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arXiv:2311.09700 · Entropy

Hybrid Classical–Quantum Branch-and-Bound Algorithm for Solving Integer Linear Problems

Claudio Sanavio, Edoardo Tignone, Elisa Ercolessi

Published Apr 19, 2024• 1 citation
arXivHTMLPDF

Abstract

Quantum annealers are suited to solve several logistic optimization problems expressed in the QUBO formulation. However, the solutions proposed by the quantum annealers are generally not optimal, as thermal noise and other disturbing effects arise when the number of qubits involved in the calculation is too large. In order to deal with this issue, we propose the use of the classical branch-and-bound algorithm, that divides the problem into sub-problems which are described by a lower number of qubits. We analyze the performance of this method on two problems, the knapsack problem and the traveling salesman problem. Our results show the advantages of this method, that balances the number of steps that the algorithm has to make with the amount of error in the solution found by the quantum hardware that the user is willing to risk. The results are obtained using the commercially available quantum hardware D-Wave Advantage, and they outline the strategy for a practical application of the quantum annealers.

DOI
10.3390/e26040345

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Quantum annealers are suited to solve several logistic optimization problems expressed in the QUBO formulation. However, the solutions proposed by the quantum annealers are generally not optimal, as thermal noise and other disturbing effects arise when the number of qubits involved in the calculation is too large. In order to deal with this issue, we propose the use of the classical branch-and-bound algorithm, that divides the problem into sub-problems which are described by a lower number of qubits. We analyze the performance of this method on two problems, the knapsack problem and the traveling salesman problem. Our results show the advantages of this method, that balances the number of steps that the algorithm has to make with the amount of error in the solution found by the quantum hardware that the user is willing to risk. The results are obtained using the commercially available quantum hardware D-Wave Advantage, and they outline the strategy for a practical application of the quantum annealers.
Abstract The transportation problem considered in this work consists of minimizing the sum of penalties for tardiness in the delivery of the product from the base to clients. We propose an innovative method for constructing hybrid algorithms using quantum annealing that guarantees the optimality of the solution. The hybrid algorithm alternately performs calculations on a classical computer equipped with a silicon CPU and a quantum annealer with a QPU processor. The entire method is based on the branch and bound scheme. The lower and upper bounds are calculated on the QPU, while the CPU is responsible for organizing the calculations, recalling the solution tree. The problem under consideration is of great practical importance because it is a component of many complex issues related to solving the traveling repairmen problem, which are very difficult or even impossible to solve today using classical methods. Undoubtedly, an important application of quantum computers in the future is NP‐hard optimization tasks related to transportation scheduling and logistics.
In this work, we attempt to solve the integer-weight knapsack problem using the D-Wave 2000Q adiabatic quantum computer. The knapsack problem is a well-known NP-complete problem in computer science, with applications in economics, business, finance, etc. We attempt to solve a number of small knapsack problems whose optimal solutions are known; we find that adiabatic quantum optimization fails to produce solutions corresponding to optimal filling of the knapsack in all problem instances. We compare results obtained on the quantum hardware to the classical simulated annealing algorithm and two solvers employing a hybrid branch-and-bound algorithm. The simulated annealing algorithm also fails to produce the optimal filling of the knapsack, though solutions obtained by simulated and quantum annealing are no more similar to each other than to the correct solution. We discuss potential causes for this observed failure of adiabatic quantum optimization.
Quantum computing has paved a new way for faster and more efficient solutions to large-scale, real-world optimization problems that are challenging for classical computing systems. For instance, selective traveling salesman problem (sTSP) that is famous in such fields as logistic optimization and has attracted increasing attention from the research community, however, is known as an NP-Hard problem. Solving the sTSP is, therefore, extremely complex because the optimization function potentially comes with an exponential number of variables that cannot be solved in polynomial time in general. To this end, we propose a quantum annealing framework for time-bounded and near-optimal solutions for the sTSP, overcoming hardware limits of near-term quantum devices. In particular, we put forth an efficient Hamiltonian (QUBO) to encode the complex decision-making for the sTSP on noisy intermediate-scale quantum (NISQ) annealer. Furthermore, experimental results we obtained on the D-Wave 2000QQQquantum hardware demonstrate that the optimal solutions for several instances can be attained.
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.
Optimization algorithms for combinatorial problems play a critical role in a wide range of applications, from logistics and scheduling to cryptography and artificial intelligence. Traditional classical methods, such as simulated annealing, genetic algorithms, and branch-and-bound techniques, have been widely employed to solve these problems, but they often face limitations in terms of computational efficiency and scalability as problem complexity grows. In recent years, quantum computing has emerged as a promising alternative, with quantum annealing offering a novel approach to solving combinatorial optimization problems. This study provides a comparative analysis of quantum annealing and classical optimization methods, highlighting their respective strengths and limitations. We explore how quantum annealing, through leveraging quantum tunnelling and superposition, has the potential to outperform classical algorithms in specific problem domains and Noisy Mean Field Annealing (NMFA) is used to comparisons of all algorithms. Benchmarking various combinatorial problems, such as the Traveling Salesman Problem (TSP), Max-Cut, Knapsack, and Quadratic Assignment Problem (QAP), this paper discusses performance metrics, computational time, and scalability. The findings suggest that while classical methods remain robust and practical for many real-world problems, quantum annealing holds significant promise, especially as quantum hardware continues to mature. However, there remain challenges in terms of noise, coherence, and problem mapping, which currently limit the full realization of quantum annealing's potential. This study offers insights into the future directions of optimization techniques and the evolving role of quantum computing in solving complex combinatorial problems.
The traveling salesman problem is a well-known NP-hard problem in combinatorial optimization. This paper shows how to solve it on an Ising Hamiltonian based quantum annealer by casting it as a quadratic unconstrained binary optimization (QUBO) problem. Results of practical experiments are also presented using D-Wave’s 5,000 qubit Advantage 1.1 quantum annealer and the performance is compared to a classical solver. It is found the quantum annealer can only handle a problem size of 8 or less nodes and its performance is subpar compared to the classical solver both in terms of time and accuracy.