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Assume $|i\rangle,i\in[1,m]$ is a set of states, like the Fock states for Fermions. The question is how to compute the quantity,$\langle i |H|j\rangle,i,j\in[1,m]$, in qiskit. The $H$ here is the qubit form Hamiltonian of some molecule obtained by the Jordan-Wigner transformation.

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  • $\begingroup$ You're looking for the matrix elements of $H$ in the basis $|i⟩$. Depending on what $|i⟩$ is, you can probably do so without qiskit. (Indeed, getting the matrix elements of $H$ is a starting point for chemistry calculations with qiskit!) $\endgroup$
    – jecado
    Oct 19 at 16:19
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As mentioned in a comment, there are many times where you might think you want to do this but actually there may be a more direct classical approach (eg. doing a single-body basis transformation of your Hamiltonian's coefficients).

But, sometimes you might need to do this quantum-ly. This paper discusses one way, which requires twice as many qubits but is otherwise pretty efficient: https://arxiv.org/abs/1909.09114

Essentially you prepare |i⟩ and |j⟩ on two different registers and perform a modified Hadamard test.

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