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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.

Martin Vesely
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Amir Naveh
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    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). – glS Jul 26 '20 at 18:51
  • @glS , Thanks for this – Amir Naveh Jul 27 '20 at 05:51
  • 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? – Amir Naveh Jul 27 '20 at 05:59
  • the same way you would do it classically. You can use (the quantum equivalent of) a classical circuit taking as input $i$ and producing $\theta_i$. – glS Jul 27 '20 at 08:43
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    Does this answer your question? Preparing a quantum state from a classical probability distribution – Condo Aug 05 '20 at 16:01
  • @Connor, Thanks, really appreciate the help! :). I think the questions are similar, but still a concrete example of some sort, even for a really simple function, would probably be the best clarification. I'm also trying to work one out on my own so if I'll share if I get to something interesting. – Amir Naveh Aug 06 '20 at 17:09
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    Aren't there too many such $\theta_i $ (namely $2^n$) to compute? Doesn't that ruin the speedup? – Hayk Tepanyan May 15 '22 at 07:57

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I think now I get @gls's point: since basic classical arithmetics (like addition, multiplication, etc.) can be done quantumly and it's assumed that the $i$ -> $\theta_i$ can be done easily classically then it should be possible to do quantumly as well. The trick is that in the quantum circuit we get the benefit of calculating $2^m$ many $\theta_i$s in step $m$, since we have that many $i$s in superposition.

As for an example, following the notations from Creating superpositions that correspond to efficiently integrable probability distributions by Grover and Rudolph, the $i$ -> $\theta_i$ transformation for the uniform distribution will be something like

$f(i) = \frac{\int_a^b p(x)dx}{\int_a^{2b} p(x)dx} = \frac{1}{2} $

$\theta_i=arccos(\sqrt{f(i)})=0.785$

so all you need is a unitary that does $| i\rangle |0 \rangle $ -> $| i\rangle |0.785 \rangle$ which can be done easily.

Hayk Tepanyan
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