In the last decade, public and industrial research funding has moved quantum computing from the early promises of Shor’s algorithm through experiments to the era of noisy intermediate scale quantum devices (NISQ) for solving real-world problems. It is likely that quantum methods can efficiently solve certain (NP-) hard optimization problems where classical approaches fail. In our perspective, we examine the field of quantum optimization, that is, solving optimization problems using quantum computers. We provide an entry point to quantum optimization for researchers from each topic, optimization or quantum computing, by demonstrating advances and obstacles with a suitable use case. We give an overview on problem formulation, available algorithms, and benchmarking. Although we show a proof-of-concept rather than a full benchmark between classical and quantum methods, this gives an idea of the current quality and capabilities of quantum computers for optimization problems. All observations are incorporated in a discussion on some recent quantum optimization breakthroughs, current status, and future directions.
In the last decade, public and industrial research funding has moved quantum computing from the early promises of Shor’s algorithm through experiments to the era of noisy intermediate scale quantum devices (NISQ) for solving real-world problems. It is likely that quantum methods can efficiently solve certain (NP-) hard optimization problems where classical approaches fail. In our perspective, we examine the field of quantum optimization, that is, solving optimization problems using quantum computers. We provide an entry point to quantum optimization for researchers from each topic, optimization or quantum computing, by demonstrating advances and obstacles with a suitable use case. We give an overview on problem formulation, available algorithms, and benchmarking. Although we show a proof-of-concept rather than a full benchmark between classical and quantum methods, this gives an idea of the current quality and capabilities of quantum computers for optimization problems. All observations are incorporated in a discussion on some recent quantum optimization breakthroughs, current status, and future directions.
Optimization problems are pervasive across various domains, including logistics, finance, machine learning, and operations research. Quantum computing has emerged as a promising frontier to address these challenges, offering potential speedups for certain classes of optimization tasks. This paper explores the development and application of quantum algorithms, such as the Quantum Approximate Optimization Algorithm (QAOA), Variational Quantum Eigensolver (VQE), and Grover’s search algorithm, tailored for optimization. Recent advancements in hardware, hybrid quantum-classical approaches, and variational techniques have enabled practical implementations on Noisy Intermediate-Scale Quantum (NISQ) devices. Challenges such as noise, scalability, and performance limitations are also discussed. Through theoretical analysis and case studies, this work demonstrates how quantum computing can complement classical methods, paving the way for breakthroughs in solving complex optimization problems.
Background: Traditional computers can be inadequate to solve computationally complex problems, generally known as NP-hard problems, for example, optimization, cryptography, and network design. Quantum Computing is referred to as a breakthrough paradigm, that exploits the principles of quantum physics for performing computation in an exceedingly efficient manner.Objective: This study aims to explore the abilities of quantum algorithms in capturing NP-hard problems and discuss their strengths and limitations. The article illustrates how algorithms like Grover's and Shor's may offer exponential or polynomial improvements under specific circumstances.Methodology: The study provides a thorough survey of present-day quantum algorithms and analyzes their capabilities as well as limitations. This article examines advancements in hardware innovation and error correction methods to assess their ability to address the challenges of limited scalability and elevated error rates currently hindering their adoption.Results: The article underscores the profound impact of quantum computing on NP-hard problems. However, significant barriers remain, such as the inherent limitations of hardware, universality with respect to programming language and robust error correction capabilities.Conclusion: The article shows important progress and open problems for solving NP-hard problems using quantum algorithms. The results reveal the capabilities to achieve significant computational speedup with algorithms such as Grover’s and Shor's especially in concert when current quantum hardware matures along with novel error correction techniques. Nevertheless, scholars have to investigate more about scalability and the creation of standard programming languages for quantum computers.
In recent years, quantum computing has emerged as a rapidly advancing interdisciplinary field, attracting growing attention from both academia and industry owing to its fundamentally novel approach to information processing and its potential to achieve computational capabilities beyond the reach of classical systems. This paper presents a structured review of quantum computing approaches applied to scheduling problems, which are among the most challenging NP-hard optimization problems. We first introduce fundamental principles of quantum computing, including gate-based models, quantum annealing, and hybrid frameworks. We then explore how various scheduling problems such as job-shop, flow-shop, and project scheduling-are encoded into quantum models, particularly using QUBO formulations suitable for current quantum devices. Finally, we discuss key limitations such as hardware constraints and model scalability, and we identify research directions to advance quantumenhanced scheduling methods. By grounding theory in empirical case studies and highlighting near-term deployable solutions, this study bridges the gap between quantum algorithm development and practical scheduling demands, offering a roadmap for researchers and practitioners in quantum optimization.
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.
Abstract Quantum computing is gaining popularity across a wide range of scientific disciplines due to its potential to solve long-standing computational problems that are considered intractable with classical computers. One promising area where quantum computing has potential is in the speed-up of NP -hard optimisation problems that are common in industrial areas such as logistics and finance. Newcomers to the field of quantum computing who are interested in using this technology to solve optimisation problems do not have an easily accessible source of information on the current capabilities of quantum computers and algorithms. This paper aims to provide a comprehensive overview of the theory of quantum optimisation techniques and their practical application, focusing on their near-term potential for noisy intermediate scale quantum devices. The paper starts by drawing parallels between classical and quantum optimisation problems, highlighting their conceptual similarities and differences. Two main paradigms for quantum hardware are then discussed: analogue and gate-based quantum computers. While analog devices such as quantum annealers are effective for some optimisation problems, they have limitations and cannot be used for universal quantum computation. In contrast, gate-based quantum computers offer the potential for universal quantum computation, but they face challenges with hardware limitations and accurate gate implementation. The paper provides a detailed mathematical discussion with references to key works in the field, as well as a more practical discussion with relevant examples. The most popular techniques for quantum optimisation on gate-based quantum computers, the quantum approximate optimisation algorithm and the quantum alternating operator ansatz framework, are discussed in detail. However, it is still unclear whether these techniques will yield quantum advantage, even with advancements in hardware and noise reduction. The paper concludes with a discussion of the challenges facing quantum optimisation techniques and the need for further research and development to identify new, effective methods for achieving quantum advantage.
Quantum computing has the potential to solve complex problems that are outside the scope of classical systems, making it a standard shifter. Current devices, which are considered as Noisy Intermediate-Scale Quantum (NISQ) systems, are insufficient in their practical utility due to hardware limitations, error rates, and qubit counts. The dynamic growths wanted to exceed the NISQ era and grow scalable quantum algorithms and architectures that can achieve useful quantum benefit in real-world applications are reviewed in this research. This research gives a comprehensive examination of the complications in scaling quantum systems, adjacent resource efficacy, qubit connectivity, and error correction. In addition to connecting architectural innovations like modular quantum computing, error-resilient circuit design, and distributed quantum systems, the paper provides new hybrid quantum-classical algorithms planned to exploit performance in noisy environments. Results from benchmarking in arenas like machine learning, optimization, and quantum chemistry show how these methods can provide short-term quantum benefit. Lastly, this paper proposes a research roadmap for quantum computing that highlights the requirement of co-designing algorithms, software, and hardware. The goal of this effort is to straighten the progress of quantum systems that can resolve problems in the real world at scale by addressing the inadequacies of NISQ devices and emphasizing scalable solutions.