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Suppose I want to estimate $Im(\langle \psi_1\lvert \sigma_x\lvert \psi_2\rangle)$ by using quantum circuit.

At first, I thought of using the Swap test, but since it gives $|\langle \psi_1|\psi_2\rangle|^2$, it won't give the imaginary part.

Then, I thought of Hadamard test + phase gate, which gives $Im(\langle \psi \lvert U |\psi \rangle)$, so that I have to find $U = \sigma_x U'$ where $U'|\psi_2\rangle = |\psi_1\rangle$. However, what if finding $U'$ is very complicated so that I want to avoid? Is there any clever way to estimate $Im(\langle \psi_1\lvert \sigma_x\lvert \psi_2\rangle)$?

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  • $\begingroup$ In which form are you given psi_1 and psi_2? $\endgroup$ Apr 15, 2022 at 17:22

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In this article, an overlap estimation algorithm (OEA) algorithm is introduced, and it can help calculate $Im(\langle\psi_1|\sigma_x|\psi_2\rangle)$. I have verified the proposal long ago.

First, prepare the initial state $|x\rangle=(|0\rangle|\psi_1\rangle-i|1\rangle|\psi_2^\prime\rangle)$, where $|\psi^\prime_2\rangle=\sigma_x|\psi_2\rangle$. Then, apply $H\otimes I$.

Finally, measuring the ancilla qubit can give you the result. Denote the probability that the measurement obtains $|1\rangle$ as $p_1$, then $Im(\langle\psi_1|\sigma_x|\psi_2\rangle)=2\times p_1-1$.

This problem is concerned by quantum expectation estimation, for other method that can measure such a value, see also this article.

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    $\begingroup$ I agree that $p_1=\||\psi_1\rangle+i X|\psi_2\rangle\|^2$, which thus gives the stated overlap, but I think it's worth pointing out that this is not an overlap between quantum states. It's an overlap between classical vectors encoded (in a specific way) in the amplitudes of a two-qubit state $\endgroup$
    – glS
    Apr 15, 2022 at 17:39
  • $\begingroup$ I am not clear about the encoding you mentioned here, if you could state with more detail it can be helpful. As far as I can see, sometimes the encoding is optional, ignoring the computational complexity you can always encapsulate quantum simulation in classical calculation, also the inverse. $\endgroup$ Apr 18, 2022 at 13:49
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Quantum states are only defined up to a phase. Thus, it is only possible to determine the absolute value of an overlap such as the one you give. The phase information -- and thus the imaginary part -- are not accessible.

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  • $\begingroup$ not that it's clear from the OP, but I think the idea here is to compute the given overlap where $|\psi_i\rangle$ are not really quantum states, but rather just vectors encoded in the amplitudes of a state (where the "state" is the larger two-qubit one, so that one can effectively encode vectors with specific global phases in it). So it won't be possible to build a circuit taking as inputs $|\psi_i\rangle$ (as states) that gives the given quantity, as that wouldn't make sense for the reasons you point out. But you can do a circuit with inputs the vector components of the $|\psi_i\rangle$ $\endgroup$
    – glS
    Apr 15, 2022 at 17:36
  • $\begingroup$ @glS Given that the OP talks about finding a unitary which maps |psi1> to |psi2> I don't see why they should really mean what you suggest. They might just as well have some basic misconception. In the end the answer will very much depend on how those states (or rather: the information about |psi1> and |psi2>) are provided. $\endgroup$ Apr 15, 2022 at 17:49

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