  Fermions are the building blocks of matter, forming atoms and nuclei, complex
materials and neutron stars. Our understanding of many-fermion systems is
however limited, as classical computers are often insufficient to handle the
intricate interplay of the Pauli principle with strong interactions. Quantum
simulators based on ultracold fermionic atoms instead directly realize
paradigmatic Fermi systems, albeit in "analog" fashion, without coherent
control of individual fermions. Digital qubit-based quantum computation of
fermion models, on the other hand, faces significant challenges in implementing
fermionic anti-symmetrization, calling for an architecture that natively
employs fermions as the fundamental unit. Here we demonstrate a robust quantum
register composed of hundreds of fermionic atom pairs trapped in an optical
lattice. With each fermion pair forming a spin-singlet, the qubit is realized
as a set of near-degenerate, symmetry-protected two-particle wavefunctions
describing common and relative motion. Degeneracy is lifted by the atomic
recoil energy, only dependent on mass and lattice wavelength, thereby rendering
two-fermion motional qubits insensitive against noise of the confining
potential. We observe quantum coherence beyond ten seconds. Universal control
is provided by modulating interactions between the atoms. Via state-dependent,
coherent conversion of free atom pairs into tightly bound molecules, we tune
the speed of motional entanglement over three orders of magnitude, yielding
$10^4$ Ramsey oscillations within the coherence time. For site-resolved
motional state readout, fermion pairs are coherently split into a double well,
creating entangled Bell pairs. The methods presented here open the door towards
fully programmable quantum simulation and digital quantum computation based on
fermions.
