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.



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