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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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1 Answers1

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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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