When Symmetry Suppresses Magic
Nonstabilizerness is a critical resource for quantum advantage, but evaluating it, especially for mixed states, requires superexponentially many samples in system size, making the problem NP-hard. While symmetries are known to reduce this complexity, it is unclear whether they also restrict the amount of magic. In this work, we provide an explicit example of such a symmetry by proving that the Robustness of Magic (RoM) for N-qubit X-states is at most $\sqrt{3}$. Leveraging this symmetry constraint, we introduce a computationally efficient method to lower-bound the RoM of arbitrary many-body states, demonstrating its utility on the ground states of a spin-1/2 Hamiltonian with system sizes well beyond the reach of exact evaluation. Furthermore, for Hamiltonians whose equilibrium states are X-states, we analytically derive the critical temperature at which magic emerges. Our work shows why certain symmetries constrain magic while others do not, provides a scalable lower-bounding method, and identifies a nontrivial regime where many-body nonstabilizerness is analytically solvable.