Large Deviation Bounds for k-designs
Richard A. Low
TL;DR
The paper develops a framework to derandomize large deviation bounds on functions of unitaries by replacing Haar randomness with approximate unitary $k$-designs, enabling efficient sampling while preserving first-moment properties. It proves a general concentration theorem for degree-$K$ polynomials under ε-approximate $k$-designs and applies it to three domains: entropy of pseudo-random states, thermalization and canonical-state emergence, and the feasibility of measurement-based quantum computing with pseudo-random states. The results show that polynomial-size designs can reproduce many Haar-like phenomena, offering practical routes for derandomized quantum information tasks, albeit with limitations on certain exponential-tail bounds. The work highlights the connection between random circuit dynamics and $k$-designs and outlines potential extensions to broader derandomization problems.
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.