The Jordan-Wigner Transformation
The most direct mapping from fermionic creation and annihilation operators to qubit Pauli operators, at the cost of long non-local strings.
The mapping
The Jordan-Wigner (JW) transformation assigns one qubit per spin-orbital, with occupation 0 or 1 stored in the qubit's computational-basis state. A creation operator becomes a_p^dagger = (Z_0 Z_1 ... Z_{p-1}) (X_p - i Y_p)/2, and annihilation similarly. The leading string of Z operators, the Jordan-Wigner string, enforces the fermionic sign under exchange.
Why the Z string is needed
Fermionic operators anticommute even for distant orbitals, but qubit operators on different qubits commute. The chain of Z operators from qubit 0 up to qubit p-1 supplies the missing sign flips, so that the encoded operators reproduce the anticommutation relations exactly.
The locality cost
The price is non-locality: an operator on orbital p touches all qubits below it. A single fermionic hopping term a_p^dagger a_q becomes a Pauli string spanning qubits p through q, whose weight grows with |p - q|. In the worst case terms act on O(N) qubits, inflating gate counts for Trotterization.
- One qubit per spin-orbital; occupation is stored directly and legibly.
- Number operators map simply: n_p = (I - Z_p)/2, purely local.
- Hopping and interaction terms carry Z strings of length up to N.
- Operator locality is poor; circuit depth suffers for large systems.
When JW is the right choice
Because occupation is stored transparently, JW is easy to reason about and to prepare reference states like Hartree-Fock in. For one-dimensional or nearest-neighbor problems the strings stay short and JW is efficient. For large, densely connected molecular Hamiltonians its O(N) string weight motivates alternatives such as Bravyi-Kitaev, which trades string length for logarithmic operator locality. JW remains the default, best-understood encoding and the baseline against which others are judged.