  We argue that all locality-preserving mappings between fermionic observables
and Pauli matrices on a two-dimensional lattice can be generated from the exact
bosonization in Ref. [1], whose gauge constraints project onto the subspace of
the toric code with emergent fermions. Starting from the exact bosonization and
applying Clifford finite-depth generalized local unitary (gLU) transformation,
we can achieve all possible fermion-to-qubit mappings (up to the re-pairing of
Majorana fermions). In particular, we discover a new super-compact encoding
using 1.25 qubits per fermion on the square lattice, which is lower than any
method in the literature. We prove the existence of fermion-to-qubit mappings
with qubit-fermion ratios $r=1+ \frac{1}{2k}$ for positive integers $k$, where
the proof utilizes the trivialness of quantum cellular automata (QCA) in two
spatial dimensions. When the ratio approaches 1, the fermion-to-qubit mapping
reduces to the 1d Jordan-Wigner transformation along a certain path in the
two-dimensional lattice. Finally, we explicitly demonstrate that the
Bravyi-Kitaev superfast simulation, the Verstraete-Cirac auxiliary method,
Kitaev's exactly solved model, the Majorana loop stabilizer codes, and the
compact fermion-to-qubit mapping can all be obtained from the exact
bosonization.
