This study develops a budget-constrained multi-item newsvendor benchmark for analysing the Quantum Approximate Optimization Algorithm (QAOA) on an inventory problem with explicit combinatorial coupling. The single-item closed-form newsvendor model is used only as a calibration benchmark, whereas the main formulation links multiple order decisions through a shared resource constraint. After binary expansion, the problem is expressed as a quadratic unconstrained binary optimization model and mapped to an Ising Hamiltonian with non-trivial pairwise interactions. QAOA is implemented in Qiskit 0.46.0 using COBYLA and SPSA under a reproducible multi-start protocol, and evaluated across depths p ∈ {1, 2, 3, 4, 6} and instance sizes from 4 to 16 qubits. Performance is assessed against exact enumeration or mixed-integer linear programming baselines using objective gap, feasible-sample mass, optimizer variability, and runtime. On the calibration benchmark, the QAOA mode recovers the classical optimum. On a representative 12-qubit coupled benchmark, increasing depth reduces the objective gap from 23.58% at p = 1 to 1.63% at p = 6. The study provides a reproducible benchmark for evaluating low-depth QAOA on coupled inventory optimization problems with direct economic interpretation.
This study develops a budget-constrained multi-item newsvendor benchmark for analysing the Quantum Approximate Optimization Algorithm (QAOA) on an inventory problem with explicit combinatorial coupling. The single-item closed-form newsvendor model is used only as a calibration benchmark, whereas the main formulation links multiple order decisions through a shared resource constraint. After binary expansion, the problem is expressed as a quadratic unconstrained binary optimization model and mapped to an Ising Hamiltonian with non-trivial pairwise interactions. QAOA is implemented in Qiskit 0.46.0 using COBYLA and SPSA under a reproducible multi-start protocol, and evaluated across depths p ∈ {1, 2, 3, 4, 6} and instance sizes from 4 to 16 qubits. Performance is assessed against exact enumeration or mixed-integer linear programming baselines using objective gap, feasible-sample mass, optimizer variability, and runtime. On the calibration benchmark, the QAOA mode recovers the classical optimum. On a representative 12-qubit coupled benchmark, increasing depth reduces the objective gap from 23.58% at p = 1 to 1.63% at p = 6. The study provides a reproducible benchmark for evaluating low-depth QAOA on coupled inventory optimization problems with direct economic interpretation.
We investigate whether the Quantum Approximate Optimization Algorithm (QAOA) can compete with classical metaheuristics for multi-product, multi-location retail inventory optimization under demand uncertainty. We formulate the binary inventory decision problem as a Quadratic Unconstrained Binary Optimization (QUBO) model and benchmark QAOA (simulated via classical statevector simulation, p=2 layers) against Mixed-Integer Linear Programming (MILP), Simulated Annealing (SA), Genetic Algorithms (GA), and Tabu Search across problem sizes from n=5 to n=200 binary variables, using Walmart sales and Kaggle retail demand datasets. Our key finding is a \textbf{negative result}: while QAOA achieves a 2.3\% optimality gap and 97\% constraint satisfaction at small scales (n≤20), it is not competitive with classical metaheuristics at practical problem sizes. Classical heuristics---particularly SA and Tabu Search---consistently outperform QAOA on all instances tested and scale efficiently to n=200, whereas QAOA simulation becomes intractable beyond n=30 due to exponential memory requirements. We do not claim any quantum speedup. This comparative study provides important insights into the limitations of current QAOA implementations for supply chain problems, including the effects of penalty parameter scaling, barren plateaus in the optimization landscape, and circuit depth constraints. We additionally present an economic sensitivity analysis linking macroeconomic indicators to optimal inventory decisions and a rolling-horizon proof-of-concept for multi-period planning.
Combinatorial optimization problems are typically formulated using Quadratic Unconstrained Binary Optimization (QUBO), where constraints are enforced through penalty terms that introduce auxiliary variables and rapidly increase Hamiltonian complexity, limiting scalability on near term quantum devices. In this work, we systematically study Quadratic Unconstrained D-ary Optimization (QUDO) as an alternative formulation in which decision variables are encoded directly in higher dimensional Hilbert spaces. We demonstrate that QUDO naturally captures structural constraints across a range of problem classes, including the Traveling Salesman Problem, two variants of the Vehicle Routing Problem, graph coloring, job scheduling, and Max-K-Cut, without the need for extensive penalty constructions. Using a qudit-level implementation of the Quantum Approximate Optimization Algorithm (qudit QAOA), we benchmark these formulations against their binary QUBO counterparts and exact classical solutions. Our study show consistently improved approximation ratios and substantially reduced computational overhead at comparable circuit depths, highlighting QUDO as a scalable and expressive representation for quantum combinatorial optimization.
The quantum approximate optimization algorithm (QAOA) is designed to determine optimum and near optimum solutions of quadratic (and higher order) unconstrained binary optimization (QUBO or HUBO) problems, which in turn accurately model unconstrained combinatorial optimization problems. While the solution space of an unconstrained combinatorial optimization problem consists of all possible combinations of the objective function's decision variables, for a constrained combinatorial optimization problem, the solution space consists only of solutions that satisfy the problem constraints. While solving such a QUBO problem optimally results in an optimal solution to the underlying constrained problem, setting the right penalty parameters is crucial. Moreover, finding suitable penalties that ensure near-optimal solutions is also challenging. In this article, we introduce our innovative profit-relaxation framework to solve constrained combinatorial optimization problems. Our effective hybrid approach transforms a selected constrained combinatorial optimization problem into an unconstrained profit problem, such that the solution spaces of the two problems are interrelated with respect to their solution qualities. This allows us to determine optimal and near-optimal solutions for the constrained problem. We use three NP-hard problems -- Minimum Vertex Cover, Maximum Independent Set, and Maximum Clique -- to demonstrate the feasibility and effectiveness of our approach. We conducted a detailed performance evaluation of our profit-relaxation framework on two different platforms: Xanadu PennyLane and Argonne QTensor Quantum Circuit Simulator.
This paper presents a Qiskit-compatible distributed quantum approximate optimization algorithm (DQAOA) simulator for quadratic unconstrained binary optimization (QUBO) problems arising in engineering design and decision applications. The open-source simulator is available through the RAISE LAB website and GitHub repository, with README documentation for installation, input formatting, configurable parameters, and example workflows. The package addresses the need for a reusable simulator that can solve and compare QUBO instances across different QAOA execution modes. It supports monolithic QAOA on a single quantum processing unit (QPU) and distributed QAOA across a user-specified number of QPUs with configurable capacities. The workflow canonicalizes the QUBO model, maps it to a cost Hamiltonian, allocates variables across QPUs, identifies local and cross-QPU couplings, and constructs the corresponding circuits. Runtime optimizations, including parameterized circuit reuse, objective reuse at fixed depth, batched evaluations, and parallel multi-start execution, reduce repeated overhead. A Streamlit graphical user interface is also provided for entering or uploading QUBO instances, configuring solver settings, running selected modes, and visualizing solution-quality metrics without editing Python scripts. The package is demonstrated on standalone QUBO benchmarks and a power generation unit commitment application. In the unit commitment case, brute force, monolithic QAOA, and distributed QAOA recover the same commitment bitstring and operating cost. Across multiple case studies, the simulator produces results consistent with classical monolithic QAOA references in terms of optimal bitstrings and costs. Staged runtime analysis shows substantial runtime reduction across implementation stages, while distributed QAOA remains more demanding because cross-QPU couplings require remote operations.
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.
This study presents a rigorous empirical evaluation of the Quantum Approximate Optimization Algorithm (QAOA) within a hybrid quantum-classical framework for solving the NP-hard Max-Cut problem. Using Python and Qiskit Aer statevector simulations, we benchmark QAOA (depths p = 2 and p = 3) against established classical algorithms, including simulated annealing, the Goemans–Williamson algorithm, and greedy heuristics, across Erdős–Rényi and regular graphs with 4 to 20 nodes (100 instances per size). Results demonstrate that low-depth QAOA suffers from significant performance degradation, achieving a 0.467 approximation ratio at 20 nodes compared to 0.856 for Goemans–Williamson, alongside a 210× runtime overhead. Convergence analysis further reveals low optimality probabilities and high iteration costs, highlighting limitations of variational optimization in NISQ-era devices. This work provides a reproducible benchmarking framework and contributes to a realistic assessment of near-term quantum advantage.