I found this article on loading a probability distribution into a quantum circuit, the Grover-Rudolph scheme: https://arxiv.org/pdf/quant-ph/0208112.pdf

I understand the idea of progressively dividing each region of the distribution into two, but I am having a hard time understanding how to actually implement this. When they describe adding the next qubit to further divide the distribution, they use f(i) as the probability of the measurement being in the left half of region i. Thus confuses me, as the region is divided into m different regions i. This doesn't seem like adding one qubit, because you would have to repeat the process for each region? Mathematically it should only take one qubit, so what am I misunderstanding?



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