What mathematics should I know to study quantum mechanics?

On Clifford Algebras.

It turns out that a set of Clifford transformations and some additional operations that don't conserve "quantum information" (roughly, phase angles between states), leads to universal quantum computation. Clifford gates in a circuit are just a physical realisation of a Clifford algebra (or ring), amirite?

This follows from the "rules" that if a set of gates are all conservative (of information), and all affine, then the set is not universal. Universal computation is a different animal than building a universal computer; current digital computers don't conserve a lot of information (so they dissipate a lot of heat because of the erasure problem); not all Boolean operations are affine, hence universality is available fairly readily at the classical level.

 

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