General background: Quantum computing (theory) is at the intersection of math, physics and computer science. (Experiment also can involve electrical engineering.) Eventually you will want to learn aspects of all of these fields, but when starting you can use any for an entry into the field. Within each field, the subjects you will want to know are:

• Physics: First learn quantum mechanics. At more advanced levels, various aspects of quantum information overlap with AMO, condensed matter and high energy.
• Math: First linear algebra and probability. Later my preferences would be to learn some group and representation theory, random matrix theory and functional analysis, but eventually most fields of math have some overlap with quantum information, and other researchers may emphasize different areas of math.
• Computer Science: Most theory topics are relevant although are less crucial at first: i.e. algorithms, cryptography, information theory, error-correcting codes, optimization, complexity, machine learning. If you haven't had any CS theory exposure, undergrad algorithms is a good place to start because it will show you CS-theory ways of thinking, including ideas like asymptotic analysis.

The canonical reference for learning quantum computing is the textbook Quantum computation and quantum information by Nielsen and Chuang. Another good book (with more of a "little yellow book" experience) is Classical and Quantum Computation by Kitaev, Shen and Vyalyi.

Online resources: There are also great free resources online.

• David Mermin's lecture notes are elementary and have a CS focus (Note: you have to click past a warning about expired certificates.)
• John Preskill's lecture notes are slightly more advanced and use a physics perspective.

If you want to get a flavor of what research is currently hot, then one place to look is at the program of the last few QIP workshops. A less curated list of interesting papers can be found at scirate.com, where looking at the most scited papers in the last year should bring up some interesting work.

Specialized sources: Some more specialized books/lecture notes are here. These are more modern and in-depth than the general resources above.

• Quantum Algorithms lecture notes by Andrew Childs
• From Classical to Quantum Shannon Theory by Mark Wilde. Thorough and detailed with plenty of exercises.
• The Functional Analysis of Quantum Information Theory written by Gupta, Mandayam and Sunder based on lectures by Effros, Paulsen, Pisier and Winter. Denser and focused on the math side more than applications.
• Alice and Bob meet Banach by Aubrun and Szarek. Incomplete textbook draft, but it looks like it'll be the definitive treatment of the probabilistic method in quantum information.
• The Mathematics of Entanglement by Brandao, Christandl, Walter and myself. Idiosyncratic and incomplete lecture notes on some of our pet topics.

If you have more resources to suggest or any comments on this page, then please email me at aram@mit.edu.