How to rewrite the quantum state to qubits?

Question

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 :)?

Answer 1

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.