Abstract
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.
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Schedule a callMore From: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
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