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

TL;DR

This paper introduces an efficient quantum algorithm for identifying hidden abelian symmetry subgroups in quantum states, with applications in learning stabilizer groups, entanglement cuts, and translation symmetries, expanding the scope of quantum learning techniques.

Contribution

It develops a generalized Fourier sampling approach for the abelian state hidden subgroup problem, enabling new quantum learning applications and unifying existing primitives.

Findings

Efficient quantum algorithm for abelian StateHSP

Applications to stabilizer, entanglement, and translation symmetries

Unification of Bell sampling methods within Fourier sampling

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 an efficient quantum algorithm that learns any hidden symmetry subgroup using a generalized form of Fourier sampling. 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. Through these applications, we reveal that well-known quantum learning primitives, such as Bell sampling and Bell difference sampling, are in fact special cases of Fourier sampling. Our results highlight the broad potential of the StateHSP framework for symmetry-based quantum learning tasks.