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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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    $\begingroup$ possible duplicate of quantumcomputing.stackexchange.com/q/11554/55 $\endgroup$
    – glS
    May 23, 2022 at 20:56
  • $\begingroup$ To make it clear, I made my question clearer. The set-up is little bit different from what you posted. $\endgroup$
    – Jon Megan
    May 23, 2022 at 21:45
  • $\begingroup$ You say $|\psi_2\rangle$ is arbitrary, but the output state implies it's orthogonal to $|\psi_1\rangle$. $\endgroup$
    – GotCarter
    May 23, 2022 at 22:12
  • $\begingroup$ @GotCarter Sorry for the confusion. I think that normalization factor shouldn't be $1/\sqrt{2}$, but just something $1/c$ that normalizes the state properly. Just edited. $\endgroup$
    – Jon Megan
    May 24, 2022 at 0:22
  • $\begingroup$ This might be helpful quantumcomputing.stackexchange.com/questions/14185/… $\endgroup$
    – Mauricio
    May 24, 2022 at 19:46

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