  Simulating fermionic systems on a quantum computer requires a high-performing
mapping of fermionic states to qubits. A characteristic of an efficient mapping
is its ability to translate local fermionic interactions into local qubit
interactions, leading to easy-to-simulate qubit Hamiltonians.
  All fermion-qubit mappings must use a numbering scheme for the fermionic
modes in order for translation to qubit operations. We make a distinction
between the unordered labelling of fermions and the ordered labelling of the
qubits. This separation shines light on a new way to design fermion-qubit
mappings by making use of the enumeration scheme for the fermionic modes. The
purpose of this paper is to demonstrate that this concept permits notions of
fermion-qubit mappings that are optimal with regard to any cost function one
might choose. Our main example is the minimisation of the average number of
Pauli matrices in the Jordan-Wigner transformations of Hamiltonians for
fermions interacting in square lattice arrangements. In choosing the best
ordering of fermionic modes for the Jordan-Wigner transformation, and unlike
other popular modifications, our prescription does not cost additional
resources such as ancilla qubits.
  We demonstrate how Mitchison and Durbin's enumeration pattern minimises the
average Pauli weight of Jordan-Wigner transformations of systems interacting in
square lattices. This leads to qubit Hamiltonians consisting of terms with
average Pauli weights 13.9% shorter than previously known. By adding only two
ancilla qubits we introduce a new class of fermion-qubit mappings, and reduce
the average Pauli weight of Hamiltonian terms by 37.9% compared to previous
methods. For $n$-mode fermionic systems in cellular arrangements, we find
enumeration patterns which result in $n^{1/4}$ improvement in average Pauli
weight over na\"ive schemes.
