# What's the circuit to create superpositions corresponding to efficiently integrable probability distributions?

See article here: https://arxiv.org/abs/quant-ph/0208112

There are two steps in this procedure that I am curious about. First off, they suppose one can construct a circuit which efficiently performs the computation

$$\sqrt{p_i^{(m)} } |i\rangle|0\cdots 0\rangle \rightarrow \sqrt{p_i^{(m)} } |i\rangle|\theta_i\rangle.$$ How would one implement such a circuit? My guess is that one implements it in a similar way that one would on a classical computer, but quantum parallelism allows for $$\theta_i$$ to be calculated for all bit-combinations simultaneously. However, I have no idea how one starts when designing such circuits. Is there any place I can learn this so I can implement this procedure for any function $$f(i) = \theta_i$$?

The next step of the procedure involves the evolution $$\sqrt{p_i^{(m)} } |i\rangle|\theta_i\rangle|0\rangle \rightarrow \sqrt{p_i^{(m)} } |i\rangle|\theta_i\rangle(\cos\theta_i|0\rangle + \sin \theta_i|1\rangle).$$ This step is also completely unfamiliar to me. How does one perform a rotation on an ancilla qubit, where the rotation angle is given by the binary number represented by the state $$|\theta_i\rangle$$?

Any literature or answers getting me closer to learning this would be highly appreciated.

• Please change the link to the abstract: arxiv.org/abs/quant-ph/0208112 instead of the PDF. Sep 8, 2020 at 14:28
• does this help quantumcomputing.stackexchange.com/q/12104/11793 Sep 8, 2020 at 15:45
• It does help, Condo. Thank you. I am however still confused regarding how one implements $U_p$. Is there any computer science litterature that can help me understand this? Sep 9, 2020 at 10:29
• Regarding the bounty: show the practical implementation of loading some log-concave distributions (eg gaussian) in some quantum circuit having a small number of qubits (eg 5) is enough to win the bounty Nov 9, 2021 at 13:22
• – glS
Aug 10 at 12:29