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In this paper, the measurement of the real part of the matrix element of a Hermitian&unitary operator $G$ between the states $U|0\rangle$ and $V|0\rangle$ (i.e. $\langle 0| V^\dagger G U | 0\rangle$) is organized by means of the Hadamard test:

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Here it is assumed that the circuits for preparing $G$, $V$, and $U$ are known.

Typically, the circuit for $G$ would be significantly shorter than those for $V$ and $U$.

I am wondering how one would accomplish the same measurement using more qubits (~ twice as many, probably) and fewer layers of gates (~ as much as needed for implementing $GV$ and $GU$), using something in the spirit of the SWAP-test?

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  • $\begingroup$ Do you have reason to think this can be done? The standard application of the swap test gets you the absolute value rather than the real component. $\endgroup$
    – DaftWullie
    Feb 2, 2022 at 8:20
  • $\begingroup$ Not really. It's just the circuit looks quite symmetric, especially for $V=U$, and I had this hope. $\endgroup$
    – mavzolej
    Feb 3, 2022 at 3:49

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