbnail_example.png
diff --git a/demonstrations_v2/QMapDataset_demo/graph_circuit.png b/_static/demonstration_assets/qubit-mapping/graph_circuit.png
similarity index 100%
rename from demonstrations_v2/QMapDataset_demo/graph_circuit.png
rename to _static/demonstration_assets/qubit-mapping/graph_circuit.png
diff --git a/demonstrations_v2/QMapDataset_demo/graph_hardware.png b/_static/demonstration_assets/qubit-mapping/graph_hardware.png
similarity index 100%
rename from demonstrations_v2/QMapDataset_demo/graph_hardware.png
rename to _static/demonstration_assets/qubit-mapping/graph_hardware.png
diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index b604b6777b..651bd52cf6 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -453,7 +453,10 @@ def edge_info(gate_dict):
 ######################################################################
 # We obtain the following graph: |My graph|
 # 
-# .. |My graph| image:: graph_circuit.png
+# .. figure:: ../_static/demonstration_assets/qubit-mapping/graph_circuit.png
+#   :alt: My graph
+#   :width: 95%
+#   :align: center
 # 
 # For the **hardware**, the nodes represent the physical qubits, and edges connect pairs of physical
 # qubits that are directly connected in the device architecture. The features of each node are the
@@ -513,8 +516,11 @@ def build_hardware_graph(hardware_data):
 plt.show()
 
 ######################################################################
-# .. figure:: graph_hardware.png
-#    :alt: My graph
+# 
+# .. figure:: ../_static/demonstration_assets/qubit-mapping/graph_hardware.png
+#   :alt: My graph
+#   :width: 95%
+#   :align: center
 # 
 #    My graph
 # 
@@ -897,4 +903,4 @@ def train_loop(model, optimizer, loss_fn, loader_train, loader_valid, n_epoch, d
 # shows encouraging results on small hardware. The next challenge is how to scale it to larger
 # hardware, such as Eagle R3 (127 qubits), Heron r1 (133 qubits), and Heron r2 (156 qubits). Maybe the
 # person who finds the solution will be you, using the dataset available in PennyLane!
-# 
\ No newline at end of file
+# 

From 80474f8e1db6d536b6887128340c9abdd8d7656e Mon Sep 17 00:00:00 2001
From: Adrien Devolder <130473693+rscadrien@users.noreply.github.com>
Date: Thu, 2 Jul 2026 15:06:18 -0400
Subject: [PATCH 19/21] Update demonstrations_v2/QMapDataset_demo/demo.py

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>
---
 demonstrations_v2/QMapDataset_demo/demo.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index 651bd52cf6..60f2455c9a 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -130,7 +130,7 @@ def  circuit():
 # We will explore in the next section the actual methods used to find the optimal assignment matrix A
 # and therefore to solve the qubit mapping problem.
 # 
-# **2.How to solve the qubit mapping problem?**
+# **2. How to solve the qubit mapping problem?**
 # ---------------------------------------------
 # 
 # The qubit mapping problem is unfortunately a NP problem and is exponentially hard to find the

From ae5c3a66e882540a40e60d6d5a5e024d9e0d9b05 Mon Sep 17 00:00:00 2001
From: Adrien Devolder <130473693+rscadrien@users.noreply.github.com>
Date: Thu, 2 Jul 2026 15:06:29 -0400
Subject: [PATCH 20/21] Update demonstrations_v2/QMapDataset_demo/demo.py

Co-authored-by: drdren <104710745+drdren@users.noreply.github.com>
---
 demonstrations_v2/QMapDataset_demo/demo.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index 60f2455c9a..933bd203f5 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -286,7 +286,7 @@ def  circuit():
 # section of the Qubit Mapping Eagle 3 page on the PennyLane website. (`Qubit Mapping Eagle
 # 3 <https://pennylane.ai/datasets/qubit-mapping-eagle3>`__ ).
 # 
-# **4.Building a machine learning model to solve the qubit mapping problem**
+# **4. Building a machine learning model to solve the qubit mapping problem**
 # --------------------------------------------------------------------------
 # 
 # In this session, we will show an example of how to preprocess data from QMapDataset and how to build

From 3999ff586910079b23dfbfbe7609654c8053a393 Mon Sep 17 00:00:00 2001
From: Adrien Devolder <adridevolder@hotmail.com>
Date: Thu, 2 Jul 2026 15:09:19 -0400
Subject: [PATCH 21/21] Simplify description of make_edge_mlp

---
 demonstrations_v2/QMapDataset_demo/demo.py | 24 ----------------------
 1 file changed, 24 deletions(-)

diff --git a/demonstrations_v2/QMapDataset_demo/demo.py b/demonstrations_v2/QMapDataset_demo/demo.py
index 933bd203f5..19d376b380 100644
--- a/demonstrations_v2/QMapDataset_demo/demo.py
+++ b/demonstrations_v2/QMapDataset_demo/demo.py
@@ -655,30 +655,6 @@ def forward(self, G_circuit, G_hardware):
 def make_edge_mlp(in_dim, out_dim, hid_dim=32):
     """
     Create a small multilayer perceptron (MLP) used to process edge features.
-
-    Parameters
-    ----------
-    in_dim : int
-        Dimension of the input edge feature vector.
-    out_dim : int
-        Dimension of the output representation (e.g., message, weight,
-        or embedding associated with an edge).
-    hid_dim : int, optional
-        Number of neurons in the hidden layer (default: 32).
-
-    Returns
-    -------
-    torch.nn.Sequential
-        Two-layer feed-forward neural network:
-            Linear(in_dim -> hid_dim)
-            ReLU activation
-            Linear(hid_dim -> out_dim)
-
-    Notes
-    -----
-    In graph neural networks, this MLP is typically applied to edge
-    attributes to generate edge embeddings or edge-dependent messages
-    that are used during the message-passing/convolution step.
     """
     return nn.Sequential(
         nn.Linear(in_dim, hid_dim),