# Trading gates for qubits in the Hadamard test for $\langle 0| V^\dagger G U | 0\rangle$... using the SWAP test?

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:

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?

• 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. Feb 2, 2022 at 8:20
• Not really. It's just the circuit looks quite symmetric, especially for $V=U$, and I had this hope. Feb 3, 2022 at 3:49