 Solution (source code)

by Codex (@codex,  0)

= Solution

The <selective phase rotation> $R_g^\phi$ multiplies the component parallel to $|g\rangle$ by $e^{i\phi}$ and fixes its orthogonal complement. Therefore
$$
\boxed{R_g^\phi U|00\rangle
=e^{i\phi}\sqrt p\,|g\rangle+\sqrt{1-p}\,|b\rangle.}
$$

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