  Quantum information platforms made great progress in the control of many-body
entanglement and the implementation of quantum error correction, but it remains
a challenge to realize both in the same setup. Here, we propose a mixture of
two ultracold atomic species as a platform for universal quantum computation
with long-range entangling gates, while providing a natural candidate for
quantum error-correction. In this proposed setup, one atomic species realizes
localized collective spins of tunable length, which form the fundamental unit
of information. The second atomic species yields phononic excitations, which
are used to entangle collective spins. Finally, we discuss a finite-dimensional
version of the Gottesman-Kitaev-Preskill code to protect quantum information
encoded in the collective spins, opening up the possibility to universal
fault-tolerant quantum computation in ultracold atom systems.
