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Mathematics of quantum computation. (English) Zbl 1077.81018
Computational Mathematics Series. Boca Raton, FL: Chapman & Hall/ CRC (ISBN 978-1-58488-282-4/hbk; 978-0-367-39635-0/pbk; 978-0-429-12279-8/ebook). xvi, 429 p. (2002).
To some extent, this book presents a state-of-the-art of quantum computing which becomes a reality as the miniaturization of electronic circuits and chips increases. It is a collection of fifteen papers written by twenty-three authors, which can be clustered into nine parts: Quantum Entanglement (Ch 1,2 and 3), Universality of Quantum Gates (Ch 4), Quantum Search Algorithms (Ch 5, 6, and 7), Quantum Computational Complexity (Ch 8), Quantum Error-Correcting Codes (Ch 9 and 10), Quantum Computing Algebraic and Geometric Structures (Ch 11 and 12), Quantum Teleportation (Ch 13), Quantum Secure Communication and Quantum Cryptography (CH 14), Commentary on Quantum Computing (Ch 15). It is not a comprehensive textboook and it is intended to quantum mechanically literate readers only.
In a more detailed manner, the topics addressed by the book are the following: Contents:
Quantum entanglement: Ch. 1: Jean-Luc Brylinski, Algebraic measures of entanglement (3–23); Ch. 2: Berthold-Georg Englert and Nasser Metwally, Kinematics of qubit pairs (25–75); Ch. 3: David A. Meyer and Noland Wallach, Invariants for multiple qubits: the case of 3 qubits (77–97).
Universality of quantum gates: Ch. 4: Jean-Luc Brylinski and Ranee Brylinski, Universal quantum gates (101–116).
Quantum search algorithms: Ch. 5: Lov K. Grover and Anirvan M. Sengupta, From coupled pendulums to quantum search (119–134); Ch. 6: Goong Chen, Stephen A. Fulling and Jeesen Chen, Generalization of Grover’s algorithm to multiobject search in quantum computing. I. Continuous time and discrete time (135–160); Ch. 7: Goong Chen and Shunhua Sun, Generalization of Grover’s algorithm to multiobject search in quantum computing. II. General unitary transformations (161–168).
Quantum computational complexity: Ch. 8: Stephen A. Fenner, Counting complexity and quantum computation (171–219).
Quantum error-correcting codes: Ch. 9: Markus Grassl, Algorithmic aspects of quantum error-correcting codes (223–252); Ch. 10: Andreas Klappenecker and Martin Rötteler, Clifford codes (253–273) .
Quantum computing algebraic and geometric structures: Ch. 11: Jean-Luc Brylinski and Ranee Brylinski, Invariant polynomial functions on $$k$$ qudits (277–286); Ch. 12: Michael H. Freedman, David A. Meyer and Feng Luo, $$Z_2$$-systolic freedom and quantum codes (287–320).
Quantum teleportation: Ch. 13: Kishore T. Kapale and M. Suhail Zubairy, Quantum teleportation (323–355).
Quantum secure communication and quantum cryptography: Ch. 14: Almut Beige, Berthold-Georg Englert, Christian Kurtsiefer and Harald Weinfurter, Communicating with qubit pairs (359–401).
Commentary on quantum computing: Ch. 15: Stephen A. Fulling, Transgressing the boundaries of quantum computation: a contribution to the hermeneutics of the NMR paradigm (405–419).
The book is easy to read, the notations are consistent from a chapter to another one, but the reference lists are not. If I were the editor I would begin the book with an introductory chapter (fifteeen-twenty pages) on the basics of quantum computation. I would suggest to the new reader to begin with the chapters 13 and 15.

##### MSC:
 81P68 Quantum computation 68Q05 Models of computation (Turing machines, etc.) (MSC2010) 68-06 Proceedings, conferences, collections, etc. pertaining to computer science 81-06 Proceedings, conferences, collections, etc. pertaining to quantum theory 00B15 Collections of articles of miscellaneous specific interest
##### Keywords:
Quantum computation
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