Quantum optimisation for supply chain: QUBO formulations and QAOA solutions for facility location and load balancing

• First use of QAOA for Facility Location (FL) and Load Balancing (LB) in supply chains optimisation. • QAOA successfully solves FL and LB using compact unbalanced-penalisation QUBO models. • Hardware-calibrated QAOA retrieves optimal FL solutions even under realistic gate and readout errors. • Provides reproducible QUBO–to–Ising transformations and full QAOA implementations in PennyLane and Qiskit. Quantum computing is emerging as a promising paradigm for solving combinatorial optimisation problems that are intractable for classical methods. Among the most prominent quantum algorithms is the Quantum Approximate Optimisation Algorithm (QAOA), which has demonstrated potential in addressing NP-hard problems using shallow circuits compatible with Noisy Intermediate-Scale Quantum (NISQ) devices. While QAOA has been applied to logistics and scheduling tasks, foundational supply chain problems such as Facility Location (FL) and Load Balancing (LB) remain largely unexplored. This paper addresses this gap by formulating FL and LB as Quadratic Unconstrained Binary Optimisation (QUBO) models and solving them using QAOA under both noiseless and hardware-calibrated noisy conditions. Two encoding methods are compared: the conventional slack-variable approach and the unbalanced penalisation technique. Results show that unbalanced penalisation consistently outperforms the slack-variable method by achieving higher feasibility and optimality rates and by avoiding the qubit inflation that limits slack-based encodings. FL instances, modelled as minimum dominating sets, exhibit strong performance in both noiseless and noisy simulations, whereas LB instances suffer from depth-induced noise due to their denser constraints. Furthermore, we outline the QUBO-to-Ising transformation process, with open-source implementations in PennyLane and Qiskit. To the best of our knowledge, this work represents the first application of QAOA to FL and LB, bridging the gap between quantum algorithm design and practical supply chain optimisation.