We start this chapter by reviewing how quantum states and spaces are described with Dirac notation. We then examine qubits, the quantum memory bits, and the states formed with multiple qubits. We next discuss quantum gates and their low-level processing of qubits. And, at last, we get on to quantum programming and QC applications. We explore the use of several software packages that simulate QC, or provide the programs to run on quantum computers. We explore an example of the latter using the IBM Quantum Platform. The assortment of examples and problems that follow includes simple circuits, Monte Carlo techniques, Schrödinger's cat, and quantum Fourier transforms.
We start this chapter by reviewing how quantum states and spaces are described with Dirac notation. We then examine qubits, the quantum memory bits, and the states formed with multiple qubits. We next discuss quantum gates and their low-level processing of qubits. And, at last, we get on to quantum programming and QC applications. We explore the use of several software packages that simulate QC, or provide the programs to run on quantum computers. We explore an example of the latter using the IBM Quantum Platform. The assortment of examples and problems that follow includes simple circuits, Monte Carlo techniques, Schrödinger's cat, and quantum Fourier transforms.
Quantum computing (QC) is an exciting new technology that completes the text with an application that relies heavily on the mathematics of linear algebra. There is more to explain than the conversion of common matrix notation into Dirac bra-ket notation. There is also the bizarre world of quantum physics that is inspiring a range of quantum information technologies. Qubits and their entangled superposition states are introduced to show that the “quantum advantage” stems from qubits storing exponentially more information that can be processed at exponentially faster rates. The process of preparing, modifying, and measuring qubit states is explained. Linear algebra skills are exercised by representing QC coding through the application of unitary operators to qubit states. QC algorithms are designed by diagramming quantum circuits, casting those circuits into sequences of operators applied to qubit states, and converting the operators into matrix forms. These are the tools of QC. Several algorithms that demonstrate a range of computational tasks from simple arithmetic through quantum Fourier analysis complete the chapter.
Quantum computing is an emerging form of computing that uses the rules of quantum physics to process information in ways that are far different from how computers do it now. Quantum computing uses quantum bits, or qubits, which can exist in superpositions of states and get entangled with each other, which is different from classical computing, which uses a bit that represents binary values (0 or 1). These characteristics allow quantum devices to execute parallel computations, enabling significant speedups for specific categories of problems considered intractable for classical machines. This chapter gives a thorough introduction to quantum computing, starting with a historical overview of its foundations in quantum physics and significant theoretical advancements. It analyzes fundamental ideas including superposition, entanglement, quantum interference, and the no-cloning theorem, explaining how these elements distinguish quantum computation from traditional approaches. The advancement of quantum hardware is examined through several architectures, trapped ions, superconducting circuits, photonics, and topological qubits, together with software frameworks like Qiskit, Cirq, and PennyLane, which support algorithm development and simulation. The chapter examines fundamental quantum algorithms, such as Grover’s algorithm for database search, Shor’s algorithm for integer factorization, and hybrid variational methods designed for noisy intermediate-scale quantum devices. The practical applications in domains including as encryption, optimization, machine learning, chemistry, and finance emphasize the real-world significance of quantum technology. Challenges like decoherence, noise, scalability, and algorithmic maturity are rigorously analyzed, followed by an anticipated discourse on future trajectories including fault-tolerant architectures, quantum networking, and the development of quantum cloud services. This chapter presents a fundamental introduction for students, researchers, and practitioners seeking to comprehend the essential principles, functionalities, and possibilities of quantum computing in the post–Moore’s law period.
These notes begin in Chapter 1 with a review of linear algebra and the postulates of quantum mechanics, leading to an explanation of single- and multi-qubit gates. Chapter 2 explores the challenge of constructing arbitrary quantum states from given initial states, and introduces circuits for building oracles. Chapter 3 presents foundational algorithms such as entanglement creation, quantum teleportation, Deutsch-Jozsa, Bernstein-Vazirani, and Simon's algorithm. Chapters 4 and 5 cover algorithms based on the quantum Fourier transform, including phase estimation, period finding, factoring, and logarithm computation. These chapters also include complexity analysis and detailed quantum circuits suitable for implementation in code. Chapter 6 introduces Grover's algorithm for quantum search and amplitude amplification, including its realization via Hamiltonian simulation and a method for derandomization. Chapter 7 discusses basic techniques for Hamiltonian simulation, such as Lie-Trotter decomposition, sparse Hamiltonians, and the linear combination of unitaries. It also provides example circuits for simulating Hamiltonians expressed as linear combinations of Pauli operators. Chapter 8 introduces variational quantum algorithms, and Chapter 9 presents an algorithm for simulating fermionic many-particle systems, with an emphasis on molecular Hamiltonians. It also outlines the key transformations needed to map a molecular Hamiltonian to a form suitable for simulation on a quantum computer.
This chapter provides a basic overview of Quantum Computers (QCs) used in Quantum Computing (Quacomp) applications. Classical computers utilize silicon-based semiconductor devices. In recent years integrated circuit lateral feature sizes have been reduced to 10 nm, increasing chip density; however, as these sizes progress to be reduced to 5 nm and smaller, silicon-based circuits become impractical. QCs make use of quantum phenomena such as entanglement and superposition to increase and improve the computation speed as compared to classical computers, enabling them to solve some computationally complex problems in practical time windows. At this juncture QCs are not able to act as replacements for general- purpose computers, but they can outperform classical computers in a set of specific computational problems, such as optimization problems or for identifying the best solution from a set of feasible solutions. This chapter covers three QC approaches: the circuit model (or gate model), the adiabatic quantum computation model, and the categorical quantum model. Quacomp has experienced relatively rapid development in recent years, starting with the basic ideation in the early 1980s, then continuing with early proof of concepts for hardware in the next two decades, and now, in the 2020s, with QC hardware that can implement over one-thousand qubits (e.g., [ TIL22 ], [ FEY82 ], [ LEU00 ], [ NEG06 ], [ PLA07 ], [ MON11 ], [ MON16 ], [ CAS23 ]). Figure 5.1 (courtesy of IBM) depicts recent illustrative achievements in practical QC development at IBM, a key player in this arena (in the fall of 2023, IBM® announced the Condor prototype, the world’s first universal QC with 1,121 qubits 1 ). The generation of QCs at the time of this writing is known as Noisy-Intermediate Scale Quantum (NISQ) devices (some proponents cite this as being the third generation). These NISQ QCs are typically cooled to cryogenic temperatures (around −273° C) in order to operate efficiently; these environmental conditions are challenging to maintain in other than laboratory settings. In practical terms, QCs remain of somewhat limited and specialized functionality and are prone to computational errors. Considering these limitations, Hybrid QC (HQC) 170 171 172 arrangements and Hybrid Quantum Applications (HQApps) endeavor to combine classical and QC systems in order to undertake calculations directed at various (but specific) optimization or related use cases; tasks such as preparing inputs for quantum algorithms, storing data, or interacting with the user are typically still performed using classical computers. The number of operations in a QC typically equates with the number of quantum gates and the number of qubits supported; however, the larger the number of gates, the more likely it is that errors, including decoherence, are introduced into the quantum state. NISQ entails a somewhat limited number of logical qubits and associated physical qubits to support error correction schemes. Fortunately, a category of algorithms running on NISQ devices can proceed with only a small number of qubits and exhibit some degree of noise resilience; typically, they are hybrid algorithms, with a portion of the computational steps performed on a quantum device and with other computational steps running on a conventional computer. The bottom line is that with NISQ devices, the number of operations is typically moderate at this juncture, thus the scope and type of algorithms that can be implemented is relatively limited. However, there is research that shows that even these QCs can outperform conventional computers on a set of mathematical problems (e.g., [ PRE18 ], [ BRO19 ], [ MAD22 ]). The next step in the evolution is the interconnection of QCs over a distance network, as discussed in Chapter 6 . A classification of fundamental QC technologies has been proposed the European standards body CEN-CENELEC, specifically the Focus Group on Quantum Technologies (FGQT), which was established in June 2020 to coordinate and support the development of standards relevant for European industry and research [ OVA22 ]. See Table 5.1 .
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.