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

arXiv:2406.08499v1 Announce Type: cross
Abstract: 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.

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