More Efficient Approximate 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 and is optimal up to logarithmic factors when ε is a constant. We obtain our results by analyzing the log-Sobolev constants of appropriate Markov chains rather than their spectral gaps.