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