# Is there an efficient circuit implementing the unitary $U|x\rangle|0\rangle=|x\rangle\Big(\sqrt{1 - x/2^n}\,|0\rangle+\sqrt{x/2^n}|1\rangle\Big)?$

Given an $$n$$-qubit register $$|x\rangle$$, does there exist an efficient circuit implementing unitary operation $$U$$ such that

$$U |x\rangle|0\rangle = |x\rangle\Big(\sqrt{1 - x/2^n}\, |0\rangle + \sqrt{x/2^n}\, |1\rangle\Big)?$$

I've found this related question from which the answer suggests to rotate and apply an $$\arccos$$ approximation (which is very complicated, and only provides an approximation). Is there not an exact circuit implementing this from simple gates plus $$R_k$$?

The context of this question is trying to implement Algorithm 1 from Quantum speedup of Monte Carlo methods by Ashley Montanaro. They say (paraphrased):

Also observe that $$U$$ can be implemented efficiently, as it is a controlled rotation of one qubit dependent on the value of $$x$$ [59]

I did not find the linked reference (Quantum algorithm for approximating partition functions by Wocjan et al) particularly enlightening. And I don't believe they used the $$\arccos$$ approximation either, as they did not include this in the error analysis. So I am confused as to how $$U$$ is actually implemented.