Suppose I have two arbitrary quantum states $\lvert \psi_1 \rangle $ and $\lvert \psi_2 \rangle$. Further suppose that we know $U_1$ such that $\lvert \psi_1 \rangle = U_1 \lvert 0 \rangle$, but we don't know $\lvert \psi_2 \rangle $ (only can use it as an initial state). Using ancilla qubits and measurements, is there a way to construct quantum circuits such that one of the wires of the output state is $\frac{1}{c}(\lvert \psi_1 \rangle + \lvert \psi_2 \rangle)$?
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