Extensive long-range magic in non-Abelian topological orders
2026-05-14 • Computational Complexity
Computational Complexity
AI summaryⓘ
The authors study special quantum states called non-Abelian topological orders and find that these states have a complex property called 'magic' that is spread out over long distances and can't be simplified by shallow local operations. They prove that certain simpler states (stabilizer states) cannot come close to these complex states, highlighting a new way to understand the complexity of topological phases. They also show that if a related system violates specific conditions, it must contain this extensive magic. Finally, they extend their findings to higher-dimensional systems, showing that complex quantum states with certain excitations also have extensive magic.
Non-Abelian topological orderMagic (quantum resource)Stabilizer statesLocal unitary circuitsString-net modelsEntanglement bootstrap axiomsAbelian phasesBraiding phasesQuantum double modelsEntanglement area law
Authors
Yuzhen Zhang, Isaac H. Kim, Yimu Bao, Sagar Vijay
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.