  Fermion-to-qubit mappings that preserve geometric locality are especially
useful for simulating lattice fermion models (e.g., the Hubbard model) on a
quantum computer. They avoid the overhead associated with geometric non-local
parity terms in mappings such as the Jordan-Wigner transformation and the
Bravyi-Kitaev transformation. As a result, they often provide quantum circuits
with lower depth and gate complexity. In such encodings, fermionic states are
encoded in the common +1 eigenspace of a set of stabilizers, akin to stabilizer
quantum error-correcting codes. Here, we discuss several known geometric
locality-preserving mappings and their abilities to correct/detect single-qubit
errors. We introduce a geometric locality-preserving map, whose stabilizers
correspond to products of Majorana operators on closed paths of the fermionic
hopping graph. We show that our code, which we refer to as the Majorana loop
stabilizer code (MLSC) can correct all single-qubit errors on a 2-dimensional
square lattice, while previous geometric locality-preserving codes can only
detect single-qubit errors on the same lattice. Compared to existing codes, the
MLSC maps the relevant fermionic operators to lower-weight qubit operators
despite having higher code distance. Going beyond lattice models, we
demonstrate that the MLSC is compatible with state-of-the-art algorithms for
simulating quantum chemistry, and can offer those simulations the same
error-correction properties without additional asymptotic overhead. These
properties make the MLSC a promising candidate for error-mitigated quantum
simulations of fermions on near-term devices.
