# 4.7. Fermion to qubit mappings

We consider a system of fermions, each labeled with an integer from to . We need a fermion to qubit mapping, a description of the correspondence between states of fermions and states of qubits, or, equivalently, between fermionic operators and multi-qubit operators. We need a mapping between the fermion creation and annihilation operators and multi-qubit operators. The Jordan-Wigner and the Bravyi-Kitaev transformations are widely used and simulate a system of electrons with the same number of qubits as electrons.

## 4.7.1. Qubit creation and annihilation operators

We define qubit operators that act on local qubits [Yepez] [Chiew], as shown in *Figure 4.22*:

The qubit operators have the anti-commutation relation: .

## 4.7.2. Jordan-Wigner transformation

The **Jordan-Wigner** (**JW**) transformation stores the occupation of each spin orbital in each qubit. It maps...