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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 $\frac{1}{c}(\lvert \psi_1 \rangle + \lvert \psi_2 \rangle)$?

Jon Megan
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    possible duplicate of https://quantumcomputing.stackexchange.com/q/11554/55 – glS May 23 '22 at 20:56
  • To make it clear, I made my question clearer. The set-up is little bit different from what you posted. – Jon Megan May 23 '22 at 21:45
  • You say $|\psi_2\rangle$ is arbitrary, but the output state implies it's orthogonal to $|\psi_1\rangle$. – GotCarter May 23 '22 at 22:12
  • @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. – Jon Megan May 24 '22 at 00:22
  • This might be helpful https://quantumcomputing.stackexchange.com/questions/14185/superposition-of-quantum-circuits – Mauricio May 24 '22 at 19:46
  • @Mauricio Thanks. Yes I saw that post before I posted this question. The set up here is different from there: here we only know $U_1$ for $|\psi_1\rangle$ but there we know both $U_1$ and $U_2$, which would ease the situation. Also, there the normalization coefficient is $1/\sqrt{2}$ which assumes that the two states are orthogonal as GotCarter pointed out above. – Jon Megan May 24 '22 at 20:26

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