INTRODUCTION TO QUANTUM COMPUTING PAPER

If you are attempting to enter the arena of quantum computation, and perhaps need a primer before tackling Nielsen and Chuang's standard text, then I would certainly consider this book. It conveys many of the illusive elements of quantum computation in an extremely clear and concise manner. When I first was getting into quantum computing, this book served as a Rosetta stone for many of the concepts that were at first foreign to me.Possessing a good knowledge of linear algebra, something that would be covered in a undergraduate course perhaps, is fairly important to ensure you get as much as you can out of reading this text. I've found this book to be an outstanding resource for learning many of the core concepts of quantum computation with an emphasis toward algorithm and circuit design.The layout of the book is as follows:1.) Introduction & Background:Here we obtain a high level view of the basics of quantum computation. Specific attention is given to various formulations of the Mach-Zehnder interferometer and how individual photons behave within this device. This lays the framework for the computational aspects related to quantum theory, and also develops enables the reader to develop an intuition early on for some of the topics ahead in the book.Preliminary concepts of computer science are also presented here with attention directed toward circuit diagrams, reversible computation, and matrix representations of computational gates.2.) Linear Algebra and the Dirac Notation:This section assumes the reader has a solid background in elementary linear algebra, and builds on top of those concepts. The Dirac notation formalism is presented and described incrementally with lots of good examples and explicit examples. One of the nice things about this book, is that it provides the reader with aThe book progresses to cover important linear algebraic concepts that are integral for comprehending the rest of the text. These include the spectral theorem, POVMs, tensor products, Schmidt decomposition, etc. Along the way, the authors provide examples (without the solutions unfortunately) to many problems to allow the reader to practice what they have read.3.) Qubits and the Framework of Quantum MechanicsThis chapter provides a gentle introduction into the notion of what a qubit is, and how one can visualize it via the Bloch sphere. It also covers how such an entity evolves through time, and how systems consisting of more than one qubit behave. I especially thought steady incline from classical to quantum notions of bit via the Bloch sphere was particularly lucid.This chapter also consists of the postulates of quantum mechanics presented in a computationally approachable manner. Some of the other mathematical necessities are also presented in this chapter, such as partial trace4.) A Quantum Model of ComputationHere, the prior notions are brought together to consider how one may perform computation on these entities. For instance, the classical/quantum gate and circuit models are compared and contrasted.5.) Superdense Coding and Quantum TeleportationThis chapter is fairly short, but delivers exactly what is says it will. The superdense protocol is conceptually introduced accompanied with an example. Following this is an explanation and application of quantum teleportation. The nice thing about this specific presentation is that a circuit model is pictorially represented for both protocols.6.) Introductory Quantum AlgorithmsHere the book considers some of the very well-known quantum algorithms in the literature. Specifically they cover, Deutsch's algorithm followed by the Deutsch-Jozsa and finally Simon's algorithm. This is a natural way that the majority of books approach presenting quantum algorithms since the difficulty level increases for each one.I personally like the treatment of the Deutsch-Jozsa algorithm compared to many others I've seen in other textbooks on quantum computing. It's accompanied by considering specific cases of the algorithm, which really brings everything into perspective. It was especially helpful to me when I was just starting out.7.) Algorithms with Superpolynomial Speed-upFor those especially interested in quantum algorithms, this chapter is of particular interest. They begin by providing the preliminary tools to approach this topic, quantum Fourier transform and quantum phase estimation, and proceed to a number of interesting algorithms. These range from eigenvalue estimation to the order-finding problem.8.) Algorithms Based on Amplitude AmplificationThe most famous quantum algorithm that manipulates amplitude amplification is Grover's algorithm, which is covered in this section. As one may assume from the title, the authors do go into more detail than most other accounts of Grover's algorithm on specifically how amplitude amplification plays a role in the algorithm.General quantum searching algorithms along with quantum counting are also briefly covered in the last few sections of the chapter.9.) Quantum Computation Complexity Theory and Lower BoundsFor those interested in the theoretical computer science aspects of quantum computing, this chapter serves as a very nice introduction. It is written in mind for someone who has had no prior exposure to such topics, and as a result introduces some of the notions of classical complexity theory before proceeding to the quantum case.The remainder of the chapter provides a nice overview of where certain problems are to be found in various complexity classes. The chapter is relatively brief, and does not go into a great amount of detail for someone who is more interested on the theoretical side of quantum computing. The chapter is still a very nice overview and synopsis.10.) Quantum Error CorrectionThe final chapter focuses on fault-tolerant quantum computation. The use and purpose of error-correcting codes are presented along with their quantum counterparts.Appendices:The appendices contain some further elaboration on the aforementioned topics such as ways to analyze probabilistic algorithms, distinguishing between two quantum states, etc.Overall, this is an exceptional introductory book on quantum computation. Emphasis is again, toward quantum algorithms and quantum circuits. I'd certainly recommend this text to anyone who wants to gain insight into the world of quantum computation. It has also served as a terrific reference source and is still today one of the best texts on the subject.

This book is geared for the reader who has an undergraduate education in a technical field and who has a solid background in linear algebra, including vector spaces and inner products. Prior familiarity with topics such as eigendecomposition and more advanced mathematical topics is not required. The book reviews all of the necessary additional material. There are some places in the book where group theory is referred to, but these sections of the book are self-contained so that the reader can skip them if needed. It is a very accessible introduction to a complex subject that is fairly detailed and complete. Exercises are integrated into the body of the text. Each exercise is designed to illustrate a particular concept, fill in the details of a calculation or proof, or to show how concepts in the book can be generalized or extended. The following is a brief overview of the book:1. Introduction and Background - Presents some fundamental notions of computation theory and quantum physics that will form the basis of what follows.2. Linear Algebra and the Dirac Notation - Familiarizes the reader with the algebraic notation used in quantum mechanics, reminds the reader of some basic facts about complex vector spaces, and introduces some notions that may not have been covered in an elementary linear algebra course.3. Qubits and the Framework of quantum Mechanics - Introduces the framework of quantum mechanics as it pertains to the types of systems that are considered in the book. Here the author also introduces the notion of a quantum bit or "qubit", which is a fundamental concept in quantum computing.4. A Quantum Model of Computation - The circuit model of classical computation can be generalized to a model of quantum circuits. In such a model you have logical qubits carried along "wires" and quantum gates that act on the qubits. For convenience, the discussion is limited to unitary quantum gates.5. Superdense Coding and Quantum Teleportation - Looks at our first protocols for quantum information. Examines two communication protocols that can be implemented using the tools which can be implemented using the tools developed in previous chapters. These protocols are known as superdense coding and quantum teleportation. Both of these are inherently quantum - there are no classical protocols that behave in the same way as these.6. Introductory Quantum Algorithms - Describes some of the early quantum algorithms that are simple and illustrate the main ingredients behind the more useful and powerful quantum algorithms described in subsequent chapters. Since quantum algorithms share some features with classical probabilistic algorithms, the chapter starts with a comparison of the two algorithmic paradigms.7. Algorithms with Superpolynomial Speed-Up - Examines one of two main classes of algorithms: quantum algorithms that solve problems with a complexity that is superpolynomially less than the complexity of the best-known classical algorithm for the same problem. That is, the complexity of the best-known classical algorithm cannot be bounded above by any poynomial in the complexity of the quantum algorithm. The chapter starts off by studying the problem of quantum phase estimation, which leads naturally to the Quantum Fourier Transform (QFT).8. Algorithms Based on Amplitude Amplification - Discusses a broadly applicable quantum algorithm - quantum search - that provides a polynomial speed-up over the best-known classical algorithms for a wide class of important problems.9. Quantum Computational Complexity Theory and Lower Bounds - Quantum computers seem to be more powerful than classical computers for certain problems. However, there are limits on the power of quantum computers. Since a classical computer can simulate a quantum one, a quantum computer can only compute the same set of functions that a classical computer can. This chapter examines this and some related issues.10. Quantum Error Correction - Quantum computers are more susceptible to errors than classical digital computers because quantum mechanical systems are more delicate and more difficult to control. If large-scale quantum computers are to be possible, a theory of quantum error correction is needed. This is the issue discussed in this chapter.Overall, I found this book well suited to self-study, particularly for someone with an engineering background. Highly recommended.

great resource ...I never depend on a single book to learn a subject on my own ...I recommned the one by Mermin and the one by Rieffel

I would not call this an "introduction" despite the title. Instead, I would call this an excellent summary reference. It covers all the important topics, but very tersely -- and I've even found tidbits (like how to extend Simon's to an arbitrary subgroup) that are not even in the Mike & Ike QC/QI "bible." In general, it covers less material than Mike & Ike though.I tried to start with this text as an "introduction" but quickly discarded it, only to return to it now months later finding it a superb reference after learning the subject material from other books.

The book certainly gives a nice and gentle introduction. But about half through it, I become more and more disappointed:- It is often not clear which theorems and statements are proved and what is merely cited from the literature. This isn't a problem at the beginning, but becomes a source of confusion in the second half of the book.- There is a large bibliography, but as far as I could see, it is very seldom referenced where it would make most sense.- Some statements are really careless. Many algorithms talk about registers without even specifying them. E.g. Simon's algorithm.

Covered a lot of interesting material--I'm a physics undergrad, and this was a very interesting excursion into computer science, that I could relate to (mainly the QM but also the mathematics and motivation). Overall, very good book--don't know how it compares to Chiang and Kiesler's grad version.