Eigenstate switching of topologically ordered states using local non-Hermitian perturbations

Topologically ordered phases have robust degenerate ground states against the local perturbations, thus it provides a promising platform for fault-tolerant quantum computation. While the non-local storage of information ensures robustness against local perturbations, it is unclear how to manipulate such an order in a versatile manner. In this work, we explore the potential of non-Hermitian exceptional points (EP) in the control of topological order. EPs arise due to non-Hermiticity, leading to the coalescence of eigenstates. We propose a novel scheme that utilizes $mathcal{PT}$-symmetric perturbations to induce eigenstate coalescence between different ground states of the toric code model. Adiabatic encircling EPs allows for the controlled switching of eigenstates, enabling dynamic manipulation of the topological degeneracy. Interestingly, we show a remarkable property of our scheme that arbitrary strengths of local perturbations can induce the EP and eigenstate switching. Finally, in a non-adiabatic regime, we also show the orientation-dependent behavior of non-adiabatic transitions (NAT) during EP encirclement. our work that controls the non-Hermiticity can serve as a promising strategy for fault-tolerant quantum information processing.