The abelian state hidden subgroup problem: Learning stabilizer groups and beyond

Abstract: Identifying the symmetry properties of quantum states is a central theme in quantum information theory and quantum many-body physics. In this work, we investigate quantum learning problems in which the goal is to identify a hidden symmetry of an unknown quantum state. Building on the recent formulation of the state hidden subgroup problem (StateHSP), we focus on abelian groups and develop a quantum algorithmic approach to efficiently learn any hidden symmetry subgroup in this case. We showcase the versatility of the approach in three concrete applications: These are learning (i) qubit and qudit stabilizer groups, (ii) cuts along which a state is unentangled, and (iii) hidden translation symmetries. Taken together, our results underscore the potential of the StateHSP framework as a foundation for broader symmetry-based quantum learning tasks.