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arXiv:2603.04089

The Steiner Tree Problem: Novel QUBO Formulation and Quantum Annealing Implementation

Dan Li, Xiang-Hui Wu, Ji-Rong Liu

quant-phSubmitted Mar 4, 2026• Announced Mar 5, 2026• 0 citations
arXivHTMLPDF

Abstract

The Steiner Tree Problem (STP) is a well-known NP-hard combinatorial optimization problem, which has wide applications in network design, integrated circuit layout, bioinformatics, and other fields. However, traditional algorithms often struggle to balance efficiency and solution quality when dealing with large-scale STP instances. In this paper, we propose a new quantum annealing-based algorithm for solving the STP: we first model the STP into a quadratic unconstrained binary optimization (QUBO) form suitable for quantum annealing, then design a corresponding encoding strategy, and finally verify the algorithm through experimental tests. The results show that our quantum annealing-based method can obtain high-quality solutions with relatively low computational overhead for moderate-scale STP instances, providing a new feasible path for handling this intractable combinatorial optimization problem.

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Quantum computers are early-stage technology, but are progressing rapidly. Here, we compare metaheuristics with a quantum annealer (QA), a quantum computer tailored to solve combinatorial optimisation problems formulated as a Quadratic Unconstrained Binary Optimisation (QUBO) problem. Formulating non-trivial optimisation problems as QUBOs is a challenge, so it is to ensure that the direct comparison of classical and quantum search algorithms is conducted fairly. This paper illustrates the use of a streamlined methodology to derive a QUBO for a classical optimisation problem – the Steiner tree problem – and provides comprehensive experiments comparing D-WAVE pure QA and hybrid QA with classical methods including Simulated Annealing and Genetic Algorithms. We find that the pure QA suffers from considerable qubit noise but the hybrid QA is on a par with traditional methods.
Quantum annealing algorithms were introduced to solve combinatorial optimization problems by taking advantage of quantum fluctuations to escape local minima in complex energy landscapes typical of NP - hard problems. In this work, we propose using quantum annealing for the theory of cuts, a field of paramount importance in theoretical computer science. We have proposed a method to formulate the Minimum Multicut Problem into the QUBO representation, and the technical difficulties faced when embedding and submitting a problem to the quantum annealer processor. We show two constructions of the quadratic unconstrained binary optimization functions for the Minimum Multicut Problem and we review several tradeoffs between the two mappings and provide numerical scaling analysis results from several classical approaches. Furthermore, we discuss some of the expected challenges and tradeoffs in the implementation of our mapping in the current generation of D-Wave machines.
This paper explores the application of quantum annealing to the challenging domain of combinatorial optimization within large-scale networks. Specifically, it investigates the feasibility of utilizing quantum annealing for problems such as network routing and scheduling, where traditional classical algorithms often struggle due to computational complexity. The core approach involves formulating the network optimization problem as a Quadratic Unconstrained Binary Optimization (QUBO) problem. This transformation enables the mapping of the problem onto the quantum annealing hardware, leveraging the quantum annealing process – characterized by its adiabatic evolution – to discover the optimal solution. We argue that this method presents a potential advantage over conventional algorithms for large-scale network scenarios, offering a pathway to tackling problems previously considered intractable. The theoretical framework and underlying principles are thoroughly discussed, outlining the methodology and highlighting the anticipated benefits.
Recent advances in quantum computing have shown great potential for solving combinatorial optimization problems. A notable example is the quadratic knapsack problem, a constrained binary optimization problem. However, current quantum-inspired algorithms require Quadratic Unconstrained Binary Optimization (QUBO) formulations. Transforming constrained optimization into QUBO impacts the algorithms' speed and efficiency. In this study, we evaluate five existing transformation methods and propose four novel approaches. We assess them by using Simulated Annealing (SA), and find that three of our approaches outperform existing methods in terms of execution time as well as quality and quantity of feasible solutions found. Also, we tested these transformations on a quantum annealer, which was unable to solve even small instances, due to limitations in connectivity and error rates. In general, our results highlight the advantages of the new approaches, which reduce the number of variables in the QUBO representation.
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.
Combinatorial optimization problems are of great significance in many fields, and this paper explores the application of quantum annealing-based QUBO (Quadratic Unconstrained Binary Optimization) model to combinatorial optimization problems, including the 0/1 knapsack problem, the traveler's problem, the maximum cut problem, and the graph coloring problem. The quantum annealing algorithm can effectively avoid falling into local optimal solutions by virtue of its unique quantum tunneling advantage, and shows significant advantages in solving complex combinatorial optimization problems, especially in dealing with large-scale data and multiple constraints, quantum annealing provides an efficient solution.