Classical computation follows the model of A. Turing,-- strings of bits, i.e., 0s and 1s; a mathematical model, now called the Turing mashine. Analogues based instead on two-level quantum systems were suggested in the 1980ties by R.P. Feynman and D. Deutsch. But it wasn't until Peter Shor's qubit-factoring algorithm in the mid 1990ties that the subject really took off, and really caught the attention of the math community. That there is a polynomial factoring algorithm shook the encryption community as well, for obvious reasons. New elements of thinking in the quantum realm, and not part of the classical framework, include superposition of (quantum) states, and (quantum) coherence. This makes a drastic change in the whole theoretical framework when one passes from the classical notion of bit-registers to that of qubit-registers. In passing from logic gates to quantum gates(unitary matrices), the concept of switching networks changes. It introduces new challenges, and new truely exciting opportunities. It is not easy for authors to make everyone happy;-- this is especially so in a new field,--one which has grabbed headlines, and one which is at the same time interdisiplinary. In this case, the authors succeed as well as anyone, I believe.-- This lovely book covers several of the appropriate areas of physics (quantum theory, (some) experiment...), of computer science (the mathematical side of the subject), and of math (operators in Hilbert space, and the theory of algorithms);-- each member of the particular scientific specialty has very definite ideas of his/her own subject,-- and that of the others. Nonetheless, in this readers opinion, the two authors did a great job;-- they explain math to the physics community,-- and they sucessfully teach quantum theory and theoretical CS to mathematicians. The book is suitable for grad students: has lots of great exercises, but it could perhaps have used some more worked examples. (Fortunately they can be found in other books on quantum computation.) The Nielsen-Chuang book is most certainly a great entry for students into this exciting new subject. There are other books,-- but they, for the most part, take a more narrow view. The material in Nielsen-Chuang is timeless,-- and I expect the book will also be popular ten years from now.
