Useful courses for topological quantum computing

[Monocles]

I recently took a great interest in topological quantum computing - so great an interest I am even considering it as a thesis topic for grad school (though I am still a junior undergrad and have awhile to figure that out). What would be some useful courses to take to pursue theoretical research in this field, besides the typical courses a physics undergrad would take (complex analysis, PDEs, numerical analysis, etc.). Would actually taking a topology class in the math department be useful, or would I take some topology-for-physics style class in grad school? Howabout modern algebra? Information theory? What about algebraic topology? As it stands, I may not be able to take topology before I graduate (it depends on what I end up doing this spring), but I would be able to take algebraic topology.

 

[fantispug]

I've never gotten into topological quantum computing, but I know enough to start learning it so I should at least be able to point you in the right direction.

As you say, all the typical courses a physics undergrad would take and especially quantum mechanics, statistical mechanics, relativity and quantum field theory. I'll just assume you're doing all the physics you need.

It's really hard to say how much maths you need, but as a theorist it will be quite a bit. Mathematicians and physicists have quite different aims, but it's not really until very end undergraduate/beginning of postgrad that the divergence becomes really significant.

Nevertheless if you have any inclination I would strongly recommend courses in (rough order of increasing difficulty/abstraction):
abstract algebra, topology, linear analysis (stuff like Hilbert spaces and Reisz representation theorem), Lie algebras.

This is a good background for the quantum mechanical part.

Information theory would no doubt be useful.

Differential geometry, up to deRham theory, vector and principal bundles and curvature would be very useful if you want to understand the physics from the modern mathematical view (and you probably do).

Algebraic topology would also be a plus, but I'd put it only after everything else I've listed (assuming they're all available). I think taking algebraic topology without taking topology could be quite difficult.

Anyway that should probably take you up until the end of your undergrad. Good luck.

 

[Monocles]

Cool, sounds fun. I'm not going to be able to get to QFT as an undergrad, but I'll have time to take a good number of those classes, which I am definitely looking forward to doing. I'm a bit surprised by the inclusion of relativity - are you talking about a general relativity class? There was a few weeks on special relativity in my modern physics class, is that enough?

 

[fantispug]

Yeah, general relativity is certainly unnecessary. The only reason I said special relativity is it's a necessary prerequisite to Quantum Field Theory, and Quantum Field Theory is good because it gives you the spin-statistics theorem (among other things). All you'd really need to know is how vectors and tensors transform under Lorentz transformations and the mass-energy relation.

These may not be absolutely essential at first, but when you get into hard research I can imagine they'd be useful. If you don't do it now you'll probably just pick it up as a postgrad.