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Hierarchical probability-distribution state preparation recursively bisects a classical probability distribution. At each level, a controlled rotation splits every parent interval's amplitude between its two children according to their conditional probabilities.

The ratio $f_i=A_i/B_i$ is the conditional probability that $X$ lies in the left half of interval $i$. After the controlled rotation and uncomputation of $|{\theta }_i⟩$, append the rotation qubit to the interval label. The amplitudes becomeButConsequently one refinement step mapsStarting from $|{\psi }_1⟩$ and repeating this hierarchical probability-distribution state preparation for $m=1,\dots ,n-1$ givesThere are $n-1=O(\log N)$ refinement levels, each of polylogarithmic size by assumption.