Large deviation bounds for k-designs

We present a technique for de-randomizing large deviation bounds of functions on the unitary group. We replace the Haar measure with a pseudo-random distribution, a k-design. k-Designs have the first k moments equal to those of the Haar measure. 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.