Random 3-bit gates in a fixed 2D nearest-neighbor brickwork produce approximate k-wise independent permutations of n bits in depth sqrt(n) e^{O(k^3)}.
Simple Permutations Mix Even Better
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Pseudorandomness Properties of Random Reversible Circuits
Random 3-bit gates in a fixed 2D nearest-neighbor brickwork produce approximate k-wise independent permutations of n bits in depth sqrt(n) e^{O(k^3)}.