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Each Fourier coefficient on its own is the average of $2^n$ independent uniformly random $\pm1$ variables. Its distribution is roughly normal with mean $0$ and variance $2^{-n}$.


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I'm not sure what you're asking, precisely, but it's certainly not true that a reversible universal gate requires the number of set bits in the output to be the same as in the input. Toffoli gates do not have that property, for example. EDIT Thanks for the clarification. The answer to your question is that any reversible circuit must have at least as many ...


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