  Quantum simulation of fermionic systems is a promising application of quantum
computers, but in order to program them, we need to map fermionic states and
operators to qubit states and quantum gates. While quantum processors may be
built as two-dimensional qubit networks with couplings between nearest
neighbors, standard Fermion-to-qubit mappings do not account for that kind of
connectivity. In this work we concatenate the (one-dimensional) Jordan-Wigner
transform with specific quantum codes defined under the addition of a certain
number of auxiliary qubits. This yields a novel class of mappings with which
any fermionic system can be embedded in a two-dimensional qubit setup,
fostering scalable quantum simulation. Our technique is demonstrated on the
two-dimensional Fermi-Hubbard model, that we transform into a local
Hamiltonian. What is more, we adapt the Verstraete-Cirac transform and
Bravyi-Kitaev Superfast simulation to the square lattice connectivity and
compare them to our mappings. An advantage of our approach in this comparison
is that it allows us to encode and decode a logical state with a simple unitary
quantum circuit.
