# "More Efficient Approximate $k$-wise Independent Permutations from Random Reversible Circuits via log-Sobolev Inequalities".

#### Speaker

Angelos Pelecanos(UC Berkeley)

#### Host

Vinod Vaikuntanathan and Yael Kalai

We prove that the permutation computed by a reversible circuit with $\widetilde{O}(nk\cdot \log(1/\epsilon))$ random $3$-bit gates is $\epsilon$-approximately $k$-wise independent.

Our bound improves on currently known bounds in the regime when the approximation error $\epsilon$ is not too small and is optimal up to logarithmic factors when $\epsilon$ is a constant.

We obtain our results by analyzing the log-Sobolev constants of appropriate Markov chains rather than their spectral gaps.

A corollary of our result concerns the \emph{incompressibility of random reversible circuits} as pointed out by concurrent work of Chen et al., who showed that a linear-in-$k$ bound for a multiplicative approximation to a $k$-wise independent permutation implies the linear growth of circuit complexity (a generalization of Shannon's argument).

Our bound improves on currently known bounds in the regime when the approximation error $\epsilon$ is not too small and is optimal up to logarithmic factors when $\epsilon$ is a constant.

We obtain our results by analyzing the log-Sobolev constants of appropriate Markov chains rather than their spectral gaps.

A corollary of our result concerns the \emph{incompressibility of random reversible circuits} as pointed out by concurrent work of Chen et al., who showed that a linear-in-$k$ bound for a multiplicative approximation to a $k$-wise independent permutation implies the linear growth of circuit complexity (a generalization of Shannon's argument).