Shuttling of $\mathbb{Z}_4$ parafermions in an electronic ladder model

Parafermions with non-Abelian statistics have been proposed as a promising platform for quantum computation, potentially enabling a broader set of topologically protected gates than Majorana fermions. The experimental and theoretical exploration of these exotic quasiparticles remains challenging, as their stability is linked to strong electron-electron interactions. A key step toward practical applications is the controlled shuttling of parafermionic modes, which is required for implementing geometric braiding operations. In the present work, we investigate the real-time dynamics of the elementary shuttling process by applying a combination of the density matrix renormalization group and the time-dependent variational principle approaches. We analyze the transport of $\mathbb{Z}_4$ parafermion edge states and assess the corresponding adiabatic speed limit under experimentally relevant conditions.