Abstract Advances in quantum computing with applications in combinatorial optimization have evolved at an increasing rate in recent years. The quadratic unconstrained binary optimization (QUBO) model is at the center of these developments and has become recognized as an effective alternative method for representing a wide variety of combinatorial optimization problems. Additional momentum has resulted from the arrival of quantum computers and their ability to solve the Ising spin glass problem, another form of the QUBO model. This paper highlights advances in solving QUBO models and extensions to more general polynomial unconstrained binary optimization (PUBO) models as important alternatives to traditional approaches. Computational experience is provided that compares the performance of unique quantum‐inspired metaheuristic solvers—the Next Generation Quantum (NGQ) solver for QUBO models and the NGQ‐PUBO solver for PUBO models—with the performance of CPLEX and the Dwave quantum advantage solver. Extensive results, including experiments with a set of large set partitioning problems representing the largest QUBO models reported in the literature to date, along with maximum diversity and max cut problem sets, disclose that our solvers outperform both CPLEX and Dwave by a wide margin in terms of both computational time and solution quality.
Abstract Advances in quantum computing with applications in combinatorial optimization have evolved at an increasing rate in recent years. The quadratic unconstrained binary optimization (QUBO) model is at the center of these developments and has become recognized as an effective alternative method for representing a wide variety of combinatorial optimization problems. Additional momentum has resulted from the arrival of quantum computers and their ability to solve the Ising spin glass problem, another form of the QUBO model. This paper highlights advances in solving QUBO models and extensions to more general polynomial unconstrained binary optimization (PUBO) models as important alternatives to traditional approaches. Computational experience is provided that compares the performance of unique quantum‐inspired metaheuristic solvers—the Next Generation Quantum (NGQ) solver for QUBO models and the NGQ‐PUBO solver for PUBO models—with the performance of CPLEX and the Dwave quantum advantage solver. Extensive results, including experiments with a set of large set partitioning problems representing the largest QUBO models reported in the literature to date, along with maximum diversity and max cut problem sets, disclose that our solvers outperform both CPLEX and Dwave by a wide margin in terms of both computational time and solution quality.
Quantum computing provides powerful algorithmic tools that have been shown to outperform established classical solvers in specific optimization tasks. A core step in solving optimization problems with known quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) is the problem formulation. While quantum optimization has historically centered around Quadratic Unconstrained Optimization (QUBO) problems, recent studies show, that many combinatorial problems such as the TSP can be solved more efficiently in their native Polynomial Unconstrained Optimization (PUBO) forms. As many optimization problems in practice also contain continuous variables, our contribution investigates the performance of the QAOA in solving continuous optimization problems when using PUBO and QUBO formulations. Our extensive evaluation on suitable benchmark functions, shows that PUBO formulations generally yield better results, while requiring less qubits. As the multi-qubit interactions needed for the PUBO variant have to be decomposed using the hardware gates available, i.e., currently single- and two-qubit gates, the circuit depth of the PUBO approach outscales its QUBO alternative roughly linearly in the order of the objective function. However, incorporating the planned addition of native multi-qubit gates such as the global Mølmer-Sørenson gate, our experiments indicate that PUBO outperforms QUBO for higher order continuous optimization problems in general.
Quadratic Unconstrained Binary Optimization (QUBO) provides a versatile framework for representing NP-hard combinatorial problems, yet existing solvers often face trade-offs among speed, accuracy, and scalability. In this work, we introduce a quantum-inspired solver (QIS) that unites branch-and-bound pruning, continuous gradient-descent refinement, and quantum-inspired heuristics within a fully adaptive control architecture. We benchmark QIS3 against eight state-of-the-art solvers, including genetic algorithms, coherent Ising machines, simulated bifurcation, parallel tempering, simulated annealing, our prior QIS2 version, D-Wave's simulated-annealing (Neal), and Gurobi on three canonical QUBO problem classes: Max-Cut, NAE-3SAT, and Sherrington-Kirkpatrick spin glass problems. Under a uniform runtime budget, QIS3 attains the best solution on nearly all instances, achieving optimality in 94% of max-cut instances. These results establish QIS3 as a robust, high-performance solver that bridges classical exact strategies and quantum-inspired heuristics for scalable QUBO optimization.
Recent advancements in quantum computing have demonstrated significant potential for solving combinatorial optimization problems, like the quadratic knapsack problem, a constrained binary optimization problem. How- ever, current quantum and quantum-inspired algorithms often require transforming these constrained problems into an unconstrained form, known as Quadratic Unconstrained Binary Optimization (QUBO). Such transfor- mations can significantly impact the algorithms’ speed and efficiency. In this study, we evaluate five existing transformation methods and propose four novel approaches. We assess all nine methods using Simulated An- nealing and find that three of our approaches outperform existing methods in terms of execution time and the quality and quantity of feasible solutions found. Additionally, we tested these transformations on quantum annealers, which were unable to solve even small problem instances, due to limitations in connectivity and error rates. However, our results highlight the advantages of the new approaches, which reduce the total num- ber of variables in the QUBO representation. This is a critical factor for enhanced performance on emerging quantum hardware, since it also reduces the required number of qubits and the embedding chain lengths.
Recent advancements in quantum computing have demonstrated significant potential for solving combinatorial optimization problems, like the quadratic knapsack problem, a constrained binary optimization problem. However, current quantum and quantum-inspired algorithms often require transforming these constrained problems into an unconstrained form, known as Quadratic Unconstrained Binary Optimization (QUBO). Such transformations can significantly impact the algorithms’ speed and efficiency. In this study, we evaluate five existing transformation methods and propose four novel approaches. We assess all nine methods using Simulated Annealing and find that three of our approaches outperform existing methods in terms of execution time and the quality and quantity of feasible solutions found. Additionally, we tested these transformations on quantum annealers, which were unable to solve even small problem instances, due to limitations in connectivity and error rates. However, our results highlight the advantages of the new approaches, which reduce the total number of variables in the QUBO representation. This is a critical factor for enhanced performance on emerging quantum hardware, since it also reduces the required number of qubits and the embedding chain lengths.
The Quadratic Unconstrained Binary Optimization (QUBO) model has gained prominence in recent years with the discovery that it unifies a rich variety of combinatorial optimization problems. By its association with the Ising problem in physics, the QUBO model has emerged as an underpinning of the quantum computing area known as quantum annealing and has become a subject of study in neuromorphic computing. Through these connections, QUBO models lie at the heart of experimentation carried out with quantum computers developed by D-Wave Systems and neuromorphic computers developed by IBM. Computational experience is being amassed by both the classical and the quantum computing communities that highlights not only the potential of the QUBO model but also its effectiveness as an alternative to traditional modeling and solution methodologies. This tutorial discloses the basic features of the QUBO model that give it the power and flexibility to encompass the range of applications that have thrust it onto center stage of the optimization field. We show how many different types of constraining relationships arising in practice can be embodied within the "unconstrained" QUBO formulation in a very natural manner using penalty functions, yielding exact model representations in contrast to the approximate representations produced by customary uses of penalty functions. Each step of generating such models is illustrated in detail by simple numerical examples, to highlight the convenience of using QUBO models in numerous settings. We also describe recent innovations for solving QUBO models that offer a fertile avenue for integrating classical and quantum computing and for applying these models in machine learning.