Ternary tree transformations are equivalent to linear encodings of the Fock basis

We consider two approaches to designing fermion-qubit mappings: (1) ternary tree transformations, which use Pauli representations of the Majorana operators that correspond to root-to-leaf paths of a tree graph and (2) linear encodings of the Fock basis, such as the Jordan-Wigner and Bravyi-Kitaev transformations, which store linear binary transformations of the fermionic occupation number vectors in the computational basis of qubits. These approaches have emerged as distinct concepts, with little notational consistency between them. In this paper we propose a universal description of fermion-qubit mappings, which reveals the relationship between ternary tree transformations and linear encodings. Using our notation, we show that every product-preserving ternary tree transformation is equivalent to a linear encoding of the Fock basis.