Relaxing group and representation assumptions for symmetry learning

Determine whether efficient quantum algorithms for quantum symmetry learning can be extended to discrete or continuous infinite groups, arbitrary finite non-abelian groups, and projective representations for Anyonic Symmetry Learning over suitable non-abelian groups.

Background

The paper's algorithms impose structural restrictions on the ambient group and representation: for example, Anyonic Symmetry Learning with linear representations is treated for polynomially near-abelian groups, projective-representation instances are treated for finite abelian groups, and Bose Symmetry Learning is treated for polynomially near-Hamiltonian groups. The authors explicitly identify relaxing these restrictions as unresolved, with particular questions concerning infinite groups, arbitrary finite non-abelian groups, and projective representations over non-abelian groups.

References

For example: \begin{itemize} \item Can we allow $G$ to be discrete but infinite, or even continuous? \item Can we allow $G$ to be any finite non-abelian group? \item Can we allow for $R$ to be projective for Anyonic Symmetry Learning over some non-abelian groups? \end{itemize}

— Learning quantum symmetries  (2609.38085 - Holt et al., 29 Sep 2026) in Section 'Outlook and future directions', paragraph 'Relaxing requirements on the group and representation'

There is an efficient quantum algorithm for #1{BoseSL} over central extensions of abelian groups.

— Learning quantum symmetries  (2609.38085 - Holt et al., 29 Sep 2026) in Section 'Symmetries of other quantum objects', subsection 'Bose symmetry learning over central extensions'