Universal quantum computation and quantum error correction with ultracold atomic mixtures

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 with 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 illustrate how to encode a qubit in the collective spin using a finite version of the Gottesman-Kitaev-Preskill code paving the way to universal faul-tolerant quantum computation in ultracold atom systems.