Extensive long-range magic in non-Abelian topological orders

TL;DR

This paper demonstrates that non-Abelian topological orders have extensive, long-range quantum magic that cannot be simplified by local unitaries, revealing new complexity and resource properties of such phases.

Contribution

It establishes that low-energy states of non-Abelian topological orders possess inherent long-range magic, refining understanding of their complexity beyond linear-depth circuit requirements.

Findings

Stabilizer states cannot approximate non-Abelian string-net ground states.

Abelian string-net phases have quantized braiding phases linked to qudit dimension.

States with nontrivial fusion spaces must exhibit extensive long-range magic.

Abstract

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.