Before revision:  In this work, we improve upon this routine in two aspects: we reduce the required qubit overhead from g to \log_2(g) in the bit precision g (at a cost of slightly increasing the count of non-Clifford operations), and show how various sets of N coefficients can be loaded significantly faster than in O(\sqrt N) rounds of amplitude amplification - up to only O(1) many - by bootstrapping the procedure with an optimised initial state .
After revision:  In this work, we improve upon this routine in two aspects: we reduce the required qubit overhead from g to \log_2(g) in the bit precision g (at a cost of slightly increasing the count of non-Clifford operations), and show how various sets of N coefficients can be loaded significantly faster than in O(\sqrt N) rounds of amplitude amplification - up to only O(1) many . This exponential speedup translates beyond the black box case .