  The mapping of fermionic states onto qubit states, as well as the mapping of
fermionic Hamiltonian into quantum gates enables us to simulate electronic
systems with a quantum computer. Benefiting the understanding of many-body
systems in chemistry and physics, quantum simulation is one of the great
promises of the coming age of quantum computers. One challenge in realizing
simulations on near-term quantum devices is the large number of qubits required
by such mappings. In this work, we develop methods that allow us to trade-off
qubit requirements against the complexity of the resulting quantum circuit. We
first show that any classical code used to map the state of a fermionic Fock
space to qubits gives rise to a mapping of fermionic models to quantum gates.
As an illustrative example, we present a mapping based on a non-linear
classical error correcting code, which leads to significant qubit savings
albeit at the expense of additional quantum gates. We proceed to use this
framework to present a number of simpler mappings that lead to qubit savings
with only a very modest increase in gate difficulty. We discuss the role of
symmetries such as particle conservation, and savings that could be obtained if
an experimental platform could easily realize multi-controlled gates.
