Extensive long-range magic in non-Abelian topological orders

Yuzhen Zhang , Isaac H. Kim , Yimu Bao , Sagar Vijay

Authors on Pith no claims yet

Pith reviewed 2026-05-15 03:19 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-elcs.CChep-th

keywords extensivemagictopologicallong-rangelow-energynon-abelianordersstates

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The pith

Non-Abelian topological orders exhibit extensive long-range magic in low-energy states, which cannot be removed by constant-depth circuits and distinguishes them via a no-go theorem from stabilizer states.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In quantum physics, topological orders are phases of matter with special properties protected by topology, useful for fault-tolerant quantum computing. Non-Abelian topological orders are particularly interesting because their excitations have fusion spaces that allow for complex braiding statistics. The paper argues that the low-energy states of these orders contain extensive magic, a measure of quantum non-stabilizerness that is spread over long distances. This magic cannot be removed by applying constant-depth local unitary circuits, which are shallow operations that don't create long-range entanglement. This finding refines our understanding of the complexity needed to prepare such states, going beyond the linear depth required for any topological phase. A key technical result is a no-go theorem showing that even stabilizer states, which are classically simulable, cannot approximate these non-Abelian states up to constant-depth circuits. For Abelian cases, the paper shows that braiding phases must be quantized by the qudit dimension for stabilizer realizability, and violations imply magic. In higher dimensions, any state with an entanglement area law and excitations with nontrivial fusion spaces must have this extensive magic, applying to quantum double models. This provides a resource-theoretic view of topological orders.

Core claim

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.

Load-bearing premise

The states satisfy the entanglement bootstrap axioms for non-Abelian string-net models.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Non-Abelian topological orders carry extensive long-range magic that constant-depth local unitaries cannot remove.

read the letter

The main thing to know is that low-energy states of non-Abelian topological orders have extensive magic that stays long-ranged and survives any constant-depth local unitary circuit. The paper backs this with a no-go theorem showing stabilizer states cannot approximate the relevant string-net models under the entanglement bootstrap axioms, plus a quantization rule for braiding phases in the Abelian case and a direct extension to higher-dimensional area-law states with nontrivial fusion spaces, including quantum doubles. What is new is the resource-theoretic framing of this magic as a long-range distinguisher that refines circuit-complexity ideas for topological phases. The no-go result is a clean technical step that does not rely on fitted parameters and follows from the stated axioms without obvious circularity. The higher-dimensional argument tracks logically from the same premises. The soft spot is limited: everything rests on the entanglement bootstrap axioms holding for the models in question, so the claim does not automatically cover every non-Abelian order outside that class, though the paper scopes it clearly and the evidence inside the scope holds up. This is for readers working on magic monotones, topological order, and many-body complexity. Someone interested in resource theories or quantum computing applications would get concrete value from the new characterization. It deserves serious peer review because the central theorem is precise and the implications are falsifiable.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that low-energy states of non-Abelian topological orders exhibit extensive long-range magic that cannot be removed by constant-depth local unitary circuits. A central result is a no-go theorem showing that stabilizer states cannot approximate low-energy states of non-Abelian string-net models satisfying the entanglement bootstrap axioms. The work further shows that Abelian string-net phases realizable by stabilizers have mutual braiding phases quantized by the on-site qudit dimension, with violations implying extensive magic, and extends the argument to higher-dimensional area-law states with nontrivial fusion spaces, including quantum double models.

Significance. If the no-go theorem is rigorously established, the result supplies a resource-theoretic characterization of non-Abelian topological order that refines circuit-depth complexity measures and identifies magic as an intrinsic, long-range feature. The extension to higher dimensions via area-law states with nontrivial fusion spaces is a natural and potentially impactful generalization.

major comments (1)

[no-go theorem] The no-go theorem (abstract and central technical result): the argument that stabilizer states cannot approximate the target states even after constant-depth local unitaries rests on the entanglement bootstrap axioms, but the manuscript does not explicitly show how these axioms produce a contradiction with zero magic for all possible constant-depth circuits; a concrete step-by-step derivation linking the axioms to the magic lower bound would make the load-bearing claim fully verifiable.

minor comments (2)

[Abstract] The abstract introduces 'extensive long-range magic' without a brief operational definition; adding one sentence clarifying the scaling with system size and the distance scale would improve readability.
[Abelian string-net section] Notation for the on-site qudit dimension and the mutual braiding phase quantization condition should be introduced once and used consistently in the Abelian string-net discussion.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper builds on established concepts in topological quantum field theory and quantum information without introducing new free parameters or entities; the central claim depends on the validity of the entanglement bootstrap framework.

axioms (1)

domain assumption The entanglement bootstrap axioms for string-net models
Invoked to establish the properties of non-Abelian topological orders for the no-go theorem.

pith-pipeline@v0.9.0 · 5483 in / 1232 out tokens · 58946 ms · 2026-05-15T03:19:06.284496+00:00 · methodology

Extensive long-range magic in non-Abelian topological orders

Core claim

Load-bearing premise

discussion (0)