Extensive long-range magic in non-Abelian topological orders
This work refines the understanding of complexity in topological phases for quantum information theorists, establishing a fundamental limitation on approximating non-Abelian topological states with stabilizer states.
The authors prove that low-energy states of non-Abelian topological orders have extensive magic that is long-ranged and cannot be removed by constant-depth local unitary circuits, providing a new resource-theoretic characterization of topological orders. They also show that any state with an entanglement area law and nontrivial fusion spaces must exhibit such magic.
We show that the low-energy states of non-Abelian topological orders possess extensive magic which is long-ranged, and cannot be eliminated by a constant-depth local unitary circuit. This refines conventional notions of complexity beyond the linear circuit depth which is required to prepare any topological phase, and provides a new resource-theoretic characterization of topological orders. A central technical result is a no-go theorem establishing that stabilizer states--even up to constant-depth local unitarie--cannot approximate low-energy states of non-Abelian string-net models which satisfy the entanglement bootstrap axioms. Moreover, we show that stabilizer-realizable Abelian string-net phases have mutual braiding phases quantized by the on-site qudit dimension, and that any violation of this condition necessarily implies extensive long-range magic. Extending to higher spatial dimensions, we argue that any state obeying an entanglement area law and hosting excitations with nontrivial fusion spaces must exhibit extensive long-range magic. This applies, in particular, to ground-states and low-energy states of higher-dimensional quantum double models.