Let $a, b \in \mathbb R$ such that $\vert a + ib \vert < 1$. Does there exists a generic $2$-qubit unitary such that $$(a |0\rangle + \sqrt{1 - a^2}| 1 \rangle) \otimes (b|0\rangle + \sqrt{1 - b^2}|1\rangle) \xrightarrow{U} ((a + ib)|0\rangle + \sqrt{1-\vert a+ib \vert^2}|1\rangle) \otimes |\text{junk}\rangle$$ i.e. it adds the amplitudes on $0$ of each qubit (with a factor of $i$ for the second one).
For context, I am trying to construct an amplitude oracle for $n$-qubit square matrices $A$ with complex coefficients. If $A_{ij}$ are all real coefficients, then it can be done assuming access to a bit oracle $O_A|0^d\rangle|i\rangle|j\rangle = |\tilde{A_{ij}}\rangle|i\rangle|j\rangle$, where $\tilde{A_{ij}}$ is a $d$-bit approximation of $A_{ij}$. With classical operations, we can turn this into the bit oracle $O_A'|0^d\rangle|i\rangle|j\rangle = |\tilde{\theta_{ij}}\rangle|i\rangle|j\rangle$ where $\theta_{ij} = \arccos(A_{ij})/\pi$. Then, apply a controlled rotation $R|0\rangle|\theta\rangle = (\cos(\pi\theta)|0\rangle + \sin(\pi\theta)|1\rangle)|\theta\rangle = (A_{ij}|0\rangle + \sqrt{1 - \vert A_{ij} \vert^2}|1\rangle)|\theta\rangle$. In the complex case, we assume access to a bit oracle which stores the real and imaginary parts of $A$ in two different registers $O_A|0^d\rangle|0^d\rangle|i\rangle|j\rangle = |Re(A_{ij})\rangle|Im(A_{ij}\rangle|i\rangle|j\rangle$. Similarly, we can classically encode these in bit oracles and apply two controlled rotations to get $$(Re(A_{ij})|0\rangle + *|1\rangle)\otimes(Im(A_{ij})|0\rangle + *|1\rangle)$$ in the first two registers. I am trying to find a circuit that will add the two amplitudes on each $|0\rangle$.
