Edit VRE_VQE_QAOA.ipynb

VRE_VQE_QAOA.ipynb

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+# **Solving the Vehicle Routing Problem (VRP) Using Quantum Computing (Qiskit AerSimulator) for Passau Location**\n
+### **A Quantum Optimization Approach with QAOA and VQE**\n
+---\n
+## 📌 **Problem Statement**\n
+The **Vehicle Routing Problem (VRP)** extends the **Traveling Salesman Problem (TSP)** by introducing multiple vehicles that must service customers while minimizing travel costs.  \n
+The goal is to **determine optimal routes for a fleet of vehicles** to serve all locations efficiently.\n
+\n
+### 🔹 **Objective**\n
+This notebook implements a **quantum approach to solving VRP** using:\n
+- **Quantum Approximate Optimization Algorithm (QAOA)**\n
+- **Variational Quantum Eigensolver (VQE)**\n
+\n
+Both algorithms leverage **quantum computing** to optimize routing problems.\n
+\n
+---\n
+## 📌 **Mathematical Formulation**\n
+### 1️⃣ **VRP as a Quadratic Unconstrained Binary Optimization (QUBO)**\n
+\n
+The **binary decision variable** is defined as:\n
+\n
+$$\n
+x_{i,j,k} =\n
+\begin{cases} \n
+1, & \text{if vehicle } k \text{ travels from node } i \text{ to node } j \\\\ \n
+0, & \text{otherwise}\n
+\end{cases}\n
+$$\n
+\n
+where:\n
+- \( d_{i,j} \) is the **distance** between locations \( i \) and \( j \).\n
+- \( x_{i,j,k} \) is **1 if vehicle \( k \) travels from \( i \) to \( j \), otherwise 0**.\n
+\n
+The **objective function** (minimizing travel cost) is given by:\n
+\n
+$$\n
+H_{\text{VRP}} = \sum_{k} \sum_{i} \sum_{j} d_{i,j} \cdot x_{i,j,k}\n
+$$\n
+\n
+### 📌 **Constraints**\n
+#### 1️⃣ **Each location is visited exactly once (except the depot):**\n
+$$\n
+\sum_{k} \sum_{j} x_{i,j,k} = 1, \quad \forall i \neq 0\n
+$$\n
+\n
+#### 2️⃣ **Each vehicle starts and ends at the depot:**\n
+$$\n
+\sum_{j} x_{0,j,k} = 1, \quad \forall k, \quad \sum_{i} x_{i,0,k} = 1, \quad \forall k\n
+$$\n
+\n
+#### 3️⃣ **Vehicle route continuity (no teleportation):**\n
+$$\n
+\sum_{j} x_{i,j,k} - \sum_{j} x_{j,i,k} = 0, \quad \forall i, k\n
+$$\n
+\n
+#### 4️⃣ **(Optional) Vehicle capacity constraint:**\n
+If each location \( i \) has a demand \( q_i \) and the vehicle has capacity \( C_k \), then:\n
+\n
+$$\n
+\sum_{i} q_i \sum_{j} x_{i,j,k} \leq C_k, \quad \forall k\n
+$$\n
+\n
+This ensures that no vehicle exceeds its allowed load.\n
+\n
+---\n
+## 📌 **Quantum Algorithms**\n
+### 1️⃣ **Quantum Approximate Optimization Algorithm (QAOA)**\n
+QAOA minimizes the **VRP cost Hamiltonian** by iteratively optimizing quantum parameters:\n
+\n
+$$\n
+|\psi(\beta, \gamma)\rangle = U(\beta, \gamma) |s\rangle\n
+$$\n
+\n
+where:\n
+\n
+$$\n
+U(\beta, \gamma) = e^{-i \beta H_M} e^{-i \gamma H_C}\n
+$$\n
+\n
+- \( H_C \) is the **VRP cost Hamiltonian**.\n
+- \( H_M \) is the **mixing Hamiltonian** for exploring different routes.\n
+- \( (\beta, \gamma) \) are **classical parameters** optimized iteratively.\n
+\n
+### 2️⃣ **Variational Quantum Eigensolver (VQE)**\n
+VQE finds the **optimal VRP route** by solving:\n
+\n
+$$\n
+E(\theta) = \langle \psi(\theta) | H | \psi(\theta) \rangle\n
+$$\n
+\n
+where:\n
+- \( H \) represents the **VRP Hamiltonian**.\n
+- \( |\psi(\theta)\rangle \) is the **quantum circuit ansatz**.\n
+- \( \theta \) are **circuit parameters optimized using a classical algorithm**.\n
+\n
+---\n
+## 📌 **Approach in This Notebook**\n
+### 🔹 **Step 1: Define the VRP**\n
+- **Select 5 locations in Passau, Germany**.\n
+- **Define the distance matrix** and construct a weighted graph.\n
+- **Convert VRP into a QUBO model**.\n
+\n
+### 🔹 **Step 2: Solve Using Quantum Algorithms**\n
+- **QAOA** minimizes the VRP Hamiltonian.\n
+- **VQE** optimizes quantum parameters to find the best route.\n
+\n
+### 🔹 **Step 3: Compare Results**\n
+- **Run QAOA and VQE** on a quantum simulator.\n
+- **Compare total travel distances & routes**.\n
+- **Visualize optimized vehicle routes**.\n
+\n
+---\n
+## 📌 **Expected Outcome**\n
+- **Quantum algorithms provide approximate solutions to VRP.**\n
+- **Compare QAOA vs. VQE performance.**\n
+- **Visualize optimized travel routes using quantum optimization.**\n
+\n
+### 🚀 **Now, let's execute the quantum VRP solver below!**\n
+
+
+
+
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