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I have a following state (indexing everything from 0 here)

$$ | \psi \rangle = \frac{1}{\sqrt{2}} \left( | \uparrow_0 \downarrow_1 \rangle - | \downarrow_0 \uparrow_1 \rangle \right) $$

and I need to be able to rewrite it in Qiskit notation in a computational basis.

I know, that Qiskit orders the spin-orbitals with all spin-up first and spin-down afterwards, e.g. when we have 2 molecular orbitals, it looks like

  • [0]: $\uparrow$ MO0
  • [1]: $\uparrow$ MO1
  • [2]: $\downarrow$ MO0
  • [3]: $\downarrow$ MO1

with qubit position given by an index in square brackets.

So, how can I approach the conversion to qubits? My idea is, that I can rewrite it in a way, where $| 0 \rangle$ will be an unoccupied spin-orbital and $|1\rangle$ an occupied orbital, i.e. the one present in $|\psi\rangle$ expression.

Thus, I'd approach it like this: $$| \psi \rangle = \frac{1}{\sqrt{2}}\left( |1001\rangle - |0110\rangle \right)$$

Is it the correct approach (and result :)?

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  • $\begingroup$ So, you have two degrees of freedom, up-down and zero-one in the first equation? $\endgroup$
    – R.W
    Sep 4, 2022 at 18:21
  • $\begingroup$ @R.W Yes, 2 spins and 2 molecular orbitals $\endgroup$
    – Eenoku
    Sep 4, 2022 at 19:59

1 Answer 1

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Yes, your approach is correct. You have implicitly made use of the Jordan-Wigner transformation to map the spin-1/2 fermions to qubits. Incidentally, there is a recent related post ($S^2$ expectation value of a circuit seems wrong) that provides a circuit to prepare this fermionic valence-bond state.

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