Abstract Current quantum computers can only solve optimization problems of a very limited size. For larger problems, decomposition methods are required in which the original problem is broken down into several smaller sub-problems. These are then solved on the quantum computer and their solutions are recombined into a final solution for the original problem. Often, these decomposition methods do not take the specific problem structure into account. In this paper, we present a tailored method using a divide-and-conquer strategy to solve the 2-way Number partitioning problem (NPP) with a large number of variables. The idea is to perform a specialized decomposition into smaller NPPs, which are solved on a quantum computer, and then recombine the results into another small auxiliary NPP. Solving this auxiliary problem yields an approximate solution of the original larger problem. We experimentally verify that our method allows to solve NPPs with over a thousand variables using the D-Wave Advantage quantum annealer (Advantage_system6.4).
Abstract Current quantum computers can only solve optimization problems of a very limited size. For larger problems, decomposition methods are required in which the original problem is broken down into several smaller sub-problems. These are then solved on the quantum computer and their solutions are recombined into a final solution for the original problem. Often, these decomposition methods do not take the specific problem structure into account. In this paper, we present a tailored method using a divide-and-conquer strategy to solve the 2-way Number partitioning problem (NPP) with a large number of variables. The idea is to perform a specialized decomposition into smaller NPPs, which are solved on a quantum computer, and then recombine the results into another small auxiliary NPP. Solving this auxiliary problem yields an approximate solution of the original larger problem. We experimentally verify that our method allows to solve NPPs with over a thousand variables using the D-Wave Advantage quantum annealer (Advantage_system6.4).
Quantum annealing is a heuristic algorithm for solving combinatorial optimization problems, and hardware for implementing this algorithm has been developed by D-Wave Systems Inc. The current version of the D-Wave quantum annealer can solve unconstrained binary optimization problems with a limited number of binary variables. However, the cost functions of several practical problems are defined by a large number of integer variables. To solve these problems using the quantum annealer, integer variables are generally binarized with one-hot encoding, and the binarized problem is partitioned into small subproblems. However, the entire search space of the binarized problem is considerably larger than that of the original integer problem and is dominated by infeasible solutions. Therefore, to efficiently solve large optimization problems with one-hot encoding, partitioning methods that extract subproblems with as many feasible solutions as possible are required. In this study, we propose two partitioning methods and demonstrate that they result in improved solutions.
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.
Abstract In this work, we investigate the capabilities of a hybrid quantum-classical procedure to explore the solution space using the D-Wave 2000QTMquantum annealer device. Here, we study the ability of the quantum hardware to solve the number partitioning problem, a well-known NP-hard optimization model that poses some challenges typical of those encountered in real-world applications. This represents one of the most complex scenario in terms of qubits connectivity and, by increasing the input problem size, we analyze the scaling properties of the quantum-classical workflow. We find remarkable results in most instances of the model; for the most complex ones, we investigate further the D-Wave Hybrid suite. Specifically, we were able to find the optimal solutions even in the worst cases by fine-tuning the parameters that schedule the annealing time and allowing a pause in the annealing cycle.
Quantum annealing is a heuristic algorithm that solves combinatorial optimization problems, and D-Wave Systems Inc. has developed hardware implementation of this algorithm. However, in general, we cannot embed all the logical variables of a large-scale problem, since the number of available qubits is limited. In order to handle a large problem, qbsolv has been proposed as a method for partitioning the original large problem into subproblems that are embeddable in the D-Wave quantum annealer, and it then iteratively optimizes the subproblems using the quantum annealer. Multiple logical variables in the subproblem are simultaneously updated in this iterative solver, and using this approach we expect to obtain better solutions than can be obtained by conventional local search algorithms. Although embedding of large subproblems is essential for improving the accuracy of solutions in this scheme, the size of the subproblems are small in qbsolv since the subproblems are basically embedded by using an embedding of a complete graph even for sparse problem graphs. This means that the resource of the D-Wave quantum annealer is not exploited efficiently. In this paper, we propose a fast algorithm for embedding larger subproblems, and we show that better solutions are obtained efficiently by embedding larger subproblems.
This paper presents the details and testing of two implementations (in C++ and Python) of the hybrid quantum-classical algorithm Quantum Annealing Learning Search (QALS) on a D-Wave quantum annealer. QALS was proposed in 2019 as a novel technique to solve general QUBO problems that cannot be directly represented into the hardware architecture of a D-Wave machine. Repeated calls to the quantum machine within a classical iterative structure and a related convergence proof originate a learning mechanism to find an encoding of a given problem into the quantum architecture. The present work considers the Number Partitioning Problem (NPP) and the Travelling Salesman Problem (TSP) for the testing of QALS. The results turn out to be quite unexpected, with QALS not being able to perform as well as the other considered methods, especially in NPP, where classical methods outperform quantum annealing in general. Nevertheless, looking at the TSP tests, QALS has fulfilled its primary goal, i.e., processing QUBO problems not directly mappable to the QPU topology.
The Benders’ decomposition algorithm is a technique in mathematical programming for complex mixed-integer linear programming (MILP) problems with a particular block structure. The strategy of Benders’ decomposition can be described as a strategy of divide and conquer. The Benders’ decomposition algorithm has been employed in a variety of applications such as communication, networking, and machine learning. However, the master problem in Benders’ decomposition is still NP-hard, which motivates us to employ quantum computing. In the paper, we propose a hybrid quantum-classical Benders’ decomposition algorithm. We transfer the Benders’ decomposition’s master problem into the quadratic unconstrained binary optimization (QUBO) model and solve it by the state-of-the-art quantum annealer. Then, we analyze the computational results and discuss the feasibility of the proposed algorithm. Due to our reformulation in the master problem in Benders’ decomposition, our hybrid algorithm, which takes advantage of both classical and quantum computers, can guarantee the solution quality for solving MILP problems.