Learning quantum symmetries

Abstract: Quantum algorithms are powerful tools for finding symmetries of classical objects, most famously through Shor's algorithm and the framework of the Hidden Subgroup Problem (HSP). In this work, we study quantum algorithms for learning symmetries of quantum objects. Our starting point is the recently introduced State Hidden Subgroup Problem (StateHSP), a quantum generalisation of HSP in which the task is to learn the symmetry subgroup of a quantum state. We obtain positive results for non-abelian StateHSP, giving efficient quantum algorithms whenever the hidden subgroup is normal and for ambient groups belonging to a broad class of non-abelian groups, extending the previous general theory beyond the abelian setting. StateHSP learns \emph{Bose} symmetries, associated with linear representations, whereas physically equivalent pure states are defined only up to global phase. Motivated by this, we introduce a new \emph{Anyonic} state symmetry learning problem based on a more physically natural notion of symmetry than that considered by StateHSP. We give an efficient quantum algorithm by reducing the problem to StateHSP, where the reduction relies on a new connection between linearisations of projective representations and linear error-correcting codes. As an application, we obtain an improved algorithm for learning stabiliser groups that applies to mixed qudit states of arbitrary local dimension. Finally, we introduce symmetry learning problems for other quantum objects, including unitaries, Hamiltonians, and finite collections of states, and obtain efficient quantum algorithms by reducing them to instances of StateHSP.