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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 $Im(\langle \psi_1\lvert \sigma_x\lvert \psi_2\rangle)$?

glS
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userflux9674
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2 Answers2

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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.

Yitian Wang
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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.

Norbert Schuch
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