Quadratic unconstrained binary optimization problems (QUBOs) are intensively discussed in the realm of quantum computing and polynomial optimization. We provide a vast experimental study of semidefinite programming (SDP) relaxations of QUBOs using sums of squares methods and on Hamiltonian Updates. We test on QUBO reformulations of industry-based instances of the (open) vehicle routing problem and the (affinity-based) slotting problem -- two common combinatorial optimization problems in logistics. Beyond comparing the performance of various methods and software, our results reaffirm that optimizing over non-generic, real-world instances provides additional challenges. In consequence, this study underscores recent developments towards structure exploitation and specialized solver development for the used methods and simultaneously shows that further research is necessary in this direction both on the classical and the quantum side.
Quadratic unconstrained binary optimization problems (QUBOs) are intensively discussed in the realm of quantum computing and polynomial optimization. We provide a vast experimental study of semidefinite programming (SDP) relaxations of QUBOs using sums of squares methods and on Hamiltonian Updates. We test on QUBO reformulations of industry-based instances of the (open) vehicle routing problem and the (affinity-based) slotting problem -- two common combinatorial optimization problems in logistics. Beyond comparing the performance of various methods and software, our results reaffirm that optimizing over non-generic, real-world instances provides additional challenges. In consequence, this study underscores recent developments towards structure exploitation and specialized solver development for the used methods and simultaneously shows that further research is necessary in this direction both on the classical and the quantum side.
In this paper, we present a new method to solve a certain type of Semidefinite Programming (SDP) problems. These types of SDPs naturally arise in the Quadratic Convex Reformulation (QCR) method and can be used to obtain dual bounds of Quadratic Unconstrained Binary Optimization (QUBO) problems. QUBO problems have recently become the focus of attention in the quantum computing and optimization communities as they are well suited to both gate-based and annealing-based quantum computers on one side, and can encompass an exceptional variety of combinatorial optimization problems on the other. Our new method can be easily warm-started, making it very effective when embedded into a branch-and-bound scheme and used to solve the QUBO problem to global optimality. We test our method in this setting and present computational results showing the effectiveness of our approach.
Recent advances in quantum computing and the increasing availability of quantum hardware have substantially enhanced the practical relevance of quantum approaches to discrete optimization. Among these, the Quadratic Unconstrained Binary Optimization (QUBO) formulation provides a unifying modeling framework for a broad class of NP-hard problems and is naturally suited to quantum computing and quantum-inspired algorithms. Location science, network design, and logistics represent core application domains of discrete optimization, combining high practical impact with substantial computational challenges. In this work, we develop QUBO formulations for several fundamental problems in these domains, including a nonlinear integer formulation of the Discrete Ordered Median Problem (DOMP). Beyond their modeling relevance, these QUBO formulations serve as representative benchmark problems for assessing quantum algorithms and quantum hardware. We further derive a tight bound for the penalty parameter ensuring equivalence between the QUBO formulation and its underlying integer program. Finally, we conduct a comprehensive computational study using QAOA, WS-QAOA, and classical heuristics for QUBO instances of the p-Median Problem and the Fixed-Charge Facility Location Problem (FCFLP), and introduce two effective warm-start strategies for WS-QAOA based on its linear programming relaxation.
Abstract The warehouse layout problem (WLP), also known as warehouse slotting problem, aims to determine the assignment of products to storage locations while minimizing the total travel distance for picking operations. It is combinatorial in nature and well-known to be NP-hard. To obtain high-quality solutions to medium and large WLP instances efficiently remains a computational challenge, which has motivated our work on applying quantum optimization algorithms for this problem. We reformulate the classical quadratic assignment programming (QAP) model of the WLP to a quadratic unconstrained binary optimization (QUBO) model. Our QUBO model avoids adding slack variables and reduces additional overheads as in the typical QUBO reformulation approach, which facilitates the application of the D-Wave Leap Hybrid solver. Computational study is performed on a use case with 30 stock-keeping-units (SKUs) and 32 storage slots, for which the QUBO formulation has 960 binary decision variables. We show that the D-Wave Leap Hybrid solver finds better solution to the base scenario than the exact IBM CPLEX mixed-integer quadratic programming (MIQP) solver operating under a 60-second wall-clock limit, one-tenth of the 600-second budget allocated to CPLEX in this experimental design. A sensitivity analysis across 30 scenarios, generated by crossing five annual pick volume profiles with six co-picking matrix density levels (0.5 to 1.0), shows that the D-Wave hybrid solver outperforms CPLEX MIQP solver in 26 of 30 cases, with solution quality improvements of 0.4 to 12.1%. D-Wave’s advantage generally increases as the density of co-picking matrix decreases. These results demonstrate quantum utility for WLP and motivate further investigation of hybrid quantum-classical methods for combinatorial logistics optimization at industrial strength scales.
A multi-objective logistics optimization problem from a real-world supply chain is formulated as a Quadratic Unconstrained Binary Optimization Problem (QUBO) that minimizes cost, emissions, and delivery time, while maintaining target distributions of supplier workshare. The model incorporates realistic constraints, including part dependencies, double sourcing, and multimodal transport. Two hybrid quantum-classical solvers are proposed: a structure-aware informed tree search (IQTS) and a modular bilevel framework (HBS), combining quantum subroutines with classical heuristics. Experimental results on IonQ's Aria-1 hardware demonstrate a methodology to map real-world logistics problems onto emerging combinatorial optimization-specialized hardware, yielding high-quality, Pareto-optimal solutions.
We give a quantum speedup for solving the canonical semidefinite programming relaxation for binary quadratic optimization. This class of relaxations for combinatorial optimization has so far eluded quantum speedups. Our methods combine ideas from quantum Gibbs sampling and matrix exponent updates. A de-quantization of the algorithm also leads to a faster classical solver. For generic instances, our quantum solver gives a nearly quadratic speedup over state-of-the-art algorithms. Such instances include approximating the ground state of spin glasses and MaxCut on Erdös-Rényi graphs. We also provide an efficient randomized rounding procedure that converts approximately optimal SDP solutions into approximations of the original quadratic optimization problem.
The increasing complexity of industrial scheduling and transport routing problems motivates the study of alternative optimization formulations and computational paradigms. In this work, we study how higher-order unconstrained binary optimization (HUBO) formulations of such problems map onto quantum optimization workflows in both noisy and fault-tolerant regimes. We consider three representative logistics and manufacturing use cases and formulate each as a HUBO problem. This captures process intricacies, such as highly correlated assembly-line scheduling rules, which are difficult to express faithfully with the standard quadratic (QUBO) form, while at the same time reducing the number of binary variables required in the quantum mapping, thus lowering qubit demand. We compare the HUBO formulations with corresponding QUBO encodings, highlighting a key trade-off: while HUBO reduces qubit requirements through compact binary encoding, it introduces higher-order interaction terms that increase circuit depth, limiting feasibility on current quantum hardware. The proposed formulations are validated using classical solvers across several problem instances and benchmark small routing problem instances using bias-field digitized counterdiabatic quantum optimization in classical simulation. We complement these results with a resource and scalability analysis, focusing on the capacitated vehicle routing problem as a representative large-scale industrial use case. Our analysis indicates that while HUBO formulations offer advantages in qubit scaling compared to QUBO encodings, their practical implementation is constrained by gate fidelity, coherence, and circuit depth, making hybrid quantum-classical workflows and early fault-tolerant quantum hardware the most plausible settings for their practical use.