Large Deviation Bounds for k-designs

TL;DR

This paper develops a method to replace Haar random unitaries with efficiently implementable k-designs for deriving large deviation bounds, with applications in quantum information and statistical mechanics.

Contribution

It introduces a technique to derandomise large deviation bounds using k-designs, enabling efficient approximations of Haar randomness in quantum applications.

Findings

Pseudo-random states have nearly maximal von Neumann entropy.

Subsystems of k-designs approximate canonical states.

Pseudo-random states are ineffective for measurement-based quantum computation.

Abstract

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.