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Journal of Physics A Mathematical and Theoretical

Principles of quantum computation and information volume II

Pieter Kok

Published Sep 18, 2007• 7 citations
Publisher

Abstract

Any new textbook in quantum information has some pretty strong competition to contend with. Not only is there the classic text by Nielsen and Chuang from 2000 [1], but also John Preskill's lecture notes, available for free online [2]. Nevertheless, a proper textbook seems more enduring than online notes, and the field has progressed considerably in the seven years since Nielsen and Chuang was published. A new textbook is a great opportunity to give a snapshot of our current state of knowledge in quantum information. Therein also lies a problem: The field has expanded so much that it is impossible to cover everything at the undergraduate level. Quantum information theory is relevant to an extremely large portion of physics, from solid state and condensed matter physics to particle physics. Every discipline that has some relation to quantum mechanics is affected by our understanding of quantum information theory. Those who wish to write a book on quantum information therefore have to make some profound choices: Do you keep the ultimate aim of a quantum computer in mind, or do you focus on quantum communication and precision measurements as well? Do you describe how to build a quantum computer with all possible physical systems or do you present only the underlying principles? Do you include only the tried and tested ideas, or will you also explore more speculative directions? You don't have to take a black-or-white stance on these questions, but how you approach them will profoundly determine the character of your book. The authors of `Principles of Quantum Computation and Information (Volume II: Basic Tools and Special Topics)' have chosen to focus on the construction of quantum computers, but restrict themselves mainly to general techniques. Only in the last chapter do they explicitly address the issues that arise in the different implementations. The book is the second volume in a series, and consists of four chapters (labelled 5 to 8) called 'Quantum Information Theory', 'Decoherence', 'Quantum Error Correction', and 'First Experimental Implementations'. The first volume covers the basics of classical computation, quantum mechanics, quantum computation, and quantum communication. Chapter five starts with the density matrix formalism, and proceeds with the development of the Kraus representation, POVMs, von Neuman entropy, quantum data compression, the Holevo bound, the partial transpose criterion, and it ends with a very nice section on the various entropies that play a role in modern physics. This includes not only the thermodynamical and statistical entropy, but also the dynamical Kolmogorov–Sinai entropy, which is used in quantum chaos in chapter 6. On the whole, I think that this is a really clear and well-presented chapter. A minor drawback is that the concept of CP maps is not explained as well as it could have been, for example by relating it to the partial transpose criterion. Chapter six continues with the high standard set in chapter five, and presents a very thorough exposition of decoherence in general. It introduces the different decoherence channels, and gives truly excellent explanations of the master equation (tied in with the Kraus representation), quantum jumps, and the quantum trajectory formalism. It also has an elegant explanation for the sensitivity of Schroedinger cats to decoherence. I do miss a proper section on the fidelity of a quantum state, though. The chapter ends with two sections on quantum chaos. Since the authors are experts in this fascinating area, this is a welcome addition to the canon of topics typically covered in quantum information. Unfortunately, the section is quite hard to follow, and as a result it is a bit of a missed opportunity. There is a section on chaos in the first volume of this series, and this may provide the required background. However, for readers who posess only volume II this is of little use. Chapter seven on quantum error correction is disappointing, and I have the feeling that the authors went through the motions without a real passion for the subject matter. The chapter describes various error correction codes, including Hamming codes and CSS codes, but it is virtually silent on fault tolerance; it does not give examples of universal sets of fault tolerant gates, and it does not mention the Solovay–Kitaev theorem. Also, it does not present the stabilizer formalism. All of these are serious omissions in a textbook on quantum information theory. Chapter eight gives a rough sketch of the early simulations and implementations of quantum gates. The readers of this journal will have no trouble following this chapter, but the undergraduate in computer science or mathematics will be completely lost. Most (but not all) physics terms do get a brief explanation, but I doubt whether that is enough to keep non-physicists on board. The chapter covers NMR, cavity QED, ion traps, solid state qubits, and optical implementations of quantum communication. I would have liked to see a more bold choice for the topics covered in the last chapter. For example, whereas liquid-state NMR was an important step in the development of quantum technologies, and many current techniques were invented for it, it does no longer play a role in the design of quantum computers. It would have been better to introduce these techniques in a section on condensed matter systems. Also, as a snapshot of our current state of knowledge in quantum information, I really miss extensive sections on the one-way model of quantum computing [3] and topological quantum computing [4]. In conclusion, the second volume of 'Principles of Quantum Computation and Information' is a partial success. The first two chapters are very good, and I would happily pay £22 for these two chapters alone. However, for a text on quantum error correction the reader is better off with Nielsen and Chuang or Preskill's lecture notes. If the reader wants an overview of quantum information in specific physical systems, there are a host of review articles to choose from, which give more details and are generally more accessible. References [1]M A Nielsen and I L Chuang 2000 Quantum Computation and Quantum Information (Cambridge University Press) [2] J Preskill, http://www.theory.caltech.edu/~preskill/ph229/ [3] R Raussendorf and H J Briegel 2001 A One-Way Quantum Computer Phys. Rev. Lett. 86 5188 [4] A Yu Kitaev 2003 Fault-tolerant quantum computation by anyons Ann. Phys. 303 2

DOI
10.1088/1751-8113/40/40/b03

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Any new textbook in quantum information has some pretty strong competition to contend with. Not only is there the classic text by Nielsen and Chuang from 2000 [1], but also John Preskill's lecture notes, available for free online [2]. Nevertheless, a proper textbook seems more enduring than online notes, and the field has progressed considerably in the seven years since Nielsen and Chuang was published. A new textbook is a great opportunity to give a snapshot of our current state of knowledge in quantum information. Therein also lies a problem: The field has expanded so much that it is impossible to cover everything at the undergraduate level. Quantum information theory is relevant to an extremely large portion of physics, from solid state and condensed matter physics to particle physics. Every discipline that has some relation to quantum mechanics is affected by our understanding of quantum information theory. Those who wish to write a book on quantum information therefore have to make some profound choices: Do you keep the ultimate aim of a quantum computer in mind, or do you focus on quantum communication and precision measurements as well? Do you describe how to build a quantum computer with all possible physical systems or do you present only the underlying principles? Do you include only the tried and tested ideas, or will you also explore more speculative directions? You don't have to take a black-or-white stance on these questions, but how you approach them will profoundly determine the character of your book. The authors of `Principles of Quantum Computation and Information (Volume II: Basic Tools and Special Topics)' have chosen to focus on the construction of quantum computers, but restrict themselves mainly to general techniques. Only in the last chapter do they explicitly address the issues that arise in the different implementations. The book is the second volume in a series, and consists of four chapters (labelled 5 to 8) called 'Quantum Information Theory', 'Decoherence', 'Quantum Error Correction', and 'First Experimental Implementations'. The first volume covers the basics of classical computation, quantum mechanics, quantum computation, and quantum communication. Chapter five starts with the density matrix formalism, and proceeds with the development of the Kraus representation, POVMs, von Neuman entropy, quantum data compression, the Holevo bound, the partial transpose criterion, and it ends with a very nice section on the various entropies that play a role in modern physics. This includes not only the thermodynamical and statistical entropy, but also the dynamical Kolmogorov–Sinai entropy, which is used in quantum chaos in chapter 6. On the whole, I think that this is a really clear and well-presented chapter. A minor drawback is that the concept of CP maps is not explained as well as it could have been, for example by relating it to the partial transpose criterion. Chapter six continues with the high standard set in chapter five, and presents a very thorough exposition of decoherence in general. It introduces the different decoherence channels, and gives truly excellent explanations of the master equation (tied in with the Kraus representation), quantum jumps, and the quantum trajectory formalism. It also has an elegant explanation for the sensitivity of Schroedinger cats to decoherence. I do miss a proper section on the fidelity of a quantum state, though. The chapter ends with two sections on quantum chaos. Since the authors are experts in this fascinating area, this is a welcome addition to the canon of topics typically covered in quantum information. Unfortunately, the section is quite hard to follow, and as a result it is a bit of a missed opportunity. There is a section on chaos in the first volume of this series, and this may provide the required background. However, for readers who posess only volume II this is of little use. Chapter seven on quantum error correction is disappointing, and I have the feeling that the authors went through the motions without a real passion for the subject matter. The chapter describes various error correction codes, including Hamming codes and CSS codes, but it is virtually silent on fault tolerance; it does not give examples of universal sets of fault tolerant gates, and it does not mention the Solovay–Kitaev theorem. Also, it does not present the stabilizer formalism. All of these are serious omissions in a textbook on quantum information theory. Chapter eight gives a rough sketch of the early simulations and implementations of quantum gates. The readers of this journal will have no trouble following this chapter, but the undergraduate in computer science or mathematics will be completely lost. Most (but not all) physics terms do get a brief explanation, but I doubt whether that is enough to keep non-physicists on board. The chapter covers NMR, cavity QED, ion traps, solid state qubits, and optical implementations of quantum communication. I would have liked to see a more bold choice for the topics covered in the last chapter. For example, whereas liquid-state NMR was an important step in the development of quantum technologies, and many current techniques were invented for it, it does no longer play a role in the design of quantum computers. It would have been better to introduce these techniques in a section on condensed matter systems. Also, as a snapshot of our current state of knowledge in quantum information, I really miss extensive sections on the one-way model of quantum computing [3] and topological quantum computing [4]. In conclusion, the second volume of 'Principles of Quantum Computation and Information' is a partial success. The first two chapters are very good, and I would happily pay £22 for these two chapters alone. However, for a text on quantum error correction the reader is better off with Nielsen and Chuang or Preskill's lecture notes. If the reader wants an overview of quantum information in specific physical systems, there are a host of review articles to choose from, which give more details and are generally more accessible. References [1]M A Nielsen and I L Chuang 2000 Quantum Computation and Quantum Information (Cambridge University Press) [2] J Preskill, http://www.theory.caltech.edu/~preskill/ph229/ [3] R Raussendorf and H J Briegel 2001 A One-Way Quantum Computer Phys. Rev. Lett. 86 5188 [4] A Yu Kitaev 2003 Fault-tolerant quantum computation by anyons Ann. Phys. 303 2
Quantum physics allows entirely new forms of computation and cryptography, which could perform tasks currently impossible on classical devices, leading to an explosion of new algorithms, communications protocols and suggestions for physical implementations of all these ideas. As a result, quantum information has made the transition from an exotic research topic to part of mainstream undergraduate courses in physics. Based on years of teaching experience, this textbook builds from simple fundamental concepts to cover the essentials of the field. Aimed at physics undergraduate students with a basic background in quantum mechanics, it guides readers through theory and experiment, introducing all the central concepts without getting caught up in details. Worked examples and exercises make this useful as a self-study text for those who want a brief introduction before starting on more advanced books. Solutions are available online at www.cambridge.org/9781107014466.
Why yet another book on quantum information theory? Like many lecturers we began writing this text because none of the alternatives seemed quite right. This book is aimed squarely at undergraduate physics students who want a brief but reasonably thorough introduction to the exciting ideas of quantum information, including its applications in computation and communication. It is based on a short course we have taught to fourth-year students at Oxford University since 2004; for the most part it only assumes knowledge of elementary quantum mechanics and linear algebra, and so could even be taught to third-year undergraduates. A brief revision guide to quantum mechanics is provided as an appendix, which should cover any minor points that have been missed. As the title implies the book is structured in three parts, starting with the basics of quantum information and then applying this to quantum computation and quantum communication. Part I is self-contained, but contains only the barest hints of the exciting applications which attract many people to this field and so might prove unsatisfying on its own. Parts II and III draw heavily on Part I, but are largely independent of each other, and it would be perfectly possible to study only one of these two without the other. As this text is aimed at physics undergraduates, we believe that it is vital to cover experimental techniques, rather than merely presenting quantum information as a series of abstract quantum operations. We have, however, concentrated on the basic ideas underlying each approach, rather than worrying about particular experimental details.
This book is intended as an introductory textbook at graduate or advanced undergraduate level. Students with diverse backgrounds, in mathematics, physics and computer science, can use it to become acquainted with the quantum theory of information. Its scope is deliberately chosen to be not too broad, while being self-consistent and usable for a semester-long course for beginners. The subject is presented using a balanced mix of intuitive arguments and rigorous formalism. The emphasis is not on the physical interpretation, but rather on the formal and mathematical features of quantum information theory, in order to be accessible from different backgrounds, the only requirement being a basic knowledge of linear algebra. In particular, a background in quantum mechanics is not needed, since quantum theory is used as a mere mathematical framework without entering into physical interpretations. The book introduces all necessary concepts from quantum theory and classical information theory, to cover the elements of quantum Shannon theory: quantum states and channels, data compression and entropy, error correcting codes and channel capacities, as well as the basics of entanglement theory and quantum cryptography. Examples are provided to help the reader in understanding the different concepts, together with exercises at the end of each chapter.
We may be on the brink of a quantum transfor -mation driven by our increasing ability to realize, detect, control, and manipulate high-purity quantum states. For decades the quantum world has crept into our lives through modern information technology, the most visible side of science that permeates almost every aspect of our daily conduct. Yet quantumness in its full glory remains a promise for the future, as existing mainstream technologies are essentially based on properties of mixed quantum states realized in, e.g., metal–oxide–semiconductor heterostructures. The scientific backbone of current technologies combines quantum theory with information theory and the theory of computation, everything being sprinkled with a healthy dose of mathematics and brought into reality by remarkable engineering achievements. This multidisciplinary approach has enabled us to understand key physical phenomena and material properties, to communicate and process information efficiently, and, ultimately, to build functional devices that provide us with unprecedented convenience and mastery over nature, possibly with the price of being in danger to become the slaves of our computers and gadgets. However, our technological success is built on a combination of essentially distinct and largely independent scientific fields. Harnessing high-purity quantum states, on the other hand, sets these previously separated “continents of knowledge” on a collision path. The main goal of the last three chapters of this book is to provide a bird's eye view of the “collision zone” and prepare the reader to explore, enjoy, and perhaps exploit the newly raising mountains. We start with a brief introduction to basic concepts in quantum information, which also includes a summary of the operational formulation of quantum mechanics (the “natural” language of quantum information theory) and a synopsis of classical information theory ( Chapter 10 ). For a thorough treatment of concepts, problems, and techniques in quantum information theory the reader is referred to the books by Nielsen and Chuang [ 389 ] and Wilde [ 548 ]. The book by Watrous [ 530 ] focuses on the mathematical theory of quantum information. An undergraduate level introduction to this subject can be found, for example, in Schumacher and Westmoreland [ 458 ]. Some of the philosophical implications of the relationship between quantum mechanics and information theory are discussed in Timpson [ 505 ]. Our survey continues with Chapter 11 , where we discuss a few key aspects pertaining to quantum computation, which can be viewed as the quintessential quantum technology involving the controlled realization and manipulation of high-purity quantum states within a large Hilbert space. Finally, Chapter 12 focuses on topological quantum computation, a promising approach to overcoming the fundamental challenge associated with quantum decoherence based on harnessing the underlying robustness of topological quantum states. This also marks our final return to the realm of topological quantum matter.
Quantum computation and information is one of the most exciting developments in science and technology of the last twenty years. To achieve large scale quantum computers and communication networks it is essential not only to overcome noise in stored quantum information, but also in general faulty quantum operations. Scalable quantum computers require a far-reaching theory of fault-tolerant quantum computation. This comprehensive text, written by leading experts in the field, focuses on quantum error correction and thoroughly covers the theory as well as experimental and practical issues. The book is not limited to a single approach, but reviews many different methods to control quantum errors, including topological codes, dynamical decoupling and decoherence-free subspaces. Basic subjects as well as advanced theory and a survey of topics from cutting-edge research make this book invaluable both as a pedagogical introduction at the graduate level and as a reference for experts in quantum information science.
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.