My first acquaintance with this book came from 
a copy which I ordered through interlibrary loan
after seeing favorable comments on the internet.
The loan period was only two weeks, so I wasn't able
to study this 600-page book in detail.  
But I learned quite a bit just by skimming it.
After I saw that it was a book that would repay study, 
I purchased it. 

The first chapter of 58 pages nicely introduces many 
of the important ideas, leaving the more difficult details
to later chapters.  
For example, I learned about quantum
teleportation, which I had never understood from popular accounts.

I read it from cover to cover and was able to follow almost 
all of it in detail.  Since I read it as someone learning this
material for the first time, I'll review it from a student's 
perspective.  A much longer review discussing technical issues 
is available on my web site. 

Chapter 2 gives a nice summary of basic quantum mechanics. 
It includes an introduction to necessary concepts from abstract
linear algebra, including important specific applications
(e.g., the Schmidt decomposition) which are not likely to be 
covered even in advanced linear algebra courses.

The third chapter gives an introduction to computer science concepts.
This gives a conceptual framework within which to present the ideas
of quantum computation.  More material is included here
than is necessary to understand the rest of the book.  Readers
may find it efficient to skim this chapter initially and return  
for more detail when necessary.

The next three chapters present the essentials of quantum circuits,
the quantum Fourier transform, and quantum search algorithms.
Here there is perhaps room for a little improvement.
I thought that important details were sometimes omitted from the
exposition, and I occasionally had to go to the original literature
to understand the ideas.    

Also, there is a bad misuse of the "Big-O" notation throughout these
chapters, startling 
in a book so generally carefully written.  Sophisticated readers 
will take this into stride, but it might demoralize beginners. 
For details, see the longer review on my web site.

The mathematics of quantum computation is easy compared 
to the problems of physically realizing it.
Chapter 7 gives an extensive discussion of these problems
and various proposals for overcoming them.  This concludes the
``quantum computation section of the book, which is a little more
than half of the 600-odd pages.

The rest deals with quantum information theory.  This is presented
in less detail than the quantum computation chapters, and demands
more from the reader.  A summary of classical information
theory is included, with sketches of proofs of important results. 
I found this very helpful in refreshing my memory of Khinchin's 
book on information theory, which I read decades ago.

Some of the more complicated proofs of quantum information results are only
sketched.  I didn't get as much from the quantum information section
of the book as from the quantum computation section.  I think it
gives a useful overview of the field, but if I wanted to learn
quantum information in detail, I would look for a book dedicated to
this topic, perhaps reading Nielsen/Chuang first as an introduction. 

The book concludes with a 12-page introduction to quantum cryptography.
I couldn't follow this section in detail.  
Perhaps it could be followed with enough work, but I wasn't motivated.  
I imagine that a proper treatment of cryptography would require many more
than 12 pages.  Again, if I wanted to learn this material,
I would seek an expository text dedicated to it. 
 
In summary, this is an exceptionally fine text which can 
be read on many levels.  The 58-page overview of quantum computation
should be comprehensible to anyone familiar with 
the basic ideas of quantum mechanics.
The rest of the book may possibly be readable with great effort
by well-prepared undergraduates, but I think a graduate-level
background in quantum mechanics and linear algebra would be 
more realistic prerequisites, and also more efficient.  These 
prerequites will have to be mastered anyway for anyone who wants
to work in a field dependent on quantum theory. 
Those who lack the prerequisites may still be able to get a
feel for the problems of quantum computation and information 
from the book, even if the details seem too difficult.  

Although this is a serious book
suitable for obtaining a professional knowledge of its subjects,
it is unusually carefully written in an expository style.  
There are many exercises interspersed with the exposition,
but no solutions are provided.  Most of them should be solvable
on sight by anyone following the presentation, so they provide
a useful check on one's understanding of the material.  
(I am a professional mathematician; students may
find the exercises less easy.) 

Each chapter ends with 
"History and further reading'' sections,  
often extensive.  I found these very helpful. 

Copyright December 17, 2007