Adaptive Preparation of the Rank-2 Toric Code: Exploring Symmetries and Duality

The adaptive preparation of long-range entangled or topologically ordered quantum states through mid-circuit measurements and feedback has recently attracted significant theoretical and experimental attention. In this work, we propose the realization of the rank-2 toric code (R2TC), a translation symmetry-enriched topological phase characterized by the conservation of dipoles of anyons, through measurements of symmetry-protected topological (SPT) state which can be efficiently prepared. This SPT state is protected by the dipolar bundle symmetries, which are central to the observed behavior. Through a duality analysis between the paramagnetic state and R2TC, we further explore the dipole conservation of anyons in R2TC and investigate its phase diagram.