More Efficient k-wise Independent Permutations from Random Reversible Circuits via log-Sobolev Inequalities

We prove that the permutation computed by a reversible circuit with Õ(nk· log(1/ε)) random $3$-bit gates is ε-approximately k-wise independent. Our bound improves on currently known bounds in the regime when the approximation error ε is not too small. We obtain our results by analyzing the log-Sobolev constants of appropriate Markov chains rather than their spectral gaps.