• 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.
• 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.
The Quantum Approximate Optimisation Algorithm (QAOA) has emerged as a leading candidate for near-term quantum advantage in combinatorial optimisation, but most evaluations remain confined to synthetic benchmarks such as random MaxCut instances. Translating QAOA to real-world problem instances -- with irregular graph structures, heterogeneous constraint types, and practical scale requirements -- introduces additional challenges including QUBO formulation overhead, hardware embedding costs, and the performance gap between QAOA on structured real instances vs. random benchmarks. This paper presents the Real-World QAOA (RW-QAOA) framework, a systematic methodology for applying QAOA to four real-world application domains: supply chain network optimisation, financial portfolio rebalancing, telecommunications frequency assignment, and traffic signal timing optimisation. RW-QAOA introduces three methodological contributions: a domain-adaptive QUBO formulator (DAQF) encoding real-world constraints as penalised QUBO terms with penalty coefficient auto-tuning; a warm-start initialisation strategy (WSIS) using classical heuristic solutions to seed QAOA parameter search; and a layered noise adaptation protocol (LNAP) adjusting circuit depth based on real-time device calibration data. RW-QAOA is evaluated on real problem instances from industry partners across all four domains on IBM Quantum Eagle (127-qubit). RW-QAOA achieves solution quality within 4.2% of classical exact solvers across all four domains, outperforming classical greedy heuristics by 12.4% (SD = 3.8%) on average. WSIS reduces QAOA parameter convergence iterations by 48.4%. LNAP improves approximation ratio by 8.4pp vs. fixed-depth QAOA under device noise variation.
This paper investigates several Quadratic Unconstrained Binary optimisation (QUBO) formulations for the parallel machine scheduling problem P m | | C max with the goal of assessing their suitability for quantum annealing and related heuristic optimisation methods. Three QUBO models are constructed, ranging from formulations derived from classical integer and one-hot encodings to a more compact load-balancing based representation. Extensive computational experiments using Simulated Annealing (SA) and Quantum Annealing (QA) reveal substantial differences in scalability and solution quality among the formulations. The alternative load-balanced model consistently yields near-optimal makespans, typically within 1%–3% of the theoretical lower bound, highlighting its stability and effectiveness compared to the more penalty sensitive formulations, whose performance deteriorates for larger instances. Sensitivity analyses further reveal that the appropriate selection of penalty weights and problem-specific parameter tuning significantly influences both solution accuracy and computational effort in QUBO formulations. While current quantum hardware limitations restrict experimentation to small problem sizes, the findings illustrate how compact QUBO designs can better exploit quantum optimisation frameworks. The insights gained in this study provide practical modelling guidelines for future quantum scheduling research and lay the groundwork for extending QUBO-based approaches to more complex scheduling environments. • Three QUBO formulations are compared for identical parallel machine scheduling. • A compact load-balancing QUBO avoids auxiliary variables. • Load balancing yields makespans on average within 1–3%. • Both simulated and quantum annealing produce near-optimal schedules. • Penalty selection and encoding strongly affect scalability and solution quality.
Recent advances in quantum computing and the increasing availability of quantum hardware have substantially enhanced the practical relevance of quantum approaches to discrete optimization. Among these, the Quadratic Unconstrained Binary Optimization (QUBO) formulation provides a unifying modeling framework for a broad class of NP-hard problems and is naturally suited to quantum computing and quantum-inspired algorithms. Location science, network design, and logistics represent core application domains of discrete optimization, combining high practical impact with substantial computational challenges. In this work, we develop QUBO formulations for several fundamental problems in these domains, including a nonlinear integer formulation of the Discrete Ordered Median Problem (DOMP). Beyond their modeling relevance, these QUBO formulations serve as representative benchmark problems for assessing quantum algorithms and quantum hardware. We further derive a tight bound for the penalty parameter ensuring equivalence between the QUBO formulation and its underlying integer program. Finally, we conduct a comprehensive computational study using QAOA, WS-QAOA, and classical heuristics for QUBO instances of the p-Median Problem and the Fixed-Charge Facility Location Problem (FCFLP), and introduce two effective warm-start strategies for WS-QAOA based on its linear programming relaxation.
Background: This paper investigates hybrid quantum–classical optimization approaches for addressing core supply chain management (SCM) problems. A unified hybrid framework is implemented and evaluated across five representative domains: vehicle routing, scheduling, facility location, inventory optimization, and demand forecasting. Methods: The framework integrates quantum algorithms—namely the Quantum Approximate Optimization Algorithm (QAOA), Quantum Annealing (QA), and the Variational Quantum Eigensolver (VQE)—with classical constraint-handling and local refinement procedures in an iterative workflow. Quantum solvers are employed for global solution exploration, while classical optimization ensures feasibility and convergence stability. Results: Experiments conducted on standardized synthetic benchmarks demonstrate that the proposed hybrid framework consistently outperforms classical-only and quantum-only baselines, achieving 12–18% reductions in operational costs and 20–35% faster convergence. In routing and fulfilment tasks, quantum-generated candidate solutions provide effective warm starts for classical refinement. Robustness analysis based on stochastic SCM simulations further indicates lower performance variance under uncertainty. Conclusions: These results demonstrate that hybrid quantum–classical optimization constitutes a practical and scalable strategy for near-term SCM decision-making under current Noisy Intermediate-Scale Quantum (NISQ) hardware constraints.
Quantum computing approaches, such as Quantum Annealing (QA) and the Quantum Approximate Optimization Algorithm (QAOA), have emerged as promising candidates for solving combinatorial optimization problems. Since these approaches typically work with Quadratic Unconstrained Binary Optimization (QUBO) problems, constraints, when present, are commonly encoded as penalty terms. Conventional approaches to incorporate inequality constraints into QUBO formulations additionally require the introduction of slack variables, increasing the required number of qubits. The Knapsack and Bin Packing problems are optimization problems with such inequality constraints. We show that the number of slack variables required to formulate the Knapsack problem as a QUBO problem can be reduced using certain properties specific to this problem. Moreover, we present two separate strategies for the Knapsack and Bin Packing problems that avoid using the costly slack variables altogether. Instead, both strategies solve several QUBO problems, with varying penalty constants, such that at least one of them is a good representation of the given optimization problem. Numerical simulations of QAOA, as well as experiments on a D-Wave quantum annealer, show that these slack-less approaches can improve performance compared to the conventional slack variable method (in terms of success probability), even when the latter is allowed to vary penalty constants too.
The Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising tool for accelerating optimization processes in the Noisy Intermediate-Scale Quantum (NISQ) era. Compared to classical methods, QAOA efficiently solves optimization problems, often formulated as Quadratic Unconstrained Binary Optimization (QUBO) problems. Classical quantum simulators are crucial for evaluating quantum algorithms due to limited quantum resources. However, QAOA's performance can vary with different simulation methods. This study analyzes QAOA's performance using various quantum simulators (e.g., density _matrix, statevector, and matrix_product_state) and demonstrates the benefits of HPC-QC integrated systems in solving QUBO problems on an active learning workflow. By simulating QAOA on dense, large-matrix QUBO problems, we evaluate accuracy and problem-solving time. We also assess QAOA's performance on local computers and HPC-QC inte-grated systems, using Oak Ridge Leadership Computing Facility (OLCF)'s Frontier supercomputer with local Qiskit Aer and remote IBM Quantum simulators.