Preparation of states that correspond to efficiently integrable probability distributions

I have been trying to implement methods from paper Creating superpositions that correspond to efficiently integrable probability distributions by Grover and Rudolph.

It is stated that there exists an efficient (polynomial) process for the preparation of certain probability density functions (e.g. log-concave distributions).

Specifically, in equation 5. It is stated that

$$sqrt{p_i^{(m)}}|irangle |0…0rangle rightarrow sqrt{p_i^{(m)}}|irangle |theta_irangle$$

Can be done efficiently under these assumptions.

I have not found any details on how this can actully be done, either with and example or with the details of how such an efficient circuit could be composed.

Would highly appreciate any insights on this.