Coding theory: Grobner bases, Pade approximation, and decoding of linear codes by J. B. Farr and S. Gao Some computational tools for estimating the parameters of algebraic geometry codes by G. L. Matthews An introduction to algebraic coding theory by H. N. Ward Recent results on $p$-ranks and Smith normal forms of some $2-(v,k,\lambda)$ designs by Q. Xiang Quantum computing: Quantum walks on graphs and quantum scattering theory by E. Feldman and M. Hillery A continuous variable Shor algorithm by S. J. Lomonaco, Jr. and L. H. Kauffman Entangled states of light by S. J. van Enk Entanglement beyond subsystems by L. Viola, H. Barnum, E. Knill, G. Ortiz, and R. Somma Generalized GHZ states and distributed quantum computing by A. Yimsiriwattana and S. J. Lomonaco, Jr.
Coding theory: Grobner bases, Pade approximation, and decoding of linear codes by J. B. Farr and S. Gao Some computational tools for estimating the parameters of algebraic geometry codes by G. L. Matthews An introduction to algebraic coding theory by H. N. Ward Recent results on p-ranks and Smith normal forms of some 2−(v,k,λ) designs by Q. Xiang Quantum computing: Quantum walks on graphs and quantum scattering theory by E. Feldman and M. Hillery A continuous variable Shor algorithm by S. J. Lomonaco, Jr. and L. H. Kauffman Entangled states of light by S. J. van Enk Entanglement beyond subsystems by L. Viola, H. Barnum, E. Knill, G. Ortiz, and R. Somma Generalized GHZ states and distributed quantum computing by A. Yimsiriwattana and S. J. Lomonaco, Jr.
Space searches with a quantum robot by P. Benioff Perturbation theory and numerical modeling of quantum logic operations with a large number of qubits by G. P. Berman, G. D. Doolen, D. I. Kamenev, G. V. Lopez, and V. I. Tsifrinovich Inconclusive rate with a positive operator valued measure by H. E. Brandt Quantum amplitude amplification and estimation by G. Brassard, P. Hoyer, M. Mosca, and A. Tapp Manipulating the entanglement of one copy of a two-particle pure entangled state by L. Hardy Geometric algebra in quantum information processing by T. F. Havel and C. J. L. Doran Quantum computing and the Jones polynomial by L. H. Kauffman Quantum hidden subgroup algorithms: A mathematical perspective by S. J. Lomonaco, Jr. and L. H. Kauffman Improved two-party and multi-party purification protocols by E. N. Maneva and J. A. Smolin Quantum games and quantum algorithms by D. A. Meyer A proof that measured data and equations of quantum mechanics can be linked only by guesswork by J. M. Myers and F. H. Madjid Quantum computation by geometrical means by J. Pachos Pauli exchange and quantum error correction by M. B. Ruskai Relative entropy in quantum information theory by B. Schumacher and M. D. Westmoreland An unentangled Gleason's theorem by N. R. Wallach Entangled chains by W. K. Wootters.
An invitation to quantum computation: A Rosetta stone for quantum mechanics with an introduction to quantum computation by S. J. Lomonaco, Jr. Qubit devices by H. E. Brandt Quantum algorithms and quantum complexity theory: Introduction to quantum algorithms by P. W. Shor Shor's quantum factoring algorithm by S. J. Lomonaco, Jr. Grover's quantum search algorithm by S. J. Lomonaco, Jr. A survey of quantum complexity theory by U. V. Vazirani Quantum error correcting codes and quantum cryptography: An introduction to quantum error correction by D. Gottesman A talk on quantum cryptography or how Alice outwits Eve by S. J. Lomonaco, Jr. More mathematical connections: Topological quantum codes and anyons by A. Kitaev Quantum topology and quantum computing by L. H. Kauffman An entangled tale of quantum entanglement by S. J. Lomonaco, Jr. Index.
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.
Acting within the framework of geometric quantum mechanics, an interpretation of quantum uncertainty is discussed in terms of Jacobi fields, and a connection with the theory of elliptic curves is outlined, via classical integrability of Schrödinger's dynamics and the cross-ratio interpretation of quantum transition probabilities. Furthermore, a thoroughly geometrical construction of all special unitary representations of the 3-strand braid group on the quantum 1-qubit space is given, and the connection of one of them with elliptic curves admitting complex multiplication automorphisms — the physically relevant one corresponding to the anharmonic ratio — is shown. Also, contact is made with the Temperley–Lieb algebra theoretic constructions of Kauffman and Lomonaco, and it is shown that the standard trace relative to one of the above representations computes the Jones polynomial for particular values of the parameter, for knots arising as closures of 3-strand braids. Subsequently, a geometric entanglement criterion (in terms of Segre embeddings) is discussed, together with a projective geometrical portrait for quantum 2-gates. Finally, Aravind's idea of describing quantum states via knot theory is critically analyzed, and a geometrical picture — involving a blend of SU(2)-representation theory, classical projective geometry, binary trees and Brunnian and Hopf links — is set up in order to describe successive measurements made upon generalized GHZ states, close in spirit to the quantum knot picture again devised by Kauffman and Lomonaco.