Abstract This guide is based on the IBM Quantum course Fundamentals of Quantum Algorithms and explores the computational advantages enabled by quantum information. The focus is on what can be achieved using a single quantum computer, rather than distributed quantum systems. The primary objective of the course is to analyze how quantum computation can outperform classical computation in specific problem classes. In particular, the material examines quantum query algorithms, which provide a structured framework for understanding quantum speedups. Key themes include: The conceptual foundations of quantum computational advantage. The role of superposition, interference, and entanglement in algorithm design. Query complexity and how quantum algorithms reduce the number of required oracle calls compared to classical approaches. Foundational algorithms that demonstrate provable quantum speedups. The central guiding question is: What advantages can a quantum computer potentially offer over a classical computer? Through hands-on experimentation on IBM Q, the work connects theoretical analysis with practical implementation, enabling deeper insight into how quantum resources translate into measurable computational gains.
Abstract This guide is based on the IBM Quantum course Fundamentals of Quantum Algorithms and explores the computational advantages enabled by quantum information. The focus is on what can be achieved using a single quantum computer, rather than distributed quantum systems. The primary objective of the course is to analyze how quantum computation can outperform classical computation in specific problem classes. In particular, the material examines quantum query algorithms, which provide a structured framework for understanding quantum speedups. Key themes include: The conceptual foundations of quantum computational advantage. The role of superposition, interference, and entanglement in algorithm design. Query complexity and how quantum algorithms reduce the number of required oracle calls compared to classical approaches. Foundational algorithms that demonstrate provable quantum speedups. The central guiding question is: What advantages can a quantum computer potentially offer over a classical computer? Through hands-on experimentation on IBM Q, the work connects theoretical analysis with practical implementation, enabling deeper insight into how quantum resources translate into measurable computational gains.
Abstract This guide is based on the IBM Quantum course Quantum Algorithmic Foundations and addresses a central question in quantum computation: Do quantum algorithms offer provable advantages beyond the oracle (black-box) query model, in more standard computational settings where inputs are given explicitly? Answering this question requires building a rigorous theoretical foundation. The course begins with a detailed discussion of computational cost in both classical and quantum models, focusing on how complexity scales with input size rather than on absolute execution time. The emphasis is placed on algorithmic efficiency, independent of hardware improvements. To ground the discussion, the material draws heavily from computational number theory, including problems such as: Integer arithmetic Greatest common divisor (GCD) computation Integer factorization Although number theory represents a specific application domain, it provides a clear and powerful framework for analyzing computational complexity and illustrating the principles underlying quantum speedups. A key theme of the lesson is the distinction between algorithmic scalability and hardware performance. The primary concern is how the cost of an algorithm grows as problem instances become larger — a perspective that remains valid across technological generations. The course concludes by addressing a critically important concept: implementing classical computations on quantum computers. The significance of this idea is not to replace classical machines, but to use classical algorithms as subroutines within quantum procedures. This hybrid approach leverages decades of classical algorithmic research while opening new pathways toward quantum computational advantage. Through practical experimentation on IBM Q, this work connects theoretical foundations with hands-on quantum programming, reinforcing the structural principles that underlie advanced quantum algorithm design.
Quantum computing represents a transformative approach to computation that exploits the principles of quantum mechanics to process information in fundamentally different ways from classical computers. Over the past few decades, the field has evolved from a primarily theoretical discipline into an active area of research spanning computer science, mathematics, physics, information theory, and engineering. As quantum technologies continue to develop, understanding the fundamental computational principles underlying quantum algorithms has become increasingly important. Foundations of Quantum Computing: The Query Model Perspective presents a systematic introduction to quantum computing with particular emphasis on the query model, a powerful framework for studying the computational complexity of quantum algorithms. The query model provides a clean and mathematically tractable setting in which the number of queries made to an unknown function or oracle can be used as the primary measure of computational cost. This perspective offers valuable insights into the capabilities and limitations of quantum computation and provides a natural foundation for understanding quantum speedups. The book introduces the fundamental concepts required to study quantum computation, including quantum states, quantum gates, quantum circuits, measurement, superposition, entanglement, and quantum algorithms. These concepts are developed alongside the query-complexity framework, enabling readers to understand how quantum algorithms exploit interference and other quantum phenomena to reduce the number of queries required to solve computational problems. Particular attention is given to important quantum query algorithms and complexity results. Classical and quantum query complexities are discussed comparatively to illustrate the sources and significance of quantum advantages. Well-known examples such as the Deutsch–Jozsa algorithm, Grover's search algorithm, and related oracle-based problems provide concrete illustrations of how quantum procedures can outperform their classical counterparts. The discussion also introduces lower-bound techniques and fundamental principles that help establish the limitations of quantum query algorithms. The query model is not only useful for understanding individual algorithms but also serves as an important bridge between quantum algorithms and computational complexity theory. By focusing on query complexity, many problems can be examined independently of the details of a particular physical implementation. This abstraction makes the framework especially useful for developing rigorous proofs, comparing computational resources, and identifying the fundamental sources of quantum advantage. This book is intended for students, researchers, educators, and professionals interested in quantum computing and theoretical computer science. It can serve as an introductory reference for readers with a background in discrete mathematics, linear algebra, algorithms, or theoretical computer science, while also providing a foundation for readers who wish to pursue more advanced studies in quantum complexity theory and quantum algorithms. The chapters have been organized to progress from foundational concepts toward more advanced aspects of the quantum query model. Mathematical formulations are accompanied by explanations and examples to help readers connect abstract theoretical ideas with computational intuition. The broader objective is to provide readers with the tools necessary not only to understand existing quantum algorithms but also to reason about new problems from a query-complexity perspective. As quantum computing continues to mature, the ability to distinguish genuine computational advantages from implementation-specific effects will become increasingly significant. The query model provides one of the clearest theoretical environments for making this distinction. By studying quantum computation through this lens, readers can develop a deeper appreciation of both the power and the limitations of quantum algorithms It is hoped that Foundations of Quantum Computing: The Query Model Perspective will serve as a useful resource for learning, teaching, and research, and will encourage further exploration into the rapidly evolving landscape of quantum computation and quantum complexity theory.
Abstract This practical guide introduces hands-on quantum computing, based on IBM Q learning resources. The exercises were completed by the author on April 28, 2024, using IBM Q cloud resources. The course provides a practical framework for understanding and experimenting with modern quantum computers, covering realistic use cases, circuit execution, and best practices for operating quantum processors with 100 or more qubits. Participants explore the concept of quantum usefulness, which refers to the practical utility of quantum computers in solving real-world problems, distinguishing it from quantum advantage, which is the point where quantum systems outperform classical computers on meaningful tasks. The course emphasizes that while classical computers remain highly powerful and adaptive, the evolving capabilities of quantum processors offer unique opportunities in computation that can complement classical methods. Hands-on exercises guide participants through practical aspects of designing and executing quantum circuits, running experiments on simulators and cloud-accessible quantum hardware, and interpreting results. This approach bridges theoretical knowledge with applied skills, enabling students and researchers to explore the current potential and limitations of quantum computing. The guide also contextualizes quantum computing within decades of classical computational development, highlighting how quantum devices may transform computational paradigms in fields such as optimization, chemistry, cryptography, and machine learning.
Quantum computing has emerged as one of the most transformative technological developments of the twenty-first century. By exploiting the fundamental principles of quantum mechanics—namely superposition, entanglement, and measurement—quantum computers offer the potential to address problems that are beyond the practical reach of classical computing systems. As the field advances from theoretical exploration to real-world implementation, the demand for clear, structured, and comprehensive educational resources continues to grow. This book presents a systematic and accessible introduction to quantum computing, beginning with the essential concepts of classical information and computation and progressively advancing toward modern quantum algorithms and communication protocols. The initial chapters establish a strong foundation in classical bits, logic gates, Boolean algebra, reversible logic, and computational complexity, providing the conceptual bridge necessary to understand quantum computation. The discussion then moves to the quantum bit (qubit), examining its behavior through superposition, measurement, Bloch sphere representation, and physical realization. Special emphasis is placed on linear algebra as the mathematical framework underpinning quantum computing. Through this approach, readers are guided to develop an intuitive yet rigorous understanding of quantum states, quantum gates, and circuit-based computation. The text further extends these ideas to multi-qubit systems, quantum entanglement, universal gate sets, and quantum error correction, integrating theoretical principles with practical insights. To support experiential learning, the book incorporates hands-on exposure to leading quantum programming platforms, including IBM Quantum, OpenQASM, and Qiskit. These sections connect theory with practice, enabling readers to design, simulate, and execute quantum circuits on both simulators and real quantum hardware. Subsequent chapters explore key quantum communication protocols such as superdense coding, quantum teleportation, and quantum key distribution, followed by an in-depth treatment of foundational quantum algorithms including Deutsch–Jozsa, Grover’s search, Simon’s algorithm, and Shor’s factoring algorithm. This text is intended for undergraduate and postgraduate students, research scholars, and industry professionals seeking a coherent and structured pathway into quantum computing. No prior knowledge of quantum mechanics is assumed, although familiarity with linear algebra and classical computation will enhance comprehension. Each chapter is carefully organized to promote conceptual clarity, supported by illustrative examples, circuit diagrams, and concise summaries. Beyond explaining the principles and mechanisms of quantum computation, this book aims to inspire readers to engage with the future of the field—whether through academic research, industrial application, or interdisciplinary innovation. As quantum technologies continue to evolve, it is hoped that this text will serve as a dependable guide and a meaningful entry point into the quantum era.
Quantum computing has emerged as one of the most transformative technological developments of the twenty-first century. By exploiting the fundamental principles of quantum mechanics—namely superposition, entanglement, and measurement—quantum computers offer the potential to address problems that are beyond the practical reach of classical computing systems. As the field advances from theoretical exploration to real-world implementation, the demand for clear, structured, and comprehensive educational resources continues to grow. This book presents a systematic and accessible introduction to quantum computing, beginning with the essential concepts of classical information and computation and progressively advancing toward modern quantum algorithms and communication protocols. The initial chapters establish a strong foundation in classical bits, logic gates, Boolean algebra, reversible logic, and computational complexity, providing the conceptual bridge necessary to understand quantum computation. The discussion then moves to the quantum bit (qubit), examining its behavior through superposition, measurement, Bloch sphere representation, and physical realization. Special emphasis is placed on linear algebra as the mathematical framework underpinning quantum computing. Through this approach, readers are guided to develop an intuitive yet rigorous understanding of quantum states, quantum gates, and circuit-based computation. The text further extends these ideas to multi-qubit systems, quantum entanglement, universal gate sets, and quantum error correction, integrating theoretical principles with practical insights. To support experiential learning, the book incorporates hands-on exposure to leading quantum programming platforms, including IBM Quantum, OpenQASM, and Qiskit. These sections connect theory with practice, enabling readers to design, simulate, and execute quantum circuits on both simulators and real quantum hardware. Subsequent chapters explore key quantum communication protocols such as superdense coding, quantum teleportation, and quantum key distribution, followed by an in-depth treatment of foundational quantum algorithms including Deutsch–Jozsa, Grover’s search, Simon’s algorithm, and Shor’s factoring algorithm. This text is intended for undergraduate and postgraduate students, research scholars, and industry professionals seeking a coherent and structured pathway into quantum computing. No prior knowledge of quantum mechanics is assumed, although familiarity with linear algebra and classical computation will enhance comprehension. Each chapter is carefully organized to promote conceptual clarity, supported by illustrative examples, circuit diagrams, and concise summaries. Beyond explaining the principles and mechanisms of quantum computation, this book aims to inspire readers to engage with the future of the field—whether through academic research, industrial application, or interdisciplinary innovation. As quantum technologies continue to evolve, it is hoped that this text will serve as a dependable guide and a meaningful entry point into the quantum era.
In this thesis we address the question, what is the resource, or property, that enables the advantage of quantum computers?The theory of quantum computers dates back to the eighties, so one would think there already is an answer to this question.There are several proposed solutions, but to this date, there is no consensus on an answer.Primarily, the advantage of quantum computers is characterized by a speedup for certain computational problems.This speedup is measured by comparing quantum algorithms with the bestknown classical algorithms.For some algorithms we assume access to an object called oracle.The oracle computes a function, and the complexity of the oracle is of no concern.Instead, we count the number of queries to the oracle needed to solve the problem.Informally, the question we ask using an oracle is: if we can compute this function efficiently, what else could we then compute.However, using oracles while measuring a quantum speedup, we assume access to vastly different oracles residing in different models of computation.For our investigation of the speedup, we introduce a classical simulation framework that imitates quantum algorithms.The simulation suggests that the property enabling the potential quantum speedup is the ability to store, process, and retrieve information in an additional degree of freedom.We then theoretically verified that this is true for all problems that can be efficiently solved with a quantum computer.In parallel to this, we also see that quantum oracles sharply specify the information we can retrieve from the additional degree of freedom, while regular oracles do not.A regular oracle does not even allow for an extra degree of freedom.We conclude that comparing quantum with classical oracle query complexity bounds does not provide conclusive evidence for a quantum advantage.I want to thank Jan-Åke Larsson, without whom this research would not have been possible.If there is any magic in the world, it shows through Jan-Åke's ability to simultaneously be a good supervisor, colleague, and friend.I would also like to thank my co-supervisor Peter Jonsson, for many valuable discussions -not only restricted to theoretical computer science.To my friends and collogues that have challenged me to become better, especially Jonathan Jogenfors