  We present a technique for derandomising large deviation bounds of functions
on the unitary group. We replace the Haar distribution with a pseudo-random
distribution, a k-design. k-designs have the first k moments equal to those of
the Haar distribution. The advantage of this is that (approximate) k-designs
can be implemented efficiently, whereas Haar random unitaries cannot. We find
large deviation bounds for unitaries chosen from a k-design and then illustrate
this general technique with three applications. We first show that the von
Neumann entropy of a pseudo-random state is almost maximal. Then we show that,
if the dynamics of the universe produces a k-design, then suitably sized
subsystems will be in the canonical state, as predicted by statistical
mechanics. Finally we show that pseudo-random states are useless for
measurement based quantum computation.
