# How to rewrite the quantum state to qubits?

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 :)?

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

## 1 Answer

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.