Jordan-Wigner Transformation
The canonical mapping that turns fermionic creation and annihilation operators into qubit Pauli operators.
Why a mapping is needed
Electrons are fermions: their operators obey anticommutation relations {a_i, a_j^dagger} = delta_ij, encoding the Pauli exclusion principle and the sign changes from exchanging particles. Qubits obey different algebra. To simulate electrons, fermionic operators must be represented by qubit operators that reproduce the anticommutation relations exactly. The Jordan-Wigner transformation is the oldest and most direct such map.
The map
Assign one qubit per spin-orbital. Occupation 1 is qubit state |1>, empty is |0>. The creation operator is a_j^dagger = (Z_0 Z_1 ... Z_{j-1}) (X_j - i Y_j)/2, and the annihilation operator a_j = (Z_0 ... Z_{j-1})(X_j + i Y_j)/2. The leading string of Z operators, the Jordan-Wigner string, tracks the parity of all lower-indexed orbitals and supplies the fermionic minus signs.
Key features
- Occupation number n_j = a_j^dagger a_j maps to (I - Z_j)/2, a purely local operator.
- The (X -/+ iY)/2 factor flips the target qubit between occupied and empty.
- The Z string enforces antisymmetry under exchange.
The nonlocality cost
The Z string means an operator on orbital j acts on all qubits below it. A two-electron term can therefore span a chain of Z operators across many qubits, producing long Pauli strings. Under Trotterized time evolution or VQE, these long strings become deep circuits, and their length grows with orbital index. This locality overhead is the main drawback of Jordan-Wigner.
Alternatives
The Bravyi-Kitaev transformation reduces the operator weight to logarithmic in the orbital count by balancing occupation and parity information across a tree structure. Parity mapping and other schemes make different trade-offs. Jordan-Wigner remains the default for its simplicity and its clean physical interpretation, and it feeds directly into the quantum chemistry workflow.