Like most physicists, I find Group Theory (GT) a majestically beautiful subject, one that is also a formidably useful tool in Physics.
However, in my 50 arXiv papers so far, I’ve only used GT in one un-outstanding paper. Not a wise move since GT shows up everywhere in Physics, and quantum computing should be no exception. To make amends, I’ve recently started trying to apply GT à la physicist (mostly representation theory) to the type of quantum computer programming I do (quantum algorithms).
Of course, some people have already begun to apply GT to quantum algorithms. (for example, it has been applied in quantum error correction, in anyonic quantum computing, and in some papers by Harrow, Bacon and Chuang to decompose n qudits into their irreps. Also, the people trying to simulate quantum field theories on a QC must use GT) I hope to build on the work of those people. As usual, my goals are very modest. I just want to understand WTF is going on, and see if I can write some simple software tools that apply the stuff.
Although I’ve been learning GT for many years, I’ve never found any practically-perfect-in-every-way textbook about the subject.
Some textbooks are just plain bad (for example, I hate Gilmore and Wybourne, two often recommended textbooks, because they have a bizarre, incomplete selection of topics and are generally unclear and hard to read).
In the case of some other books, I like some sections in them, but I find them woefully incomplete. (for example, I LOVE Cvitanovic’s diagrammatic stuff, but it’s wise to learn also the canonical Cartan approach to classifying semi-simple Lie algebras.) Completeness is hard to achieve for GT textbooks because it’s such a vast subject.
Most books written by mathematicians take me too much time to understand and usually fail to explain the practical, physicky and calculational side of things. On the other hand, many textbooks written by physicists dwell too much on the applications (quark models and the like) and never explain clearly the underlying theory.
Due to the lack of really clear and complete (for my taste) GT books, what has happened over the years is that I’ve ended up buying about a dozen GT books, and whenever I’m trying to figure out the answer to a GT question that is nagging me, I just jump from one book to another or to Google until something clicks.