  Simulating the properties of many-body fermionic systems is an outstanding
computational challenge relevant to material science, quantum chemistry, and
particle physics. Although qubit-based quantum computers can potentially tackle
this problem more efficiently than classical devices, encoding non-local
fermionic statistics introduces an overhead in the required resources, limiting
their applicability on near-term architectures. In this work, we present a
fermionic quantum processor, where fermionic models are locally encoded in a
fermionic register and simulated in a hardware-efficient manner using fermionic
gates. We consider in particular fermionic atoms in programmable tweezer arrays
and develop different protocols to implement non-local tunneling gates,
guaranteeing Fermi statistics at the hardware level. We use this gate set,
together with Rydberg-mediated interaction gates, to find efficient circuit
decompositions for digital and variational quantum simulation algorithms,
illustrated here for molecular energy estimation. Finally, we consider a
combined fermion-qubit architecture, where both the motional and internal
degrees of freedom of the atoms are harnessed to efficiently implement quantum
phase estimation, as well as to simulate lattice gauge theory dynamics.
