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.
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.
Due to the expected disparity in quantum vs. classical clock speeds, quantum advantage for branch and bound algorithms is more likely achievable in settings involving large search trees and low operator evaluation costs. Therefore, in this paper, we describe and experimentally validate an exact classical branch and bound solver for quadratic unconstrained binary optimization (QUBO) problems that matches these criteria. Our solver leverages cheap-to-implement bounds from the literature previously proposed for Ising models, including that of Hartwig, Daske, and Kobe from 1984. We detail a variety of techniques from high-performance computing and operations research used to boost solver performance, including a global variable reordering heuristic, a primal heuristic based on simulated annealing, and a truncated computation of the recursive bound. We also outline a number of simple and inexpensive bound extrapolation techniques. Finally, we conduct an extensive empirical analysis of our solver, comparing its performance to state-of-the-art QUBO and MaxCut solvers, and discuss the challenges of a speedup via quantum branch and bound beyond those faced by any quadratic quantum speedup.
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.
An Ising machine (IM), as a type of analog computer tailored for tackling intractable combinatorial optimization problems, has attracted remarkable attention in recent years. In contrast to the blossoming field of bespoke IM hardware, developing metaheuristics from IMs remains largely uninvestigated. Here, we propose a physics-inspired evolutionary computation paradigm, termed the Ising optimization framework (IOF); it comprises a unique Ising algorithm and a hybrid annealing scheme, which together are well-suited for solving quadratic unconstrained binary optimization (QUBO) problems embedded in Ising system energy. The Ising algorithm leverages a set of iterated self-mapping functions to evolve an Ising-spin swarm, enabling efficient energy minimization in artificial Ising systems while mitigating detrimental chaos. Complementing the algorithm, a hybrid annealing scheme integrating singular value dropout, bifurcation control, and a nudging strategy, is devised to augment the overall optimization capacity. The effectiveness of the IOF is validated on various Ising and Max-cut problems with decision variables ranging from 625 to 5000 in number. In comparison to four major types of methods for solving QUBOs, including IM simulations, nature-inspired algorithms, a state-of-the-art heuristic, and the commercial solver Gurobi, the IOF consistently demonstrates notable optimization quality and computational efficiency. This paper provides a theoretical foundation and practical guidelines for bridging Ising-inspired approaches with evolutionary computation, offering an evolutionary perspective on Ising optimizations and suggesting a fertile avenue for future research and application.
We present a heuristic algorithm designed to solve Quadratic Unconstrained Binary Optimization (QUBO) problems efficiently. The algorithm, referred to as IC-D2S, leverages a hybrid approach using Ising and classical machines to address very large problem sizes. Considering the practical limitation on the size of the Ising machine(IM), our algorithm partitions the QUBO problem into a collection of QUBO subproblems (called subQUBOs) and utilizes the IM to solve each subQUBO. Our proposed heuristic algorithm uses a set of control parameters to generate the subQUBOs and explore the search space. Also, it utilizes an annealer based on cosine waveform and applies a mutation operator at each step of the search to diversify the solution space and facilitate the process of finding the global minimum of the problem. We have evaluated the effectiveness of our IC-D2S algorithm on three large-sized problem sets and compared its efficiency in finding the (near-)optimal solution with three QUBO solvers. One of the solvers is a software-based algorithm (D2TS), while the other one (D-Wave) employs a similar approach to ours, utilizing both classical and Ising machines. The results demonstrate that for large-sized problems (>= 5000) the proposed algorithm identifies superior solutions. Additionally, for smaller-sized problems (= 2500), IC-D2S efficiently finds the optimal solution in a significantly faster manner.
We present a heuristic algorithm designed to solve Quadratic Unconstrained Binary Optimization (QUBO) problems efficiently. The algorithm, referred to as IC-D2S, leverages a hybrid approach using Ising and classical machines to address very large problem sizes. Considering the practical limitation on the size of the Ising machine (IM), our algorithm partitions the QUBO problem into a collection of QUBO subproblems (called subQUBOs) and utilizes the IM to solve each subQUBO. Our proposed heuristic algorithm uses a set of control parameters to generate the subQUBOs and explore the search space. Also, it utilizes an annealer based on cosine waveform and applies a mutation operator at each step of the search to diversify the solution space and facilitate the process of finding the global minimum of the problem. We have evaluated the effectiveness of our IC-D2S algorithm on three large-sized problem sets and compared its efficiency in finding the (near-)optimal solution with three QUBO solvers. One of the solvers is a software-based algorithm (D2TS), while the other one (D-Wave) employs a similar approach to ours, utilizing both classical and Ising machines. The results demonstrate that for large-sized problems (≥ 5000) the proposed algorithm identifies superior solutions. Additionally, for smaller-sized problems (= 2500), IC-D2S efficiently finds the optimal solution in a significantly faster manner.
Quadratic Unconstrained Binary Optimization (QUBO) is a broad class of optimization problems with many practical applications. To solve its hard instances in an exact way, known classical algorithms require exponential time and several approximate methods have been devised to reduce such cost. With the growing maturity of quantum computing, quantum algorithms have been proposed to speed up the solution by using either quantum annealers or universal quantum computers. Here we apply the divide-and-conquer approach to reduce the original problem to a collection of smaller problems whose solutions can be assembled to form a single Polynomial Binary Unconstrained Optimization instance with fewer variables. This technique can be applied to any QUBO instance and leads to either an all-classical or a hybrid quantum-classical approach. When quantum heuristics like the Quantum Approximate Optimization Algorithm (QAOA) are used, our proposal leads to a double advantage: a substantial reduction of quantum resources, specifically an average of ~42% fewer qubits to solve MaxCut on random 3-regular graphs, together with an improvement in the quality of the approximate solutions reached.