Jordan–Wigner Fermion Parity

The Jordan–Wigner transformation maps fermionic creation and annihilation operators to spin (or qubit) operators. A core requirement of fermions is that they obey canonical anticommutation relations. Because spatial sites (or modes) can be occupied or empty, swapping two fermionic operators introduces a minus sign.

Experiment

You can interact with the fermionic parity logic using the module:
https://riano.app/module/jordan-wigner-fermion-parity

1. Set Modes to "3".
2. Set State index to "3" to load the |011〉 state.
3. Set Mode j to "2".
4. Set Operation to "Annihilate".
5. Observe that the Action coefficient becomes -1 and the Action state is "010", matching our worked example.
6. Change Mode j to "0" and Operation to "Create".
7. Observe that the Action coefficient is 1 (since there are 0 occupied modes before mode 0) and the Action state becomes "111".