Fujii, Keisuke Quantum computation with topological codes. From qubit to topological fault-tolerance. (English) Zbl 1339.81005 SpringerBriefs in Mathematical Physics 8. Singapore: Springer (ISBN 978-981-287-995-0/pbk; 978-981-287-996-7/ebook). x, 138 p. (2015). The book addresses topological quantum computation from the information theoretic perspective, namely focusing on topological fault-tolerant quantum computation using the surface code.Chapter 1 provides for a brief introduction to quantum computation, including the concept of a quantum bit (qubit), elementary operations, the quantum circuit diagrams and multiple qubit gates, as well as fundamental quantum algorithms such as the Solovay-Kitaev algorithm, indirect measurement and the Haddamard test, phase estimation, quantum Fourier transform and Shor’s prime factoring algorithm as well as Aharonov-Jones-Landau algorithm for approximating the Jones polynomial. The final section of Chapter 1 ends with a brief review of quantum noise.Chapters 2 and 3 introduce the stabilizer formalism and the topological stabilizer codes, including the concept of surface code (introduced in Chapter 3), these are two key chapters that lay down the groundwork for the last two and main chapters of the book: Chapter 4, which addresses topological quantum computation with surface codes, and Chapter 5, which deals with topologically protected measurement-based quantum computation.It is important to stress that the book itself does not cover the full scope of topological quantum computation, a point that is stressed by the author. The book’s focus is on the quantum information aspect of topological quantum computation.In terms of connections with main textbooks on Quantum Computation and Quantum Information, this work is a good companion to M. A. Nielsen and I. L. Chuang’s [Quantum computation and quantum information. Cambridge: Cambridge University Press (2000; Zbl 1049.81015); 10th anniversary edition (2010; Zbl 1288.81001)] as well as J. K. Pachos’ [Introduction to topological quantum computation. Cambridge: Cambridge University Press (2012; Zbl 1247.81003)], a point that is also stressed by the author.However, the present work “Quantum computation with topological codes”, stands on its own in the sense that the author provides all the elements needed for the reader to understand the main points of the formalism and information theoretic aspects of topological quantum computation.This is largely achieved by the fact that the author uses many familiar examples and shows the correspondence between the quantum circuit formalism and the topological diagrams. This is achieved, in particular, in Chapter 4 where topological quantum computation on the surface code is described in terms of the circuit-based model and the topological diagrams are introduced to express the logical operations on the surface code diagrammatically.Given the way in which it is written the work provides a good reference for quantum computation and quantum information courses, allowing for students to become familiar with major points on the quantum information theoretical aspects of topological quantum computation and the advantages of topological quantum computation for quantum noise resistance.The book is also of interest to anyone doing research on quantum computation, quantum information and quantum error correction. Reviewer: Carlos Pedro Gonçalves (Lisboa) Cited in 3 Documents MSC: 81-02 Research exposition (monographs, survey articles) pertaining to quantum theory 81P45 Quantum information, communication, networks (quantum-theoretic aspects) 81P68 Quantum computation 68Q12 Quantum algorithms and complexity in the theory of computing 81P70 Quantum coding (general) Keywords:fault-tolerant quantum computation; topological quantum computation; surface code; quantum information Citations:Zbl 1049.81015; Zbl 1288.81001; Zbl 1247.81003 PDFBibTeX XMLCite \textit{K. Fujii}, Quantum computation with topological codes. From qubit to topological fault-tolerance. Singapore: Springer (2015; Zbl 1339.81005) Full Text: DOI arXiv