Abstract Optimisation appears in the workflow of virtually every application and is often a computationally expensive step. Therefore, exploring whether quantum computing can offer an advantage in optimisation – even if only in certain circumstances – is extremely well-motivated from a practical perspective. Unlike the other algorithms covered in the book, which are expressed and analysed as quantum circuits, the most natural starting point for understanding quantum optimisation is the adiabatic model of quantum computation, which is introduced in this chapter. The quantum approximate optimisation algorithm, a gate-based algorithm inspired by quantum annealing (itself closely-related to adiabatic quantum computation), is also covered.
Abstract Optimisation appears in the workflow of virtually every application and is often a computationally expensive step. Therefore, exploring whether quantum computing can offer an advantage in optimisation – even if only in certain circumstances – is extremely well-motivated from a practical perspective. Unlike the other algorithms covered in the book, which are expressed and analysed as quantum circuits, the most natural starting point for understanding quantum optimisation is the adiabatic model of quantum computation, which is introduced in this chapter. The quantum approximate optimisation algorithm, a gate-based algorithm inspired by quantum annealing (itself closely-related to adiabatic quantum computation), is also covered.
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.
Optimisation and AI: What are companies doing today?The earlier chapter on quantum applications discussed whether quantum computers can offer practical speedups for optimisation and AI.We concluded that this is a subtle case and that it is still unclear how much utility quantum computers can provide in these fields.Still, a large body of literature claims some form of near-term speedup in specific quantum applications.What's really going on here?This chapter aims to build a more detailed intuition of how different organisations are exploring quantum applications.We will assess some example research papers, examine the problems they tackle, and analyse how convincingly they point to quantum utility.Moreover, we will look at the most fruitful directions for finding new and useful quantum algorithms.But before we dive into these details, let's take a step back and ask ourselves: what must a quantum algorithm do to be a genuine improvement over its classical counterpart?
Quantum computing offers a fundamentally new paradigm for solving complex optimisation and learning tasks by exploiting superposition, entanglement, and other quantum phenomena. Over the past decade, research has increasingly focused on quantum-enhanced approaches to optimisation and machine learning (ML), showing the potential to outperform classical methods in specific settings (Zaman, 2023; Peral-García, 2024). This paper presents a comprehensive review of quantum algorithms for optimisation (e.g., the Quantum Approximate Optimization Algorithm and quantum annealing) and quantum machine learning frameworks (such as quantum support vector machines and quantum neural networks), analysing their theoretical underpinnings, implementation status, and applicability to real-world tasks. Key challenges including hardware noise, limited qubit counts, and algorithmic scalability are examined (Chen, 2024). We also explore hybrid quantum-classical architectures as a near-term route to quantum advantage and propose future research directions aimed at bridging the current gap between quantum algorithm theory and large-scale deployment. Through this synthesis, we aim to provide both academic and practitioner audiences with a clear roadmap for leveraging quantum algorithms in optimisation and ML workflows
Abstract Having spent almost an entire book on working out that Hopfield nets and adiabatic quantum computing are closely related concepts, this chapter departs from our main story line of problem solving as energy minimization and concludes our study with an outlook to the possibilities of quantum gate computing. Indeed, the central ideas behind quantum gate computing differ from most of what we studied so far but our study still has us prepared for getting to grips with the basics of this paradigm. As this paradigm posits that quantum algorithms are sequences of unitary operators which each act on the joint state of a qubit system, we will look at the circuit model of quantum computing and see that quantum circuits consist of quantum gates which are interconnected by quantum wires where the latter represent the states of qubits on which the former perform unitary operations. As a practical example, we discuss how such circuits could compute univariate Boolean functions. This then seamlessly allows us to finally study Deutsch’s problem and algorithm and therefore to see an instance of a computational problem for which quantum computing algorithms can be exponentially faster than their classical counterparts.
Abstract The discovery of physical systems that exhibit what we now call quantum mechanical behaviour was unquestionably one of the most astonishing events of modern physics. No less remarkable was the realisation that quantum mechanics presents a much more powerful paradigm for computation and information processing than its classical counterpart. But this power is somewhat subtle, and only by building up a rigorous model of quantum computation from the fundamental physics can it be fully understood and analysed. Developing and then using the quantum circuit model as a suitable representation of quantum mechanics, this textbook presents a canon of the most important quantum algorithms. Starting with historically important query algorithms, ‘classic’ results such as Grover’s and Shor’s algorithm are then covered, followed by more recent breakthroughs including the quantum algorithm for solving linear systems (HHL) and the quantum singular value transformation. The thread that runs through the book is the hunt for quantum advantage, in all of its manifestations, and to this end the final chapter moves from the idealised setting of closed quantum systems, to address real ‘noisy’ open quantum systems. Quantum error correction is introduced, culminating in the seminal result, the threshold theorem, which roughly says that the idealised performance predicted by the analysis of quantum mechanics as a model of computation can actually be realised on real-world quantum computers.
Quantum Computing The twenty-first century has witnessed remarkable advancements in computing technologies, transforming every aspect of modern society. As conventional computing systems approach their physical and computational limits, the search for new paradigms capable of solving increasingly complex problems has led to the emergence of Quantum Computing. By harnessing the principles of quantum mechanics, quantum computing represents a revolutionary approach that has the potential to redefine computation, communication, and information processing on an unprecedented scale. Quantum Computing is designed to provide readers with a comprehensive understanding of the concepts, principles, technologies, and applications that form the foundation of this rapidly evolving field. This book aims to bridge the gap between theoretical quantum mechanics and practical computational applications, enabling students, researchers, academicians, and professionals to explore the transformative potential of quantum technologies. The chapters in this book cover a broad spectrum of topics, including the fundamentals of quantum mechanics, quantum bits (qubits), quantum gates, superposition, entanglement, quantum circuits, quantum algorithms, quantum cryptography, quantum communication, and quantum error correction. The book also explores the architecture of quantum computers and examines the challenges involved in building scalable and fault-tolerant quantum systems. As industries increasingly seek innovative solutions to complex computational problems, quantum computing is emerging as a promising technology with applications in cryptography, artificial intelligence, machine learning, optimization, financial modeling, healthcare, pharmaceutical research, materials science, logistics, and climate modeling. The ability of quantum computers to process vast amounts of information simultaneously offers opportunities that were previously beyond the reach of classical computing systems. Special attention has been given to contemporary developments in quantum technologies, including quantum supremacy, hybrid quantum-classical computing, cloud-based quantum platforms, and the growing role of quantum computing in scientific discovery and industrial innovation. The book also discusses ethical, security, and societal implications associated with the advancement of quantum technologies. The primary objective of this book is to equip readers with both foundational knowledge and practical insights into the field of quantum computing. Through a balanced presentation of theory, applications, case studies, and emerging research trends, the book seeks to inspire curiosity, innovation, and further exploration in this transformative domain. We extend our sincere gratitude to the authors, reviewers, researchers, academicians, and industry experts whose valuable contributions have enriched the quality and relevance of this publication. Their dedication and expertise have helped create a resource that reflects the latest developments and future possibilities of quantum computing. We hope that Quantum Computing will serve as a valuable guide for students, educators, researchers, and technology professionals seeking to understand the next frontier of computational science. May this book inspire new ideas, foster innovation, and contribute to the advancement of knowledge in one of the most exciting technological revolutions of our time. Mr. M. Rajeshkumar Dr. R. Mariappan Mr. E.Janakiraman,