Recent advancements in quantum computing suggest the potential to revolutionize computational algorithms across various scientific domains including oceanography and atmospheric science. The field is still relatively young and quantum computation is so different from classical computation that suitable frameworks to represent oceanic and atmospheric dynamics are yet to be explored. Quantum annealing (QA), one of the major paradigms, focuses on combinatorial optimization tasks. Given its potential to excel in NP-hard problems, QA may significantly accelerate the calculation of ocean and atmospheric systems described by the Navier-Stokes equations in the future. In this paper, we apply both QA and simulated annealing (SA), its classical counterpart, to a simplified ocean model known as the Stommel problem. We use the Stommel problem, which is not an NP problem and therefore does not benefit from QA today, just as an example, a first step in exploring QA for more intricate problems governed by the Navier-Stokes equations. We cast the linear partial differential equation governing the Stommel model into an optimization problem by the least-squares method and discretize the cost function in two ways: finite difference and truncated basis expansion. In either case, SA successfully reproduces the expected solution when appropriate parameters are chosen. In contrast, QA using the D-Wave quantum annealing machine fails to obtain good solutions for some cases owing to hardware limitations; in particular, the highly limited connectivity graph of the machine limits the size of the solvable problems, at least under currently available algorithms. Either expanding the machine's connectivity graph or improving the graph-embedding algorithms would probably be necessary for quantum annealing machines to be usable for oceanic and atmospheric dynamics problems.
Recent advancements in quantum computing suggest the potential to revolutionize computational algorithms across various scientific domains including oceanography and atmospheric science. The field is still relatively young and quantum computation is so different from classical computation that suitable frameworks to represent oceanic and atmospheric dynamics are yet to be explored. Quantum annealing (QA), one of the major paradigms, focuses on combinatorial optimization tasks. Given its potential to excel in NP-hard problems, QA may significantly accelerate the calculation of ocean and atmospheric systems described by the Navier-Stokes equations in the future. In this paper, we apply both QA and simulated annealing (SA), its classical counterpart, to a simplified ocean model known as the Stommel problem. We use the Stommel problem, which is not an NP problem and therefore does not benefit from QA today, just as an example, a first step in exploring QA for more intricate problems governed by the Navier-Stokes equations. We cast the linear partial differential equation governing the Stommel model into an optimization problem by the least-squares method and discretize the cost function in two ways: finite difference and truncated basis expansion. In either case, SA successfully reproduces the expected solution when appropriate parameters are chosen. In contrast, QA using the D-Wave quantum annealing machine fails to obtain good solutions for some cases owing to hardware limitations; in particular, the highly limited connectivity graph of the machine limits the size of the solvable problems, at least under currently available algorithms. Either expanding the machine's connectivity graph or improving the graph-embedding algorithms would probably be necessary for quantum annealing machines to be usable for oceanic and atmospheric dynamics problems.
Partial differential equations (PDEs) present important computational challenges in several scientific disciplines. Classical numerical methods, while effective, often encounter scalability issues and high computational costs, particularly for complex problems. Quantum computing seems to address these challenges, offering a potential exponential speedup over classical methods. The presented work studies the application of quantum computing, specifically Quantum Annealing, to the computational complexities inherent in solving the Navier-Stokes equations within Computational Fluid Dynamics (CFD). Our study examines the quantum annealing methodology, conducting two different experiments and analyzing the error between the simulation and the analytical solution. The results obtained demonstrate that quantum annealing paradigm can solve this problems and the errors obtained are acceptable in comparison with other methods in the state of the art. By combining theoretical insights with practical implementation, this research clarifies the potential of quantum annealing, providing a comprehensive overview of the simulation framework, analyzing the obtained results, and suggesting future research avenues.
Quantum computing has emerged as a promising computational paradigm with the potential to solve problems that are intractable for classical computers. This paper provides an overview of the current state of quantum computing, including its historical development, hardware advancements, applications, and challenges. Special attention is given to combinatorial optimization, which is considered one of the most promising application domains. The limitations of current quantum systems and future research directions are also discussed. This paper presents a comparative study between a conventional D-Wave-based simulated annealing approach and a proposed hybrid optimization model for solving largescale combinatorial optimization problems. The study focuses on benchmark problems, including AES, TSP gr24, and a real-world load balancing problem, all formulated as QUBO models. The D-Wave approach employs the dwave-neal simulated annealing engine with various parameter settings, while the proposed model introduces a decomposition-based framework combined with adaptive parameter control and iterative refinement. The experimental results demonstrate that the proposed model consistently outperforms the conventional approach in terms of solution quality, achieving significantly lower energy values across all problem instances. In particular, for the TSP gr24 problem, the proposed model achieves near-optimal solutions with substantially reduced computation time. For large-scale problems such as AES and load balancing, the proposed model maintains scalability and robustness, whereas the conventional approach suffers from increased computational cost and performance degradation. Although the proposed method introduces additional computational steps, its hybrid structure effectively mitigates limitations related to local minima, parameter sensitivity, and scalability. These findings indicate that integrating classical optimization techniques with quantum-inspired methods is essential for practical applications. The proposed model provides a promising framework for solving complex real-world optimization problems and highlights the importance of hybrid approaches in advancing quantum annealing technologies.
Critical decision-making issues in science, engineering, and industry are based on combinatorial optimization; however, its application is inherently limited by the NP-hard nature of the problem. A specialized paradigm of analogue quantum computing, quantum annealing (QA), has been proposed to solve these problems by encoding optimization problems into physical energy landscapes and solving them by quantum tunnelling systematically through exploration of solution space. This is a critical review that summarizes the current applications of quantum annealing to combinatorial optimization and includes a theoretical background, hardware designs, algorithm implementation strategies, encoding and embedding schemes, protocols to benchmark quantum annealing, areas of implementation, and links with the quantum algorithms implementation with gate-based hardware and classical solvers. We develop a unified framework, relating adiabatic quantum dynamics, Ising and QUBO models, stoquastic and non-stoquastic Hamiltonians, and diabatic transitions to modern flux-qubit annealers (Chimera, Pegasus, Zephyr topologies), and emergent architectures (Lechner-Hauke-Zoller systems, Rydberg atom platforms), and hybrids of quantum and classical computation. Through our analysis, we find that overhead in embedding and encoding is the largest determinant of the scalability and performance (this is not just the number of qubits). Minor embeddings also usually have a physical qubit count per logical variable of between 5 and 12 qubits, which limits effective problem capacity by 80-92% and, due to chain-breaking errors, compromises the quality of solutions.
Combinatorial optimization problems are frequently classified as NP-hard, which means that the time needed to find the optimal solution generally increases exponentially with the problem size. However, the existing research gap highlights the necessity to develop combinatorial optimization methods capable of scaling to large problems while maintaining low solution latency. In this paper, we introduce stochastic simulated quantum annealing (SSQA) for fast solution of combinatorial optimization problems. SSQA is designed based on stochastic computing, which can simulate quantum annealing (QA) by using multiple replicas of spins (probabilistic bits) in classical computing. The use of stochastic computing leads to an efficient parallel spin-state update algorithm, enabling quick search for a solution around the global minimum energy. Therefore, SSQA realizes quantum-like annealing for large-scale problems and can handle fully connected models in combinatorial optimization, unlike QA. The proposed method is evaluated in MATLAB on graph isomorphism problems, which are typical combinatorial optimization problems. It can handle a 100-times larger problem size compared to QA and a 25-times larger problem size compared to a traditional SA method, respectively, for similar convergence probabilities. The time to solution of SSQA is 11.5 times and 423 times smaller than that of the conventional stochastic simulated annealing method and traditional SA, respectively, when solving a 2,025-node combinatorial optimization problem.
Combinatorial optimization is a foundation of numerous scientific, industrial, and social decision-making questions, but its usefulness is mostly restricted by the exponential complexity of the problem of classical computability. The quantum annealing (QA) has become one of the leading analogue quantum systems to address such problems, whereby optimization problems are converted into physical energy structures and quantum fluctuations are utilized to perform systematic exploration. It provides a critical, synthesized, and comprehensive review of quantum annealing as applied to combinatorial optimization, including the theoretical basis, hardware designs, algorithm methods, issues of embedding and encoding, benchmarking procedures, applications, and integration with quantum algorithms based on gates and solvers on a classical computer. We build a comprehensive taxonomy between adiabatic dynamics, Ising and QUBO models, stoquastic and non-stoquastic Hamiltonians, and diabatic transitions to state-of-the-art flux-qubit annealers, new architectures, and hybrid quantum classical pipelines. We show that the overhead of embedding and encoding and the benchmarking approach taken are the most significant factors in determining scalability and performance, but not the raw number of qubits. Transportation, energy systems, robotics, finance, pharmacological discovery, and machine learning are just some examples of the domain-specific case studies that use QA with minimal empirical utility as a hybrid refinement engine and not an independent solver. Moreover, we introduce a stringent evaluation of the existing benchmarking culture, clarify the causal links between QA and QAOA as well as between VQE and structural predicaments, and reveal the institutional roots of the existing restrictions. We conclude by identifying a research roadmap in the future, focusing on the annealing hardness characterization, Stoquastic control manipulation, embedding rules automation, architecture design, and development of principled quantum advantage. It creates a benchmark reference to those in the field and research and deployment of scalable, reliable, and application-relevant quantum optimization
The recent availability of the first commercial quantum computers has provided a promising tool to tackle NP hard problems which can only be solved heuristically with present techniques. However, it is unclear if the current state of quantum computing already provides a quantum advantage over the current state of the art in classical computing. This article assesses the performance of the D-Wave 2X quantum annealer on two NP hard graph problems, in particular clique finding and graph partitioning. For this, we provide formulations as Qubo and Ising Hamiltonians suitable for the quantum annealer and compare a variety of quantum solvers (Sapi, QBSolv, QSage provided by D-Wave Sys, Inc.) to current classical algorithms (METIS, Simulated Annealing, third-party clique finding and graph splitting heuristics) on certain test sets of graphs. We demonstrate that for small graph instances, classical methods still outperform the quantum annealer in terms of computing time, even though the quality of the best solutions obtained is comparable. Nevertheless, due to the limited problem size which can be embedded on the D-Wave 2X chip, the aforementioned finding applies to most of problems of general nature solvable on the quantum annealer. For instances specifically designed to fit the D-Wave 2X architecture, we observe substantial speed-ups in computing time over classical approaches.