Edit VRE_VQE_QAOA.ipynb

VRE_VQE_QAOA.ipynb

@@ -162,128 +162,7 @@
     "**Now, let's execute the quantum VRP solver below!**\n"
    ]
   },
-  # **Solving the Vehicle Routing Problem (VRP) Using Quantum Computing (Qiskit AerSimulator) for Passau Location**
-  ### **A Quantum Optimization Approach with QAOA and VQE**
-  ---
-  ## 📌 **Problem Statement**
-  The **Vehicle Routing Problem (VRP)** extends the **Traveling Salesman Problem (TSP)** by introducing multiple vehicles that must service customers while minimizing travel costs.  
-  The goal is to **determine optimal routes for a fleet of vehicles** to serve all locations efficiently.
   
-  ### 🔹 **Objective**
-  This notebook implements a **quantum approach to solving VRP** using:
-  - **Quantum Approximate Optimization Algorithm (QAOA)**
-  - **Variational Quantum Eigensolver (VQE)**
-  
-  Both algorithms leverage **quantum computing** to optimize routing problems.
-  
-  ---
-  ## 📌 **Mathematical Formulation**
-  ### 1️⃣ **VRP as a Quadratic Unconstrained Binary Optimization (QUBO)**
-  
-  The **binary decision variable** is defined as:
-  
-  $$
-  x_{i,j,k} =
-  \begin{cases} 
-  1, & \text{if vehicle } k \text{ travels from node } i \text{ to node } j \\
-  0, & \text{otherwise}
-  \end{cases}
-  $$
-  
-  where:
-  - \( d_{i,j} \) is the **distance** between locations \( i \) and \( j \).
-  - \( x_{i,j,k} \) is **1 if vehicle \( k \) travels from \( i \) to \( j \), otherwise 0**.
-  
-  The **objective function** (minimizing travel cost) is given by:
-  
-  $$
-  H_{\text{VRP}} = \sum_{k} \sum_{i} \sum_{j} d_{i,j} \cdot x_{i,j,k}
-  $$
-  
-  ### 📌 **Constraints**
-  #### 1️⃣ **Each location is visited exactly once (except the depot):**
-  $$
-  \sum_{k} \sum_{j} x_{i,j,k} = 1, \quad \forall i \neq 0
-  $$
-  
-  #### 2️⃣ **Each vehicle starts and ends at the depot:**
-  $$
-  \sum_{j} x_{0,j,k} = 1, \quad \forall k, \quad \sum_{i} x_{i,0,k} = 1, \quad \forall k
-  $$
-  
-  #### 3️⃣ **Vehicle route continuity (no teleportation):**
-  $$
-  \sum_{j} x_{i,j,k} - \sum_{j} x_{j,i,k} = 0, \quad \forall i, k
-  $$
-  
-  #### 4️⃣ **(Optional) Vehicle capacity constraint:**
-  If each location \( i \) has a demand \( q_i \) and the vehicle has capacity \( C_k \), then:
-  
-  $$
-  \sum_{i} q_i \sum_{j} x_{i,j,k} \leq C_k, \quad \forall k
-  $$
-  
-  This ensures that no vehicle exceeds its allowed load.
-  
-  ---
-  ## 📌 **Quantum Algorithms**
-  ### 1️⃣ **Quantum Approximate Optimization Algorithm (QAOA)**
-  QAOA minimizes the **VRP cost Hamiltonian** by iteratively optimizing quantum parameters:
-  
-  $$
-  |\psi(\beta, \gamma)\rangle = U(\beta, \gamma) |s\rangle
-  $$
-  
-  where:
-  
-  $$
-  U(\beta, \gamma) = e^{-i \beta H_M} e^{-i \gamma H_C}
-  $$
-  
-  - \( H_C \) is the **VRP cost Hamiltonian**.
-  - \( H_M \) is the **mixing Hamiltonian** for exploring different routes.
-  - \( (\beta, \gamma) \) are **classical parameters** optimized iteratively.
-  
-  ### 2️⃣ **Variational Quantum Eigensolver (VQE)**
-  VQE finds the **optimal VRP route** by solving:
-  
-  $$
-  E(\theta) = \langle \psi(\theta) | H | \psi(\theta) \rangle
-  $$
-  
-  where:
-  - \( H \) represents the **VRP Hamiltonian**.
-  - \( |\psi(\theta)\rangle \) is the **quantum circuit ansatz**.
-  - \( \theta \) are **circuit parameters optimized using a classical algorithm**.
-  
-  ---
-  ## 📌 **Approach in This Notebook**
-  ### 🔹 **Step 1: Define the VRP**
-  - **Select 5 locations in Passau, Germany**.
-  - **Define the distance matrix** and construct a weighted graph.
-  - **Convert VRP into a QUBO model**.
-  
-  ### 🔹 **Step 2: Solve Using Quantum Algorithms**
-  - **QAOA** minimizes the VRP Hamiltonian.
-  - **VQE** optimizes quantum parameters to find the best route.
-  
-  ### 🔹 **Step 3: Compare Results**
-  - **Run QAOA and VQE** on a quantum simulator.
-  - **Compare total travel distances & routes**.
-  - **Visualize optimized vehicle routes**.
-  
-  ---
-  ## 📌 **Expected Outcome**
-  - **Quantum algorithms provide approximate solutions to VRP.**  
-  - **Compare QAOA vs. VQE performance.**  
-  - **Visualize optimized travel routes using quantum optimization.**  
-  
-  ---
-  
-  ### 🚀 **Now, let's execute the quantum VRP solver below!**  
-
-
-
   {
    "cell_type": "code",
    "execution_count": 1,