Two randomized Hadamard transforms suffice to make coordinate marginals O(d^{-1/2})-close to Gaussian for most quantization methods, with three needed for vector quantization to match uniform random rotations asymptotically.
A tight Gaussian bound for weighted sums of Rademacher random variables.Bernoulli, 21(2):1231 – 1237
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Quantizing With Randomized Hadamard Transforms: Efficient Heuristic Now Proven
Two randomized Hadamard transforms suffice to make coordinate marginals O(d^{-1/2})-close to Gaussian for most quantization methods, with three needed for vector quantization to match uniform random rotations asymptotically.