In the effort to develop useful quantum computers, simulating quantum machines with conventional classical computing resources is a key capability. Such simulations will always face limits, preventing the emulation of quantum computers at substantial scale; however, by pushing the envelope through optimal choices of algorithms and hardware, the value of simulator tools can be maximized. This work reviews state-of-the-art numerical simulation methods, i.e., classical algorithms that emulate quantum computer evolution under specific operations. We focus on the mainstream state-vector and tensor-network paradigms, while briefly mentioning alternative methods. Moreover, we review the diverse applications of simulation across different facets of quantum computer development, including understanding the fundamental differences between quantum and classical computations, exploring algorithmic design for quantum advantage, predicting quantum processor performance at the design stage, and efficiently characterizing fabricated devices for rapid iterations. This review complements recent surveys of current tools and implementations; here, we aim to provide readers with an essential understanding of the theoretical basis of classical simulation methods, a detailed discussion of their advantages and limitations, and an overview of the demands and challenges arising from practical use cases.
In the effort to develop useful quantum computers, simulating quantum machines with conventional classical computing resources is a key capability. Such simulations will always face limits, preventing the emulation of quantum computers at substantial scale; however, by pushing the envelope through optimal choices of algorithms and hardware, the value of simulator tools can be maximized. This work reviews state-of-the-art numerical simulation methods, i.e., classical algorithms that emulate quantum computer evolution under specific operations. We focus on the mainstream state-vector and tensor-network paradigms, while briefly mentioning alternative methods. Moreover, we review the diverse applications of simulation across different facets of quantum computer development, including understanding the fundamental differences between quantum and classical computations, exploring algorithmic design for quantum advantage, predicting quantum processor performance at the design stage, and efficiently characterizing fabricated devices for rapid iterations. This review complements recent surveys of current tools and implementations; here, we aim to provide readers with an essential understanding of the theoretical basis of classical simulation methods, a detailed discussion of their advantages and limitations, and an overview of the demands and challenges arising from practical use cases.
In recent years, the rapidly advancing field of quantum computing has become a focal point of attention. However, issues such as scarcity of quantum hardware and noise pose challenges, leading to the reliance on quantum simulators running on classical computers for tasks like researching quantum algorithms and validating quantum chips. This paper discusses the main simulation methods used by different quantum simulators and explores various optimizations for mainstream full-amplitude state vector simulators and tensor network-based quantum simulators. Finally, we summarize the current status and future directions of quantum simulators. This paper also discusses algorithm-level and system-level optimizations that enhance simulator performance. Looking ahead, the development of quantum simulators faces challenges such as maximizing the use of quantum state properties and creating algorithms to overcome exponential growth issues in simulations. The future will likely focus on integrating high-performance computing resources, advancing optimization strategies, and tailoring simulations to specific applications, aiming for higher efficiency and precision in quantum simulations.
Quantum computing is rapidly evolving as a paradigm that promises to solve complex computational problems far beyond the capabilities of classical systems. This article explores the current challenges facing quantum computation, particularly in algorithm development, error correction, and hardware scalability. The paper also outlines the prospects of next-generation quantum algorithms, especially those applicable to cryptography, optimization, and machine learning. Recent advances in quantum error mitigation and hybrid quantum-classical models are reviewed, emphasizing the transition from theoretical constructs to practical deployment. This study aims to bridge the gap between quantum algorithm research and its implementation, offering insights into the interdisciplinary efforts needed to make quantum advantage a reality.
Quantum computing has the potential to revolutionise multiple fields by solving complex problems that cannot be solved in reasonable time with current classical computers. Nevertheless, the development of quantum computers is still in its early stages and the available systems have still very limited resources. As such, currently, the most practical way to develop and test quantum algorithms is to use classical simulators of quantum computers. In addition, the development of new quantum computers and their components also depends on simulations. Given the characteristics of a quantum computer, their simulation is a very demanding application in terms of both computation and memory. As such, simulations do not scale well in current classical systems. Thus different optimisation and approximation techniques need to be applied at different levels. This review provides an overview of the components of a quantum computer, the levels at which these components and the whole quantum computer can be simulated, and an in-depth analysis of different state-of-the-art acceleration approaches. Besides the optimisations that can be performed at the algorithmic level, this review presents the most promising hardware-aware optimisations and future directions that can be explored for improving the performance and scalability of the simulations.
Quantum mechanics, a fundamental theory in physics that describes the behavior of nature at atomic and subatomic scales, offers significant advantages over classical physics in various contexts. These advances have laid the foundation for the field of quantum computing, which leverages the unique properties of quantum mechanics to solve complex computational problems more efficiently than classical computers. At the core of quantum computing are qubits, the quantum analogs of classical bits. Unlike classical bits, which are restricted to binary states (0 or 1), qubits utilize superposition and entanglement, enabling them to exist in multiple states simultaneously and interact in ways that amplify computational capabilities. This study presents an overview of quantum algorithms and tools for quantum machine learning, with a focus on recent advancements. Through a comparative analysis and empirical evidence, the research highlights the potential advantages of quantum algorithms in various applications, such as data processing, pattern recognition, and algorithmic complexity. The findings suggest that quantum methods may surpass their classical counterparts in certain domains. However, a key challenge remains: the current limitations of quantum hardware. Despite the theoretical benefits, practical implementation is constrained by the noisy and error-prone nature of quantum devices. Consequently, the experiments conducted in this study were performed in simulation environments, which demonstrated potential improvements when applying quantum paradigms.
Quantum computing, grounded in the principles of superposition and entanglement, has emerged as a revolutionary paradigm with the potential to outperform classical systems in specific computational tasks. This paper presents a comprehensive survey of the foundational concepts, algorithmic developments, hardware architectures, and programming ecosystems in quantum computing. We analyze the current progress in key application areas such as cryptography, machine learning, optimization, and quantum chemistry. Furthermore, we review recent advances between 2023 and 2025, including developments in fault-tolerant architectures and quantum error correction. The paper also discusses the significant technical challenges that hinder large-scale practical deployment, including qubit scalability, decoherence, and programming limitations. By synthesizing literature across disciplines, this review aims to provide a holistic understanding of the quantum computing landscape and outline critical directions for future research and integration with classical systems.
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.