Maximizing gas throughput in transmission networks under hydraulic and operational constraints is a combinatorial problem whose complexity grows exponentially with network size, making it computationally intensive to solve exactly. This paper addresses the graph-based optimization problem by optimizing nodal-pressure assignments under the Panhandle-B hydraulic equation. By framing the problem as a search over discretized nodal-pressure assignments coupled with a cost Hamiltonian that encodes both the delivery objective and physical-constraint penalties, we establish a unified formulation suitable for the Quantum Approximate Optimization Algorithm (QAOA). The mathematical model is adapted to a Quadratic Unconstrained Binary Optimization (QUBO) formulation and implemented using the Classiq quantum software platform. In simulator-based experiments, QAOA recovered the maximum-throughput valid operating point, consistent with classical exhaustive evaluation and classical hydraulic simulation reference solutions. A distinctive contribution of this work is the end-to-end execution of a reduced problem instance on the IonQ Forte-1 trapped-ion quantum processor. Remarkably, the hardware implementation used only $p=2$ QAOA layers, substantially fewer than the $p=30$ layers used in the simulator-based study. Despite this significant reduction in circuit depth, the QPU produced physically valid and interpretable candidate solutions that bracketed the continuous classical optimum, with each located within one pressure-discretization step of it. These results demonstrate that meaningful gas-network optimization behavior can be obtained using considerably shallower QAOA circuits than initially expected and provide an end-to-end proof of concept for near-term quantum-assisted gas-network optimization.
Maximizing gas throughput in transmission networks under hydraulic and operational constraints is a combinatorial problem whose complexity grows exponentially with network size, making it computationally intensive to solve exactly. This paper addresses the graph-based optimization problem by optimizing nodal-pressure assignments under the Panhandle-B hydraulic equation. By framing the problem as a search over discretized nodal-pressure assignments coupled with a cost Hamiltonian that encodes both the delivery objective and physical-constraint penalties, we establish a unified formulation suitable for the Quantum Approximate Optimization Algorithm (QAOA). The mathematical model is adapted to a Quadratic Unconstrained Binary Optimization (QUBO) formulation and implemented using the Classiq quantum software platform. In simulator-based experiments, QAOA recovered the maximum-throughput valid operating point, consistent with classical exhaustive evaluation and classical hydraulic simulation reference solutions. A distinctive contribution of this work is the end-to-end execution of a reduced problem instance on the IonQ Forte-1 trapped-ion quantum processor. Remarkably, the hardware implementation used only p=2 QAOA layers, substantially fewer than the p=30 layers used in the simulator-based study. Despite this significant reduction in circuit depth, the QPU produced physically valid and interpretable candidate solutions that bracketed the continuous classical optimum, with each located within one pressure-discretization step of it. These results demonstrate that meaningful gas-network optimization behavior can be obtained using considerably shallower QAOA circuits than initially expected and provide an end-to-end proof of concept for near-term quantum-assisted gas-network optimization.
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.
Single-path transmission optimization is a core task for resource scheduling and operation of complex networks, which requires coordinated optimization of transmission cost and flow. Classical algorithms bear heavy computational loads in high-dimensional decision spaces as networks grow. This paper constructs a hybrid quantum model integrating quantum approximate optimization algorithm (QAOA) and cubic spline interpolation. Paths, discrete flows, and trade-off coefficients are unified within a quadratic unconstrained binary optimization (QUBO) model. Least-squares fitting converts native parameters into QUBO coefficients, whose fitting errors are measured to verify robustness and penalty sensitivity, and auxiliary variables eliminate high-order terms to exponentially cut qubit consumption. QAOA narrows the feasible range via global coarse search, and cubic spline interpolation further yields precise continuous flow values. Powered by quantum superposition for parallel full-space exploration, the framework avoids repeated modeling for separate bias coefficients. Mixed integer programming (MIP) and genetic algorithm (GA) are adopted as comparative benchmarks. For the small-scale network instance, the relative error between the proposed method and the global optimum solved by MIP is less than 1%. For the large-scale case, the overall error of our approach remains within an acceptable range even when discrepancies exist between results yielded by classical algorithms.
This study explores quantum computing for combinatorial optimization, focusing on the Traveling Salesman Problem (TSP). Using Qiskit, the Quantum Approximate Optimization Algorithm (QAOA) is implemented on simulators and quantum hardware, and compared with classical optimization using Gurobi. TSP instances are encoded as Quadratic Unconstrained Binary Optimization (QUBO) problems derived from the Miller–Tucker–Zemlin formulation, with quadratic penalties enforcing degree and subtour constraints. Results indicate that QAOA consistently generates feasible tours, but classical branch-and-bound methods achieve better solution quality and runtime. On Noisy Intermediate-Scale Quantum (NISQ) devices, performance degrades with problem size due to sensitivity to parameter tuning, penalty scaling, and hardware noise. Among the tested classical optimizers, COBYLA provides the best balance of efficiency and noise tolerance. Overall, this work establishes a reproducible quantum-classical workflow, provides performance baselines, and identifies critical factors such as encoding design, optimizer choice, and error mitigation for improving near-term quantum heuristic performance.
Quantum computers are expected to offer advantages in solving optimization problems challenging for classical computers. Quadratic Unconstrained Binary Optimization (QUBO) problems represent an important class of problems with relevance in finance and logistics. The Quantum Approximate Optimization Algorithm (QAOA) is a prominent candidate for solving QUBO problems on near-term quantum devices. In this paper, we evaluate the performance of both the standard QAOA and the adaptive derivative assembled problem tailored QAOA (ADAPT-QAOA) to solve QUBO problems of varying sizes and hardnesses for financial feature selection problems. Our main observation is that ADAPT-QAOA achieves substantially higher approximation ratios than standard QAOA for harder feature-selection problems (α = 0.6) with statistically significant improvements observed for problem sizes n = 6 and 14. However, the standard QAOA remains competitive for simpler problems. Additionally, we evaluate the practical feasibility and limitations of QAOA through a hardware-aware scaling analysis based on the real-device calibration data for various hardware platforms. We estimate that standard QAOA implementation on superconducting quantum computers provides a shorter time-to-solution compared to trapped-ion devices, while trapped-ion devices yield more favorable error rates. Our findings provide a comprehensive overview of the challenges, trade-offs, and strategies for deploying QAOA-based methods on near-term quantum hardware.
The Quantum Approximate Optimization Algorithm (QAOA) is a promising variational quantum algorithm introduced to tackle classically intractable combinatorial optimization problems. This tutorial offers a comprehensive, first-principles introduction to QAOA and its properties, focusing on its application to Quadratic and Polynomial Unconstrained Binary Optimization (QUBO and PUBO) problems. The tutorial begins by outlining variational quantum circuits and QUBO problems, focusing on their key properties and the encoding of problem constraints through quadratic penalty terms. Next, it explores the QAOA in detail, covering its Hamiltonian formulation, gate decomposition, and example applications, along with their implementation and performance results. This is followed by an analysis of the algorithm's energy landscape, where proofs are provided for its symmetry and periodicity, and where a resulting parameter space reduction is proposed. Finally, the tutorial extends these concepts to PUBO problems by generalizing the results to higher-order Hamiltonians and discussing the associated symmetries and circuit construction.
Multi-objective optimization (MOO) problems are common in logistics, where routing decisions must balance conflicting objectives such as travel distance, delivery time, and operational risk. A recently proposed Quantum Approximate Optimization Algorithm (QAOA) parameter-transfer strategy solves multi-objective MAX-CUT problems by reusing parameters trained on smaller instances, avoiding costly reoptimization for each scalarized problem. However, its effectiveness has only been demonstrated on proof-of-concept instances tailored to quantum hardware connectivity. In this work, we evaluate the applicability of this strategy to realistic routing problems. We formulate the Traveling Salesman Problem (TSP) and Vehicle Routing Problem (VRP) as Quadratic Unconstrained Binary Optimization (QUBO) models, reduce them to MAX-CUT, and assess the parameter-transfer framework under conditions matching the original study. Validation is performed through classical simulations and experiments on IBM quantum hardware. The resulting Pareto fronts are compared with those obtained using an adapted classical ε-constraint method, using hypervolume as the primary quality metric. Results indicate that parameter transfer remains effective, frequently achieving higher hypervolume and often finding competitive solutions earlier. However, performance depends on problem structure. The approach is consistently effective for TSP instances but less stable for the more constrained VRP, suggesting that QAOA parameter transferability decreases as the optimization landscape becomes more complex. These results provide the first comprehensive evaluation of QAOA parameter transfer on realistic multi-objective routing problems and demonstrate its potential beyond proof-of-concept MAX-CUT benchmarks.