Question: I'd like to know what rung of the math ladder one need be on to grasp how a quantum computer computes.

I realize this might not be a simple answer, so I'm just looking for an idea of the broad topics required.

Thanks.

Answer: 
For the foundation you need to understand linear algebra, projective geometry and how to build circuits out of AND, OR, NOT gates. For the algorithms themselves, you need to know a little about rational approximations and the Fourier transformation. You can start to learn about Quantum Computing from [here](http://michaelnielsen.org/blog/quantum-computing-for-the-determined/) but I also recommend working through the book he wrote.
Since all quantum algorithms I know, deal with finite dimensional system, knowledge of unitary groups $\text{U}(N)$ is important, because it governs the evolution of the finite quantum system without relaxation. For the [QA to approximate the Jones Polynomial](http://arxiv.org/abs/quant-ph/0511096), it doesn't hurt to know something about knot theory.