Here's Scott Aaronson's answer to a similar question on his blog a couple of weeks ago:
Yes, Nielsen and Chuang is the standard textbook for quantum computing; it’s excellent (even though 16 years old by now). If you wanted to start with something shorter and more introductory, you could try David Mermin’s “Introduction to Quantum Computer Science.”
Lipton wrote a pretty decent and short intro to QC that builds the theory a little differently from usual. It is aimed more towards Mathematicians and Computer Scientists than Physicists, I believe, but I think it is pretty understandable in general.
I think Sipser is probably the best introduction to Computation if you do not have a background in Computability and Complexity. Arora and Barak's text has a much larger breadth and is much more detailed as far as Complexity goes. It is the more appropriate of the two if you are just interested in Computational Complexity.
Yes, Nielsen and Chuang is the standard textbook for quantum computing; it’s excellent (even though 16 years old by now). If you wanted to start with something shorter and more introductory, you could try David Mermin’s “Introduction to Quantum Computer Science.”