Quantum annealing is a recent advancement in the field of optimization metaheuristics. Quantum annealers such as the D-Wave machines are primarily designed for solving discrete optimization problems. Solving nonlinear continuous optimization problems is also a very challenging task, in particular when nonconvex functions and multiple local optima are involved; nevertheless, convex problems can be solved efficiently using standard classical methods. It has not been explored in sufficient detail how quantum annealing could help in solving continuous optimization problems. In this work, we develop a prototype quantum-inspired solver for nonlinear continuous optimization problems. The solver is hybrid, as it has two pathways - classical and quantum-inspired. Given a continuous optimization problem via the web-based graphical interface that enables users to define and edit optimization problems, our solver computes a nonconvexity index for the problem, which determines the solution pathway. Problems with low nonconvexity indices are sent through the classical pathway and solved by standard separable programming techniques, while the others are transformed into mixed integer linear programming (MILP) problems via separable reformulations, which are restated as quadratic unconstrained binary optimization (QUBO) problems - the problem format compatible for quantum annealers, and eventually sent through the quantum pathway using D-Wave’s Neal sampler.
Quantum annealing is a recent advancement in the field of optimization metaheuristics. Quantum annealers such as the D-Wave machines are primarily designed for solving discrete optimization problems. Solving nonlinear continuous optimization problems is also a very challenging task, in particular when nonconvex functions and multiple local optima are involved; nevertheless, convex problems can be solved efficiently using standard classical methods. It has not been explored in sufficient detail how quantum annealing could help in solving continuous optimization problems. In this work, we develop a prototype quantum-inspired solver for nonlinear continuous optimization problems. The solver is hybrid, as it has two pathways - classical and quantum-inspired. Given a continuous optimization problem via the web-based graphical interface that enables users to define and edit optimization problems, our solver computes a nonconvexity index for the problem, which determines the solution pathway. Problems with low nonconvexity indices are sent through the classical pathway and solved by standard separable programming techniques, while the others are transformed into mixed integer linear programming (MILP) problems via separable reformulations, which are restated as quadratic unconstrained binary optimization (QUBO) problems - the problem format compatible for quantum annealers, and eventually sent through the quantum pathway using D-Wave’s Neal sampler.
Quantum annealers such as the commercially available D-Wave machines are designed to natively solve quadratic unconstrained binary optimization (QUBO) problems. While most of the well-known NP-hard optimization problems can easily be formulated as quadratic binary problems, such formulations also contain constraints, which commonly are added to the objective function in the form of penalties to obtain a QUBO version. However, the standard method for defining such penalties leads to QUBOs that are dense and therefore take too much of the resources of the quantum annealer. In this paper, we describe an alternative approach to the constraint embedding problem that uses mixed-integer linear programming (MILP) and is scalable to problems of arbitrary number of variables.
Mixed-integer linear programming problems are extensively used in industry for a wide range of optimization tasks. However, as they get larger, they present computational challenges for classical solvers within practical time limits. Quantum annealers can, in principle, accelerate the solution of problems formulated as quadratic unconstrained binary optimization instances, but their limited scale currently prevents achieving practical speedups. Quantum-classical algorithms have been proposed to take advantage of both paradigms and to allow current quantum computers to be used in larger problems. In this work, a hardware-agnostic Benders' decomposition algorithm and a series of enhancements with the goal of taking the most advantage of quantum computing are presented. The decomposition consists of a master problem with integer variables, which is reformulated as a quadratic unconstrained binary optimization problem and solved with a quantum annealer, and a linear subproblem solved by a classical computer. The enhancements consist, among others, of different embedding processes that substantially reduce the pre-processing time of the embedding computation without compromising solution quality, a conservative handling of cut constraints, and a stopping criterion that accounts for the limited size of current quantum computers and their heuristic nature. The proposed algorithm is benchmarked against classical approaches using a D-Wave quantum annealer for a scalable family of transmission network expansion planning problems.
Quantum computing is rapidly advancing, harnessing the power of qubits' superposition and entanglement for computational advantages over classical systems. However, scalability poses a primary challenge for these machines. By implementing a hybrid workflow between classical and quantum computing instances, D-Wave has succeeded in pushing this boundary to the realm of industrial use. Furthermore, they have recently opened up to mixed integer linear programming (MILP) problems, expanding their applicability to many relevant problems in the field of optimisation. However, the extent of their suitability for diverse problem categories and their computational advantages remains unclear. This study conducts a comprehensive examination by applying a selection of diverse case studies to benchmark the performance of D-Wave's hybrid solver against that of industry-leading solvers such as CPLEX, Gurobi, and IPOPT. The findings indicate that D-Wave's hybrid solver is currently most advantageous for integer quadratic objective functions and shows potential for quadratic constraints. To illustrate this, we applied it to a real-world energy problem, specifically the MILP unit commitment problem. While D-Wave can solve such problems, its performance has not yet matched that of its classical counterparts.
For various optimization problems, the classical time to solution is super-polynomial and intractable to solve with classical bit-based computing hardware to date. Digital and quantum annealers have the potential to identify near-optimal solutions for such optimization problems using a quadratic unconstrained binary optimization (QUBO) problem formulation. This work benchmarks two use cases to evaluate the utility of QUBO solvers for combinatorial optimization problems, in order to determine if a QUBO formulation and annealing-based algorithms have an advantage over classical mixed-integer programming (MIP) and constraint programming (CP) solvers. Various QUBO and solver metrics such as problem mapping, quantitative interconnectivity, penalty structure, solver minimum cost (obtained optimal value) and solver time to solution have been applied to evaluate different QUBO problems. Constrained and unconstrained QUBO solvers are compared including the Fujitsu digital annealer (DA), various D-Wave hybrid quantum annealing solvers (QA, HQA), and the classical MIP/CP solvers HiGHS, Gurobi, SCIP, and CP-SAT. The two industrially relevant use cases are reaction network pathway analysis and mRNA codon selection. For reaction pathway analysis, classical MIP/CP solvers (especially Gurobi and CP-SAT) are observed to solve the problem to optimality in reasonable time frames. For mRNA codon selection, Gurobi outperformed all other solvers in time to solution for all problem sizes, followed by CP-SAT and the D-Wave Nonlinear (NL) HQA solver.
We present a hybrid classical-quantum framework based on the Frank-Wolfe algorithm, Q-FW, for solving quadratic, linearly-constrained, binary optimization problems on quantum annealers (QA). The computational premise of quantum computers has cultivated the re-design of various existing vision problems into quantum-friendly forms. Experimental QA realizations can solve a particular non-convex problem known as the quadratic unconstrained binary optimization (QUBO). Yet a naive-QUBO cannot take into account the restrictions on the parameters. To introduce additional structure in the parameter space, researchers have crafted ad-hoc solutions incorporating (linear) constraints in the form of regularizers. However, this comes at the expense of a hyper-parameter, balancing the impact of regularization. To date, a true constrained solver of quadratic binary optimization (QBO) problems has lacked. Q-FW first reformulates constrained-QBO as a copositive program (CP), then employs Frank-Wolfe iterations to solve CP while satisfying linear (in)equality constraints. This procedure unrolls the original constrained-QBO into a set of unconstrained QUBOs all of which are solved, in a sequel, on a QA. We use D-Wave Advantage QA to conduct synthetic and real experiments on two important computer vision problems, graph matching and permutation synchronization, which demonstrate that our approach is effective in alleviating the need for an explicit regularization coefficient.
Low-thrust trajectory optimization is a central task in interplanetary mission design, but its nonlinear dynamics and operational constraints often lead to challenging non-convex optimal-control problems. Sequential convex programming has emerged as an effective approach to address these problems, while quantum annealing offers a complementary paradigm for solving quadratic unconstrained binary optimization problems. This paper introduces quSCP, a quantum-based sequential convex programming framework that reformulates each convex subproblem as a quadratic unconstrained binary optimization problem suitable for quantum and hybrid quantum--classical solvers. Equality, inequality, and trust-region constraints are embedded through quadratic penalty terms, while an iterative refinement strategy is used to reduce the accuracy loss introduced by binary discretization. The method is assessed on a fuel-optimal Earth--Mars low-thrust transfer by comparing standard sequential convex programming, a continuous quadratic unconstrained formulation, direct quantum processing unit sampling, and D-Wave hybrid solvers. Results show that quSCP produces physically consistent trajectories with propellant consumption and nonlinear constraint violations close to classical benchmarks. Direct quantum annealing is feasible only for small instances because of embedding overhead and hardware connectivity limits, whereas hybrid solvers scale to larger discretizations. Although no computational quantum advantage is demonstrated with current hardware, the results show that quantum and hybrid quantum--classical optimization can already provide competitive solutions for demanding trajectory design problems.