Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, Fernando G. S. L. Brandão
This chapter covers applications of quantum computing in the area of combinatorial optimization. This area is related to operations research, and it encompasses many tasks that appear in science and industry, such as scheduling, routing, and supply chain management. We cover specific problems where a quadratic quantum speedup may be available via Grover’s quantum algorithm for unstructured search. We also cover several more recent proposals for achieving superquadratic speedups, including the quantum adiabatic algorithm, the quantum approximate optimization algorithm (QAOA), and the short-path algorithm.
This chapter covers applications of quantum computing in the area of combinatorial optimization. This area is related to operations research, and it encompasses many tasks that appear in science and industry, such as scheduling, routing, and supply chain management. We cover specific problems where a quadratic quantum speedup may be available via Grover’s quantum algorithm for unstructured search. We also cover several more recent proposals for achieving superquadratic speedups, including the quantum adiabatic algorithm, the quantum approximate optimization algorithm (QAOA), and the short-path algorithm.
Combinatorial optimization problems play a vital role in computer science and engineering, with applications in logistics, network design, finance, and resource allocation. However, many of these problems are NP-hard, making them computationally expensive to solve using classical optimization techniques, especially for large-scale instances. In recent years, quantum computing has emerged as a promising paradigm capable of addressing these limitations by leveraging quantum mechanical principles such as superposition and entanglement. This study investigates the application of quantum computing for solving combinatorial optimization problems, with a focus on algorithm design, complexity analysis, and real-world applicability. The research primarily employs the Quantum Approximate Optimization Algorithm (QAOA) and Grover's search algorithm to solve representative problems such as Max-Cut and the Knapsack problem. The performance of these algorithms is evaluated using key metrics including accuracy, execution time, and approximation ratio, and is compared with classical optimization techniques. The results demonstrate that QAOA is capable of generating near-optimal solutions, achieving accuracy levels of up to 93% for small problem instances. However, the study also highlights the challenges associated with current quantum systems, including noise, limited qubit availability, and increased execution time. Comparative analysis reveals that while classical methods remain more efficient for small-scale problems, quantum algorithms show significant potential for scalability and handling complex optimization tasks. The findings suggest that hybrid quantum-classical approaches offer a practical solution in the current Noisy Intermediate-Scale Quantum (NISQ) era. The study concludes that although quantum computing is still evolving, it holds strong potential for transforming combinatorial optimization in the future.
Quantum computing has emerged as a transformative technology, particularly for solving optimization problems, which are notoriously difficult for classical computers. Quantum algorithms, leveraging quantum parallelism and entanglement, promise significant speedups for a wide range of optimization tasks, from combinatorial optimization to machine learning. This paper reviews recent advancements in quantum algorithms for optimization problems, highlighting key algorithms such as Grover’s search algorithm, the Quantum Approximate Optimization Algorithm (QAOA), and the Quantum Alternating Operator Ansatz (QAOA). We also explore the challenges and potential of quantum computing in addressing real-world optimization problems, such as in logistics, finance, and machine learning. The paper discusses the current state of quantum hardware, as well as theoretical and practical limitations in implementing these quantum algorithms. Finally, we propose future research directions to overcome existing challenges and enhance the application of quantum algorithms in solving complex optimization problems
Combinatorial Optimization (CO) is one of the most important areas in the field of optimization, with practical applications found in every industry, including both the private and public sectors. In recent years, it was discovered that a mathematical formulation known as QUBO (Quadratic Unconstrained Binary Optimization) problem can embrace an exceptional variety of important CO problems found in the industry. In this work, we explore the Quantum Approximate Optimization Algorithm (QAOA) as a quantum computing approach to solve the Knapsack Problem efficiently by formulating the problem as a QUBO model. Here we implemented QAOA on a quantum emulator, leveraging quantum superposition and entanglement to explore multiple solutions simultaneously. The method offers a promising route for solving large, complex scheduling and resource allocation problems that are tough for classical algorithms, with potential applications in logistics, HPC resource scheduling, and energy optimization.
The research explores the application of quantum computing to manufacturing and supply chain optimization in an effort to increase productivity, reduce costs, and improve product quality. Quantum algorithms, specifically the Quantum Approximate Optimization Algorithm (QAOA), are developed and evaluated to solve complex optimization problems in these domains. Quantum computing approaches are contrasted with traditional optimization techniques to demonstrate the potential advantages of quantum algorithms in terms of solution quality and working time efficiency. Practical implementation considerations of data availability, algorithm scalability, and system integration are also discussed. This research shows that quantum algorithms can effectively optimize production scheduling, resource allocation, and supply chain management, resulting in shorter production schedules and improved operational performance. This research recognizes the limitations of current quantum hardware, the complexity of the problem domain, and the difficulty of implementation. Despite these limitations, this research lays the foundation for further investigation and innovation in quantum computing for manufacturing and supply chain optimization, highlighting the potential for long-term transformative effects on industrial operations.
Quantum search is one of the most important algorithmic developments in quantum computing because it demonstrates how quantum principles can improve the efficiency of solving unstructured search problems. Grover’s algorithm provides a quadratic speed-up over classical linear search by reducing query complexity from O(N) to O(√N), where N represents the size of the search space. This review examines the principles, performance, and applications of Grover’s algorithm, with emphasis on its theoretical foundation and practical relevance. The discussion covers the roles of superposition, quantum interference, oracle operations, diffusion operators, and amplitude amplification in increasing the probability of measuring a desired solution. The manuscript also reviews the computational performance of quantum search in comparison with classical search methods and considers its relevance to database searching, combinatorial optimisation, cryptography, machine learning, bioinformatics, quantum chemistry, and materials science. The review highlights that Grover’s algorithm is not limited to database retrieval, as many computationally intensive problems can be reformulated as search or optimisation tasks. However, practical implementation remains constrained by current quantum hardware limitations, including decoherence, quantum noise, limited qubit availability, circuit depth, and the difficulty of designing efficient oracles. Despite these challenges, continuing progress in quantum hardware, error correction, cloud-based quantum platforms, and hybrid quantum-classical architectures suggests that quantum search will remain an important area of research. Overall, Grover’s algorithm represents a foundational quantum algorithm with significant theoretical value and potential long-term application in computational domains involving large and unstructured search spaces.
A new paradigm in computing, quantum computing has the ability to solve computational problems that were previously considered impossible for classical computers. Optimization problems are fundamental to many disciplines, including operations research, finance, machine learning, and logistics; this article investigates how quantum computing methods can be used to solve these problems. In order to conduct calculations in a way that is radically distinct from classical computers, quantum computers utilize quantum mechanical phenomena like entanglement and superposition. Because of this, quantum algorithms might potentially achieve exponential speedups over classical algorithms for specific optimization tasks by simultaneously exploring large solution spaces. A number of well-known quantum algorithms have been reviewed in this abstract, including Grover's algorithm and quantum annealing methods created by D-Wave Systems, among others. These algorithms aim to solve a wide variety of optimization problems, such as scheduling issues, optimization of portfolios, and combinatorial optimization. Their computational complexity, theoretical foundations, and real-world applications on existing and future quantum hardware are covered in detail.