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.

To reproduce these minus signs using ordinary qubits, the transformation attaches a "Z-string" to each operator. This string counts the parity of occupied modes before the target mode in a chosen ordering.

Worked Example

Consider a system with 3 modes and a basis state index of 3. In our binary convention (with mode 0 as the most significant bit on the left), 3 in binary is 011. This means modes 0, 1, and 2 have occupation numbers 0, 1, and 1, respectively.

We want to apply the annihilation operator for mode 2, denoted as c₂, to the state |011〉.

Identify the target mode: Mode j = 2.
Find the state of the target mode: The bit at index 2 is '1' (occupied), so annihilation is possible.
Calculate the left occupation parity: We count the number of occupied modes strictly before mode 2. Mode 0 is '0' and Mode 1 is '1'. The count is 1.
Determine the sign: The sign is given by (-1)^count = (-1)¹ = -1.
Determine the new state: Annihilating mode 2 changes the bit at index 2 from '1' to '0'. The new state is |010〉.

Thus, c₂ |011〉 = -1 |010〉.

Experiment

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

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

Limitations

This model illustrates the parity counting mechanism and basis vector transformations for a small, finite number of modes (up to 3). It clamps state indices strictly to the maximum representable integer (e.g., index 7 for 3 modes). Operations on empty modes for annihilation, or occupied modes for creation, strictly return a coefficient of 0 and a literal "Zero vector" state. The visualization does not encompass Fermi occupation thermodynamics, complex amplitudes, or full matrix representations of the operators.

References

Google Quantum AI, "Jordan-Wigner and Bravyi-Kitaev Transforms". Available at:https://quantumai.google/openfermion/tutorials/jordan_wigner_and_bravyi_kitaev_transforms. This tutorial explains the theoretical foundation for mapping fermionic operators to qubits using the Jordan-Wigner transform, supporting the parity sign calculations demonstrated here.

Jordan–Wigner Fermion Parity