Interaction effects in passively stabilized spin-cat encodings

There is a significant push towards implementing robust, error-correctable encodings of qubits in trapped atoms, especially by choosing robust sets of states and leveraging ubiquitous tools such as laser driving. One promising encoding is to employ large hyperfine manifolds of fermionic atoms to create an autonomously-stabilized qubit encoding. In this approach, errors incurred by e.g. stray magnetic fields are passively mitigated through a combination of coherent laser driving and spontaneous emission. Here we explore multi-body aspects and extensions of this approach. We study the interplay of autonomous stabilization with specific types of interatomic interactions. Incorporating these offers prospects for both entangling logical gates, and for the generation of useful entangled multi-atom states.