Implement Reverse Jordan-Wigner Transformation

What is the expected behavior?

While Jordan-Wigner transformation is implemented in the mappers framework (JordanWignerMapper), the reverse transformation is missing.

Inspiration can be drawn from https://github.com/quantumlib/OpenFermion/blob/master/src/openfermion/transforms/opconversions/reverse_jordan_wigner.py

PS: If maintainers agree with this issue, it will be solved by collaboration among Quantum Hackathon Korea participants.

[0sophy1]

Thank you for creating the issue! We are mentors of this event (Also Qiskit Advocates) - https://github.com/qiskit-community/quantum-hackathon-korea-21 and while handling the questions, we found there is no reverse Jordan Wigner function in Qiskit. We hope to contribute to Qiskit Nature by providing this function with our attendees.

[mrossinek]

Hi @0sophy1! It has been a while and Qiskit Nature just underwent yet another large refactoring (hopefully the last one as big as this for a long time).

Are you still interested in adding the reverse Jordan-Wigner mapping to Qiskit Nature?

I see that the implementation was basically already done (@BStar14). Instead of the version you proposed there I would suggest to add the reverse map to the existing JordanWignerMapper (because the QubitMapper class dictates a SparseLabelOp -> PauliSumOp direction which your mapper would not match). But that is a design detail which should be easy to figure out 👍