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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

$$\sqrt{p_i^{(m)}}|i\rangle |0...0\rangle \rightarrow \sqrt{p_i^{(m)}}|i\rangle |\theta_i\rangle$$

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.

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    $\begingroup$ I think it's because it's a purely classical operation. You can use regular logic to read the value of the first register and output the state $\theta_i$ on the second register, assuming you know classically $\theta_i$ (i.e. $f(i)$ here). $\endgroup$
    – glS
    Jul 26 '20 at 18:51
  • $\begingroup$ @glS , Thanks for this 👍👍 $\endgroup$
    – Amir Naveh
    Jul 27 '20 at 5:51
  • $\begingroup$ Lets indeed assume that θi is easy to known. My understanding is that now we wish to put this value in the register. How can this be done efficiently? Is there an easy way to load a value into a register? $\endgroup$
    – Amir Naveh
    Jul 27 '20 at 5:59
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    $\begingroup$ Does this answer help quantumcomputing.stackexchange.com/a/12120/11793 $\endgroup$
    – Condo
    Jul 29 '20 at 20:18
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    $\begingroup$ Does this answer your question? Preparing a quantum state from a classical probability distribution $\endgroup$
    – Condo
    Aug 5 '20 at 16:01

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