This study focuses on reformulating the shortest path problem in undirected weighted graphs into a QUBO problem, leveraging the D-Wave quantum annealing hybrid solver for execution. While A * search with heuristics, a widely used algorithm for solving shortest path problem, serves as a baseline for comparison, our primary contribution lies in analyzing the accuracy and benefits of the proposed QUBO formulation. Our analysis reveals that the proposed QUBO formulation together with selected quantum annealing hybrid solver is loosing its efficiency with increasing number of nodes. This is caused by the size of the problem and related penalty calculation but also by the architecture of the D-Wave QPU and design of the hybrid solver. We are proposing techniques to improve the results, such as graph pruning during preprocessing or penalty parameter tuning to achieve better, more accurate, results. This study provides valuable insights into the limitations of current approaches and explores the future applications of quantum computing in solving real-world challenges.
This study focuses on reformulating the shortest path problem in undirected weighted graphs into a QUBO problem, leveraging the D-Wave quantum annealing hybrid solver for execution. While A * search with heuristics, a widely used algorithm for solving shortest path problem, serves as a baseline for comparison, our primary contribution lies in analyzing the accuracy and benefits of the proposed QUBO formulation. Our analysis reveals that the proposed QUBO formulation together with selected quantum annealing hybrid solver is loosing its efficiency with increasing number of nodes. This is caused by the size of the problem and related penalty calculation but also by the architecture of the D-Wave QPU and design of the hybrid solver. We are proposing techniques to improve the results, such as graph pruning during preprocessing or penalty parameter tuning to achieve better, more accurate, results. This study provides valuable insights into the limitations of current approaches and explores the future applications of quantum computing in solving real-world challenges.
The problem of finding the maximum weighted cycle in a directed graph maps to optimization problems in NP, implying that approaches in classical computing are inefficient. Here, Quantum computing might be a promising alternative. Many current approaches on the quantum computer are based on a Quadratic Unconstrained Binary Optimization (QUBO) problem formulation. This paper develops four different QUBO approaches to this problem. The first two take the starting vertex and the number of vertices used in the cycle as given, while the latter two loosen the second assumption of knowing the size of the cycle. A QUBO formulation is derived for each approach. Further, the number of binary variables required to encode the maximum weighted cycle problem with one or both assumptions for the respective approach is made explicit. The problem is motivated by finding the maximum weighted debt cycle in a debt graph. This paper compares classical computing versus currently avaiable (hybrid) quantum computing approaches for various debt graphs. For the classical part, it investigated the Depth-First-Search (DFS) method and Simulated Annealing. For the (hybrid) quantum approaches, we utilized direct embedding on the quantum annealer and two types of quantum hybrid solvers. Simulated Annealing and the usage of the hybrid CQM (Constrained Quadratic Model) machine functioned promising. The DFS method, direct QPU, and hybrid BQM (Binary Quadratic Model), on the other hand, performed less, due to memory issues, surpassing the limit of decision variables, and finding the right penalty values, respectively.
This article addresses the formulation for implementing a single source, single-destination shortest path algorithm on a quantum annealing computer. Three distinct approaches are presented. In all the three cases, the shortest path problem is formulated as a quadratic unconstrained binary optimization problem amenable to quantum annealing. The first implementation builds on existing quantum annealing solutions to the traveling salesman problem, and requires the anticipated maximum number of vertices on the solution path |P| to be provided as an input. For a graph with |V| vertices, |E| edges, and no self-loops, it encodes the problem instance using |V||P| qubits. The second implementation adapts the linear programming formulation of the shortest path problem, and encodes the problem instance using |E| qubits for directed graphs or 2|E| qubits for undirected graphs. The third implementation, designed exclusively for undirected graphs, encodes the problem in |E| + |V| qubits. Scaling factors for penalty terms, complexity of coupling matrix construction, and numerical estimates of the annealing time required to find the shortest path are made explicit in the article.
QUBO formulations of combinatorial optimization problems allow for solving them using various quantum heuristics. While large-scale quantum computations are currently still out of reach, we can already numerically test such QUBO formulations on a perhaps surprisingly large scale. In this work, we benchmark Fujitsu’s Digital Annealer (DA) on the Max-Cut problem, which captures the main complexity of the QUBO problem. We make a comprehensive benchmark against leading other heuristic algorithms on graphs with up to 53,000 variables by focusing on the wall-clock time. Moreover, we compare the DA performance against published performance results of the D-Wave hybrid quantum-classical annealer and the recently proposed QIS3 heuristic. Based on performance statistics for over 2,000 graphs from the MQLib, we find that the DA yields competitive results. We hope that this benchmark demonstrates the extent to which large QUBO instances can be heuristically solved today, yielding consistent results across different solvers.
This paper evaluates quantum annealing as a potential method for solving multi-path routing in large-scale quantum communication networks. The routing problem was formulated as a quadratic unconstrained binary optimization (QUBO) problem that minimizes negative throughput while satisfying quantum network constraints. The experimental framework tests four solution approaches on randomly generated networks: (1) the classical quantum fidelity rate maximizing (Q-FiRM) algorithm, (2) classical simulated annealing, (3) D-Wave hybrid solver, and (4) pure quantum annealing on D-Wave Advantage2 hardware. Results show that while quantum annealers produce feasible solutions, the classical Q-FiRM algorithm achieves superior throughput and faster computation times for this problem class. Among QUBO approaches, hybrid solvers demonstrate computational advantages over pure classical simulated annealing while maintaining comparable solution quality. These findings provide practical insight into the current capabilities and limitations of quantum optimization for quantum internet infrastructure design.
Combinatorial problems pose significant challenges, especially for large-scale instances. A promising approach to addressing these challenges involves quadratic unconstrained binary optimization (QUBO) models, a key framework in quantum computing. QUBO formulations require tailored approaches owing to their unique characteristics and the hardware constraints of quantum systems. This paper focuses on the QUBO formulations of the Hamiltonian cycle problem (HCP). We analyse three QUBO formulations for the HCP, including two inspired by Lagrangian relaxation techniques, where penalties may result in negative costs for certain solutions but are balanced within the model. A preprocessing methodology is also introduced. To evaluate these models, we employ D-Wave quantum annealers, which feature a fixed qubit topology and impose strict resource constraints. The mapping of QUBOs on to this architecture highlights the importance of efficient model design. Through experimental studies, we analyse the performance of the proposed QUBOs with and without preprocessing. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.
Algorithms and data structures are important in information technology (particularly in 6G terahertz wireless communications), logistics, transportation, and route planning in medicine. One of the fundamental algorithms that find application in many practical problems is the shortest path algorithm. The shortest path algorithm is used to find the optimal route between two vertices while minimizing the length or number of intermediate nodes. This problem is one of the 21 NP-hard problems formulated by Richard Karp in 1972. It still poses a serious challenge to science, since there is no polynomial algorithm that can provide an exact solution to this problem. The article considers approaches to solving the problem of finding the shortest path on a graph, and discusses their advantages and disadvantages. Various algorithms, such as the Bellman-Ford algorithm, Dijkstra, and the A* algorithm, are considered; their efficiency and applicability in various situations are analyzed. Issues of optimization of shortest path algorithms are discussed, such as the use of heuristics, graph preprocessing, and other methods that can improve the performance and accuracy of algorithms. The article proposes an optimization of Dijkstra’s algorithm with quantum acceleration. The stability of the quantum algorithm and the magnitude of distortions in a circuit with a different number of qubits are estimated. It is shown that with an increase in the intensity of distortions, the probability of detecting the desired element decreases, and the time spent on its search increases. When using Grover’s algorithm to optimize a path in a quantum network, these interferences cause an increase in the time it takes to find the optimal path and an increase in the probability of choosing an incorrect route. It is shown that an increase in the number of qubits and, accordingly, the number of iterations makes the algorithm more stable. It is demonstrated that the stability of Grover’s algorithm for circuits with a large number of qubits has a similar nature. The work of quantum circuits on 9, 10 and 11 qubits, the difference in the probabilities of correct operation differs by less than 10–3. Thus, from the obtained dependencies, it is possible to predict the nature of such an influence on circuits with an even larger number of qubits, the stability of which cannot be studied due to the insufficient computing power of classical computers. The most efficient way to modify Dijkstra’s algorithm is to replace the priority queue in the classical version of the algorithm with some quantum data structure that finds the minimum element for the next iteration of the algorithm. Despite the fact that quantum search does not provide a potential gain in asymptotics, another quantum modification of Dijkstra’s algorithm can be used. Namely, instead of maintaining and searching for a minimum in the priority queue, use the quantum algorithm immediately to search for a vertex that needs to be added to the shortest path tree, which will additionally acceleration the graph traversal by Dijkstra’s algorithm.