An Entropic Orbit-Stabiliser Theorem for Symmetry-Aware Quantum Learning

The classical orbit-stabiliser theorem describes exact symmetry through membership in the stabiliser subgroup. We define an orbit stabilisation entropy $H_{\mathrm{stab}}(\psi )$ and an average stabilisation weight $\overline{W}(\psi )$, and establish the new bound $H_{\mathrm{stab}}(\psi )\ge \log |{\mathrm{Stab}}_G([\psi ])|$, giving an entropic version of the orbit-stabiliser theorem. We further show that $\overline{W}(\psi )$ equals the purity of the group-twirled state. In a controlled quantum learning setting, we study whether state-level symmetry is sufficient to produce symmetry of the learned function. We first encourage state symmetry through soft regularisation using $H_{\mathrm{stab}}(\psi )$ and $\overline{W}(\psi )$, but do not observe a statistically robust improvement in generalisation to unseen symmetry-transformed inputs. We then derive sufficient conditions for exact function-level symmetry and implement them through hard-equivariant architectures, specifically through parameter tying. We test this across five finite groups ${\mathbb{Z}}_2$, ${\mathbb{Z}}_4$, $V_4$, ${\mathbb{Z}}_6$, and $S_3$. The additional orbit-transfer gain from symmetry-enforcing parameter tying is significant for ${\mathbb{Z}}_2$ and $S_3$, but not for the other three groups. Three controlled experiments test possible explanations for this group-dependent effect. Finally, we apply our entropic bound to prune candidate stabiliser subgroups. In subgroup-rich group families, this can substantially reduce the number of candidates that require direct verification, giving a measured $4.12\times$ end-to-end speedup at group order $256$. We also extend our framework to mixed states and study its behaviour under controlled noise and symmetry breaking. Our work opens the door to further study of entropic symmetry measures and their applications in symmetry-aware quantum learning.