Solving The Vehicle Routing Problem via Quantum Support Vector Machines

TL;DR

The paper tackles VRP by deploying a hybrid quantum machine learning approach using QSVMs trained with VQE, evaluated on $3$-city and $4$-city instances with $6$- and $12$-qubit circuits. It systematically compares four quantum encoding strategies—AE, AgE, HO, and IQP—and assesses fixed versus variable Hamiltonians across multiple IBM Qiskit optimizers. Key findings show that AE and AgE offer high accuracy with manageable circuit depth, HO can excel in certain settings but deteriorates with depth, and IQP underperforms overall; the approach can outperform conventional VQE/QAOA-based methods for these small VRP cases. The work highlights the importance of encoding choice and optimizer selection in achieving practical performance on near-term quantum devices, and it points to scalable future work toward larger VRP problems and more efficient quantum kernels.

Abstract

The Vehicle Routing Problem (VRP) is an example of a combinatorial optimization problem that has attracted academic attention due to its potential use in various contexts. VRP aims to arrange vehicle deliveries to several sites in the most efficient and economical manner possible. Quantum machine learning offers a new way to obtain solutions by harnessing the natural speedups of quantum effects, although many solutions and methodologies are modified using classical tools to provide excellent approximations of the VRP. In this paper, we implement and test hybrid quantum machine learning methods for solving VRP of 3 and 4-city scenarios, which use 6 and 12 qubit circuits, respectively. The proposed method is based on quantum support vector machines (QSVMs) with a variational quantum eigensolver on a fixed or variable ansatz. Different encoding strategies are used in the experiment to transform the VRP formulation into a QSVM and solve it. Multiple optimizers from the IBM Qiskit framework are also evaluated and compared.

Figures (7)

• Figure 1: Schematic diagram depicting quantum circuit for Kernel estimation.
• Figure 2: (a) Circuit example illustrating gate operations for ${H}_{\mathrm{cost}}$. (b) Circuit example displaying gate selections with an additional $u$ gate for ${H}_{\mathrm{mixer}}$.
• Figure 3: Plot illustrating different encoding methods for two qubits. (a) Amplitude encoding, (b) angle encoding, (c) Higher order encoding, (d) IQP encoding.
• Figure 4: Plot illustrating Amplitude encoding results for QSVM solution of VRP. (a) Amplitude encoding $6$ qubits Fix Hamiltonian, (b) Amplitude encoding $6$ qubits Variable hamiltonian
• Figure 5: Plot illustrating angle encoding results for QSVM solution of VRP. (a) Angle encoding $6$ qubits Fix hamiltonian, (b) Angle encoding $12$ qubits Fix hamiltonian, (c) Angle encoding $6$ qubits Variable hamiltonian, (d) Angle encoding $12$ qubits Variable hamiltonian.