We propose a complete quantum-classical hybrid branch-and-bound algorithm (QCBB) to solve binary linear programs with equality constraints. That includes bound calculation, convergence metrics and optimality guarantee to the quantum optimization based algorithm, which makes our method directly comparable to classical methods. Key aspects of the proposed algorithm are (i) encapsulation of the quantum optimization method, (ii) utilization of noisy samples for problem reduction, (iii) classical approximation based bound calculation, (iv) branch and bound traits like gap-based stopping criterion and monotonic increase in solution quality, (v) integrated composition of many different solutions that can be improved individually. We show numerical results on set partitioning problem instances and provide many details about the characteristics of the different steps of the algorithm.
We propose a complete quantum-classical hybrid branch-and-bound algorithm (QCBB) to solve binary linear programs with equality constraints. That includes bound calculation, convergence metrics and optimality guarantee to the quantum optimization based algorithm, which makes our method directly comparable to classical methods. Key aspects of the proposed algorithm are (i) encapsulation of the quantum optimization method, (ii) utilization of noisy samples for problem reduction, (iii) classical approximation based bound calculation, (iv) branch and bound traits like gap-based stopping criterion and monotonic increase in solution quality, (v) integrated composition of many different solutions that can be improved individually. We show numerical results on set partitioning problem instances and provide many details about the characteristics of the different steps of the algorithm.
We present a method that integrates any quantum algorithm capable of finding solutions to integer linear programs into the branch-and-price algorithm, which is regularly used to solve large-scale integer linear programs with a specific structure. The role of the quantum algorithm is to find integer solutions to subproblems appearing in branch-and-price. Obtaining optimal or near-optimal integer solutions to these subproblems can increase the quality of solutions and reduce the depth and branching factor of the branch-and-price algorithm and hence reduce the overall running time. We investigate the viability of the approach by considering the tail assignment problem and the quantum approximate optimization algorithm (QAOA). Here, the master problem is the optimization problem set partitioning or its decision version exact cover and can be expressed as finding the ground state of an Ising spin glass Hamiltonian. For exact cover, our numerical results indicate that the required algorithm depth decreases with the number of feasible solutions for a given success probability of finding a feasible solution. For set partitioning, on the other hand, we find that for a given success probability of finding the optimal solution, the required algorithm depth can increase with the number of feasible solutions if the Hamiltonian is balanced poorly, which in the worst case is exponential in the problem size. We therefore address the significance of properly balancing the objective and constraint parts of the Hamiltonian. We empirically find that the approach is viable with QAOA if polynomial algorithm depth can be realized on quantum devices.
Abstract We propose a new approach to utilize quantum computers for binary linear programming (BLP), which can be extended to general integer linear programs (ILP). Quantum optimization algorithms, hybrid or quantum-only, are currently general purpose, standalone solvers for ILP. However, to consider them practically useful, we expect them to overperform the current state of the art classical solvers. That expectation is unfair to quantum algorithms: in classical ILP solvers, after many decades of evolution, many different algorithms work together as a robust machine to get the best result. This is the approach we would like to follow now with our quantum ‘solver’ solutions. In this study we wrap any suitable quantum optimization algorithm into a quantum informed classical constraint generation framework. First we relax our problem by dropping all constraints and encode it into an Ising Hamiltonian for the quantum optimization subroutine. Then, by sampling from the solution state of the subroutine, we obtain information about constraint violations in the initial problem, from which we decide which coupling terms we need to introduce to the Hamiltonian. The coupling terms correspond to the constraints of the initial binary linear program. Then we optimize over the new Hamiltonian again, until we reach a feasible solution, or other stopping conditions hold. Since one can decide how many constraints they add to the Hamiltonian in a single step, our algorithm is at least as efficient as the (hybrid) quantum optimization algorithm it wraps. We support our claim with results on small scale minimum cost exact cover problem instances.
We present a method that integrates any quantum algorithm capable of finding solutions to integer linear programs into the Branch-and-Price algorithm, which is regularly used to solve large-scale integer linear programs with a specific structure. The role of the quantum algorithm is to find integer solutions to subproblems appearing in Branch-and-Price. Obtaining optimal or near-optimal integer solutions to these subproblems can increase the quality of solutions and reduce the depth and branching factor of the Branch-and-Price algorithm and hence reduce the overall running time. We investigate the viability of the approach by considering the Tail Assignment problem and the Quantum Approximate Optimization Algorithm (QAOA). Here, the master problem is the optimization problem Set Partitioning or its decision version Exact Cover and can be expressed as finding the ground state of an Ising spin glass Hamiltonian. For Exact Cover, our numerical results indicate that the required algorithm depth decreases with the number of feasible solutions for a given success probability of finding a feasible solution. For Set Partitioning, on the other hand, we find that for a given success probability of finding the optimal solution, the required algorithm depth can increase with the number of feasible solutions if the Hamiltonian is balanced poorly, which in the worst case is exponential in the problem size. We therefore address the importance of properly balancing the objective and constraint parts of the Hamiltonian. We empirically find that the approach is viable with QAOA if polynomial algorithm depth can be realized on quantum devices.
Quantum optimization provides a promising approach for solving large-scale combinatorial problems through quadratic unconstrained binary optimization (QUBO) formulations. However, integrating QUBO-based solvers into structured optimization frameworks while preserving solution guarantees remains a fundamental challenge. This paper develops a hybrid quantum-classical column-and-constraint generation (QCCG) framework for solving two-stage adaptive robust optimization problems with binary first-stage decisions and linear recourse under polyhedral uncertainty. The proposed approach reformulates the restricted master problem as a QUBO and solves it approximately using a quantum optimizer, while retaining a classical adversarial subproblem to compute worst-case recourse and certify solution quality. We construct a constraint-preserving QUBO encoding for inequality-constrained master problems using slack variables and penalty terms, enabling general mixed-integer structures to be mapped to quantum-compatible representations. To address inexactness arising from discretization, penalty modeling, and quantum optimization, we introduce a bound-adjustment mechanism that yields valid lower and upper bounds and provides a certified stopping criterion. We show that the proposed framework generalizes classical column-and-constraint generation and retains its convergence properties when the master problem is solved exactly. Numerical experiments on two-stage robust location-transportation problems demonstrate that the proposed hybrid approach achieves solution quality comparable to classical methods while reducing the computational burden associated with solving mixed-integer master problems, highlighting the potential of hybrid quantum-classical optimization for scalable decision-making under uncertainty.
Quantum and physics-inspired solvers provide specialized sampling interfaces for Ising and quadratic unconstrained binary optimization models, but constrained optimization requires broader coordination and certification. This survey uses classical decomposition to analyze how such solvers can be integrated into hybrid quantum-classical algorithms. We distinguish formal decomposition, reformulation and reduction, and computational orchestration. For each interface, we ask what the transformation preserves, which assumptions make the outer method converge, and which certificates the implemented inner routine supplies. The central questions are whether that information meets the classical algorithm's requirements and whether hybridization improves solution quality or total cost enough to justify its coordination overhead. Benders decomposition, column generation, graph reduction, and copositive methods illustrate this distinction by separating heuristic proposals from classical certification. We also distinguish component effects within a fixed wrapper, reduction--solver interactions, and comparisons between complete quantum-enabled and classical systems. The analyzed studies provide conditional component benefits and favorable co-design evidence, but do not establish transferable end-to-end superiority over strong classical comparators. The resulting framework identifies opportunities for constraint programming and operations research in structure-preserving design, safeguarded hybrid algorithms, and controlled benchmarking.
Quantum algorithms have shown promise in solving Quadratic Unconstrained Binary Optimization (QUBO) problems, benefiting from their connection to the transverse field Ising model.Various Ising solvers, both classical and quantum, have emerged to tackle such problems efficiently but lack global optimality guarantees and oftentimes sufer from hardware limitations such as limited qubit availability.In this work, we propose a hybrid branch-andbound (B&B) framework that integrates Ising solvers as heuristics within a classical B&B algorithm.Unlike prior theoretical studies, our work presents a practical implementation, available as open-source at GitHub.We explore when and where to apply Ising solvers in the search tree, and introduce a custom branching rule optimized QUBO embedding.Our method is evaluated on hundreds of QUBO instances from QUBOLib.jl using Gurobi and the D-Wave quantum annealer.Our Results show up to 11% less solution time and 17% fewer nodes compared to default Gurobi, an off-the-shelf commercial optimization solver.These findings demonstrate the value of hybrid quantum-classical strategies for enhancing exact optimization.