Quantum complexity of topological phases. Or lack thereof

Topological phases are expected to carry enhanced computational power, stemming from the long-range entanglement that characterizes them. Here we investigate whether this advantage is reflected in quantum magic, quantified through stabilizer Rényi entropies (SREs). We analyze one-dimensional models with an exact duality between a symmetry-protected topological phase and a trivial one. Surprisingly, a finite (parameters dependent) asymmetry appears only when boundary conditions break the duality, and emerges as well in the transverse-field Ising chain, which lacks topological order. These results complement recent analysis in the literature and show that the total magic of a state is unable to detect topological phases, indicating either that a different property is required to capture their unique quantum contribution, or that not all forms of long-range entanglement constitute a genuine computational resource.