Quantum-Assisted Vehicle Routing: Realizing QAOA-based Approach on Gate-Based Quantum Computer

TL;DR

This work presents a quantum-assisted framework that integrates the Quantum Approximate Optimization Algorithm with a link-based formulation of VRP, and encodes flow conservation and subtour elimination directly into the cost Hamiltonian, preserving graph structure while minimizing resource requirements for practical hardware implementation.

Abstract

The Vehicle Routing Problem (VRP) is a fundamental combinatorial optimization challenge with broad applications in logistics and transportation. In this work, we present a quantum-assisted framework that integrates the Quantum Approximate Optimization Algorithm (QAOA) with a link-based formulation of VRP. Our approach encodes flow conservation and subtour elimination directly into the cost Hamiltonian, preserving graph structure while minimizing resource requirements for practical hardware implementation. We design and implement the full pipeline on a gate-based quantum computer, including problem formulation, encoding, circuit synthesis, and execution on IBM Quantum System One. Experimental results on small VRP instances highlight the effects of penalty scaling, coefficient normalization, and circuit depth on solution feasibility under hardware noise. While scalability remains constrained by circuit complexity and decoherence, the study demonstrates a practical pathway for implementing VRP on quantum hardware and identifies methodological directions for advancing near-term quantum optimization.

Figures (11)

• Figure 1: Solving the VRP using QAOA involves a formulation of the link based constrained minimization form to the unconstrained quadratic binary representation, which is then cast as an Ising Hamiltonian required for QAOA. The high level circuit is transpiled into one and two qubit basic gates for implementation on the IBM-Rensselaer quantum computer. The circuit is run repeatedly to optimize the variational parameters, and a final sampling provides the solution. The problem graph shown here is for illustrative purposes only. Section 4 describes a specific example problem.
• Figure 2: Two states showing even superposition on the Bloch sphere. The Z direction is upwards.
• Figure 3: Quantum circuit for creating the Bell state. Qubits start in the $|00\rangle$ state. The element H is a Hadamard gate applied to qubit-0, while the blue symbol with $+$ sign indicates a CNOT gate, with $q_0$ controlling the NOT operation on $q_1$. The two gray elements on the right depict measurement.
• Figure 4: QAOA circuit with the left part showing the mapping to physical qubits on the quantum computer; the initialization $R_Z(\pi/2)$ refers to the Hadamard gate for superposition. On the right is the measurement of each qubit. The middle part has been truncated for space.
• Figure 5: Convergence of QAOA parameters over optimization iterations