ABSTRACT Quantum annealing is emerging as a practical tool for large‐scale combinatorial optimization. We map continuum displacements and element densities to binary variables, turning both deformation analysis and topology optimization into quadratic unconstrained binary optimization (QUBO) problems that run on today's annealers. Two formulations are compared: one derived from the stiffness equation and another from the minimum total potential energy. Benchmarks show the energy‐based QUBO matches finite‐element displacements and stresses more closely and avoids the rapid term‐count growth that slows the stiffness‐based form. The numerical demonstrations reported here include deformation analysis of a 100‐element plate model and topology optimization of cantilever beams discretized with up to 60 triangular elements or 32 rectangular elements. These examples are intended as proof‐of‐concept demonstrations under current hardware and encoding limitations rather than as large‐scale industrial benchmarks. Leveraging this model, a single QUBO embedding density variables–implemented with combinatorial random‐number sums and a quadratic volume penalty–yields two‐dimensional beam layouts that are quantitatively compared with those obtained using the classical Optimality Criteria (OC) method.
ABSTRACT Quantum annealing is emerging as a practical tool for large‐scale combinatorial optimization. We map continuum displacements and element densities to binary variables, turning both deformation analysis and topology optimization into quadratic unconstrained binary optimization (QUBO) problems that run on today's annealers. Two formulations are compared: one derived from the stiffness equation and another from the minimum total potential energy. Benchmarks show the energy‐based QUBO matches finite‐element displacements and stresses more closely and avoids the rapid term‐count growth that slows the stiffness‐based form. The numerical demonstrations reported here include deformation analysis of a 100‐element plate model and topology optimization of cantilever beams discretized with up to 60 triangular elements or 32 rectangular elements. These examples are intended as proof‐of‐concept demonstrations under current hardware and encoding limitations rather than as large‐scale industrial benchmarks. Leveraging this model, a single QUBO embedding density variables–implemented with combinatorial random‐number sums and a quadratic volume penalty–yields two‐dimensional beam layouts that are quantitatively compared with those obtained using the classical Optimality Criteria (OC) method.
Critical decision-making issues in science, engineering, and industry are based on combinatorial optimization; however, its application is inherently limited by the NP-hard nature of the problem. A specialized paradigm of analogue quantum computing, quantum annealing (QA), has been proposed to solve these problems by encoding optimization problems into physical energy landscapes and solving them by quantum tunnelling systematically through exploration of solution space. This is a critical review that summarizes the current applications of quantum annealing to combinatorial optimization and includes a theoretical background, hardware designs, algorithm implementation strategies, encoding and embedding schemes, protocols to benchmark quantum annealing, areas of implementation, and links with the quantum algorithms implementation with gate-based hardware and classical solvers. We develop a unified framework, relating adiabatic quantum dynamics, Ising and QUBO models, stoquastic and non-stoquastic Hamiltonians, and diabatic transitions to modern flux-qubit annealers (Chimera, Pegasus, Zephyr topologies), and emergent architectures (Lechner-Hauke-Zoller systems, Rydberg atom platforms), and hybrids of quantum and classical computation. Through our analysis, we find that overhead in embedding and encoding is the largest determinant of the scalability and performance (this is not just the number of qubits). Minor embeddings also usually have a physical qubit count per logical variable of between 5 and 12 qubits, which limits effective problem capacity by 80-92% and, due to chain-breaking errors, compromises the quality of solutions.
Abstract Quantum annealing aims at solving optimization problems of practical relevance using quantum-computing hardware. Problems of interest are typically formulated in terms of quadratic unconstrained binary optimization (QUBO) Hamiltonians. However, many optimization problems are much more naturally formulated in terms of polynomial unconstrained binary optimization (PUBO) functions of higher order. As we show with various problem examples, leveraging the PUBO formulation can bring considerable savings in terms of required number of qubits. Moreover, in numerical benchmarks for the paradigmatic 3-SAT problem, we find scenarios where the scaling of the minimum energy gap during the optimization sweep differs significantly, suggesting the possibility of an exponentially faster annealing time when using the PUBO as compared to the QUBO formulation. This advantage persists even when considering the overhead caused by the higher-order interactions necessary for PUBO cost Hamiltonians. As an interesting side effect, the analysis on minimum energy gaps of different 3-SAT instance generators reveals different degrees of hardness, which will be of interest also for classical benchmark calculations. Our findings show a promising path to improving the resource efficiency and sweeping speed of quantum annealing protocols on both analog and digital platforms, which are important prerequisites when aiming at solving larger optimization problems with relevance to industry.
Abstract Layout optimization problems involve finding the optimal arrangement of elements in order to maximize efficiency. For instance, the wind farm layout optimization (WFLO) problem consists of the best turbine placement to maximize energy production while minimizing wake losses. As its nonlinear and combinatorial nature makes it challenging for traditional optimization methods, alternative approaches such as quantum annealing and quantum‐classical hybrid methods offer a promising alternative for tackling such complex problems. Here, WFLO is formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem using the Jensen wake model. A quantum annealer is compared, the Gurobi solver, and the Quantum Approximate Optimization Algorithm (QAOA). The quantum annealer provides solutions one order of magnitude faster than Gurobi with at most 3% lower power output, making it suitable for rapid suboptimal approximations. These findings highlight the trade‐off between the quality of the solution and the computational time and demonstrate how quantum methods, especially when combined with classical solvers, can contribute to efficient renewable energy optimization.
This paper explores the application of Quantum Annealing (QA) to the field of combinatorial optimization. The core idea is to leverage the principles of quantum mechanics, specifically quantum tunneling, to efficiently solve complex optimization problems. We present a framework for utilizing QA to tackle problems represented as quadratic unconstrained binary optimization (QUBO) models. The methodology focuses on mapping the problem's constraints and objective function into a form suitable for QA hardware. We discuss the challenges involved in this mapping process, including the limitations of current QA devices and the need for careful problem formulation. Furthermore, we analyze the potential advantages of QA over classical optimization techniques, particularly concerning the exploration of a wider solution space. The research highlights the promise of QACO as a viable approach for tackling computationally demanding combinatorial optimization tasks, offering a novel method to enhance optimization efficiency.
Resource constrained project scheduling and related allocation tasks can be expressed as Quadratic Unconstrained Binary Optimization (QUBO) models and submitted to quantum annealers. This paper develops the QUBO formulation in detail: choice of binary variables, linear and quadratic objective terms, conversion of precedence and resource constraints into penalty functions, and calibration of penalty weights. The resulting Hamiltonian is embedded onto quantum hardware, annealed, and post-processed. Hybrid classical–quantum workflows that wrap the QUBO solver inside Monte Carlo risk analysis or a digital twin are examined. Supporting process flowcharts and energy-landscape figures illustrate every stage. The treatment remains grounded in construction and logistics applications while remaining transferable to other combinatorial domains such as ULD configuration. Keywords: QUBO formulation; quantum annealing; Ising model; penalty methods; resource-constrained project scheduling; hybrid quantum-classical; minor embedding; Monte Carlo; digital twin.
The quantum annealing devices, which encode the solution to a computational problem in the ground state of a quantum Hamiltonian, are implemented in D-Wave systems with more than 2,000 qubits. However, quantum annealing can solve only a classical combinatorial optimization problem such as an Ising model, or equivalently, a quadratic unconstrained binary optimization (QUBO) problem. In this paper, we formulate the QUBO model to solve elliptic problems with Dirichlet and Neumann boundary conditions using the finite element method. In this formulation, we develop the objective function of quadratic binary variables represented by qubits and the system finds the binary string combination minimizing the objective function globally. Based on the QUBO formulation, we introduce an iterative algorithm to solve the elliptic problems. We discuss the validation of the modeling on the D-Wave quantum annealing system.