We study long-range nonstabilizerness (LRN), namely the obstruction to remove nonstabilizerness with shallow-depth local quantum circuits. In one-dimensional settings, the mutual information between disconnected spatial regions has proven to be a powerful tool to diagnose LRN. In this work, we focus on encoded states of two-dimensional topologically-ordered systems, and explore the ability of the mutual information to serve as a diagnostic of LRN. Focusing on the concrete setting of lattice models defined on a torus, we show that information about LRN can be gained from the analysis of the mutual information between non-overlapping regions containing non-contractible loops, and of the change of such mutual information under modular real-space transformations. We exemplify this idea in the toric code and the non-abelian string-net model with doubled Fibonacci topological order. In the former case, we show that the mutual information provides a full classification, certifying LRN for all encoded non-stabilizer states. In the latter case, instead, our approach does not lead to a full classification, as it detects LRN for all states except from a finite subset with special transformation properties under the modular group. Finally, we discuss how our results on LRN constrain the logical gates that can be implemented fault-tolerantly on the torus.
We study long-range nonstabilizerness (LRN), namely the obstruction to remove nonstabilizerness with shallow-depth local quantum circuits. In one-dimensional settings, the mutual information between disconnected spatial regions has proven to be a powerful tool to diagnose LRN. In this work, we focus on encoded states of two-dimensional topologically-ordered systems, and explore the ability of the mutual information to serve as a diagnostic of LRN. Focusing on the concrete setting of lattice models defined on a torus, we show that information about LRN can be gained from the analysis of the mutual information between non-overlapping regions containing non-contractible loops, and of the change of such mutual information under modular real-space transformations. We exemplify this idea in the toric code and the non-abelian string-net model with doubled Fibonacci topological order. In the former case, we show that the mutual information provides a full classification, certifying LRN for all encoded non-stabilizer states. In the latter case, instead, our approach does not lead to a full classification, as it detects LRN for all states except from a finite subset with special transformation properties under the modular group. Finally, we discuss how our results on LRN constrain the logical gates that can be implemented fault-tolerantly on the torus.
Long-range nonstabilizerness can be defined as the amount of nonstabilizerness that cannot be removed by shallow local quantum circuits. We study long-range nonstabilizerness in the context of many-body quantum physics, a task with possible implications for quantum-state preparation protocols and implementation of quantum-error correcting codes. After presenting a simple argument showing that long-range nonstabilizerness is a generic property of many-body states, we restrict to the class of ground states of gapped local Hamiltonians. We focus on one-dimensional (1D) systems and present rigorous results in the context of translation-invariant matrix product states (MPSs), with our analysis extending to all gapped 1D phases under widely accepted physical assumptions. By analyzing the fixed points of the MPS renormalization-group flow, we provide a sufficient condition for long-range nonstabilizerness, which depends entirely on the local MPS tensors. Physically, our condition captures the fact that the mutual information between distant regions of stabilizer fixed points is quantized and remains so after applying shallow quantum circuits.
Licensing: paper, figures and data are under CC BY 4.0; the source code is under the Apache License, Version 2.0. The paper text names the code licence "Apache License 2.0" — this is the same licence. Mutual information between distant regions of a topologically ordered state can witness long-range nonstabilizerness ("long-range magic"). For anyon states on a torus it takes an operational form: the Shannon entropy of the flux (anyon-type) statistics along a loop, evaluated in every frame generated by the modular transformations S and T. The witness is one-sided: a non-integer entropy certifies magic, an integer entropy certifies nothing. We ask whether it can certify the states that were actually prepared in the trapped-ion demonstration of non-Abelian D(D_4) topological order. In exact arithmetic, under a model of the prepared states (below), we find that it cannot. Modeling the 22 ground states of that experiment as the joint eigenbasis of the abelian Wilson loops, we find all of them blind to the witness in every frame, because S and T permute them up to phases and each is uniform over two or eight anyon types, so the entropy is 1 or 3 bits throughout. We then give a protocol that leaves the blind set: one logical string operator, controlled by an ancilla and applied at a rotation angle t≠π/4. In simulation, and in an exploratory robustness analysis, at least 42.5 % of all angles give a distance of at least 0.25 bit from the nearest integer; the budget is 200 to 300 flux samples (about 300 with a conservative decision rule). Because the target states are blind, even 1 % depolarizing noise produces false certificates at a few thousand samples unless the decision rule is protected; in our model the protected rule tolerates up to 6 % in the best case. One pre-registered prediction was refuted and one proved ill-posed; both are reported as such. A short chapter derives requirements for a photonic realization with qudits of dimension 8: announced state-independent loss is harmless, and state-dependent loss shifts the tested state only at second order. Scripts, result files, and the pre-registration are deposited with this paper. This record contains the manuscript, the 13 scripts (verification, analysis, and the figure), their result files, and the pre-registration with its deviations. The README explains how to re-run the scripts and which result files are reproduced bit for bit after normalizing line endings to LF (Python on Windows writes CRLF). Use of AI tools: In the research, processing, and writing of this paper and its results I worked together with generative AI tools, in practice a system of multiple coordinated AI instances that I set up and orchestrate (large language models, mainly Claude, by Anthropic, inside Claude Code). At their current context sizes I found it far more effective to work with several specialized instances, each with its own role and its own harness of rules and parameters that I designed and refined through feedback, than to load a single instance with all of the material; for my workflow that would have been inefficient, though this depends on the individual implementation. I lead this collaboration: I choose the research directions, set the goals, and make the final decisions in open exchange with the AI, learning actively as the work proceeds. The AI carries out the drafting, including the mathematical and technical parts, the numerical computation, and the literature search, under my direction. The AI works autonomously only task by task, within the structure I develop through feedback: it completes a task, and at open questions that need me it stops until the point is settled before the next step. Along the way I witness and take many of the decisions that shape the path, and it is common for me to spot things that need improvement. The work spans many separate runs, and a single simulation or build task alone can take up to an hour, so it could not happen all together in one autonomous run; and had I let the AI do all of it together alone, even if it is possible, it would no longer be my work but the AI's. I run multiple verifications at the different stages of the work and one before release, and all references are checked against the original sources. In the end what matters are human eyes, a principle that is itself written into the parameters of my system: I reach out to experts after publishing for review and feedback, so I learn what is solid and what must be corrected or falsified. My scripts for reproduction and review are released with this record. These tools are not authors; I am the author, and I take full responsibility for all scientific content and decisions leading to these results and their publication.
We study nonstabilizerness on the information lattice and demonstrate that noninteger local information directly indicates nonstabilizerness. For states with a clear separation of short- and large-scale information, a noninteger total information at large scales Γ serves as a witness of long-range nonstabilizerness. We propose a folding procedure to separate the global and edge-to-edge contributions to Γ. As an example we show that the ferromagnetic ground state of the spin-1/2 three-state Potts model has long-range nonstabilizerness originating from global correlations, while the paramagnetic ground state has at most short-range nonstabilizerness.
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.
Licensing: paper, figures and data are under CC BY 4.0; the source code is under the Apache License, Version 2.0. The paper text names the code licence "Apache License 2.0" — this is the same licence. Korbany, Ellison, Stephen and Piroli showed that a non-integer value, in some modular frame, of the Shannon entropy of the topological-sector weights of a topologically ordered ground state, which equals the mutual information between separated regions, certifies long-range nonstabilizerness. They treated the toric code and doubled Fibonacci. We ask where this witness is blind on doubled SU(2)_k string nets. For minimal-entropy states whose modular-data columns are flat, we prove that the witness is integer in every modular frame if and only if the column supports are powers of two. The proof embeds the modular representation into the Clifford/Weil representation of a qudit of dimension 2(k+2). We map all minimal-entropy states for k ≤ 32 (the flat-column characterization was registered before the computations for 17 ≤ k ≤ 32), show that the doubled-Ising state σσ̄ is bit-blind in exact arithmetic although the ground-space dimension criterion of Wei and Liu certifies strong long-range magic in a qubit embedding, and show that the blind spots depend on the encoding of the local degrees of freedom. The modular-invariant diagonal state is blind exactly when the number of anyons is a power of the local prime. We argue that it is the ground state obtained from the empty string-net configuration, and confirm this numerically on the doubled-Fibonacci honeycomb torus. This record contains the manuscript, the 16 verification scripts and their result files, the pre-registration documents (one of them redacted, see README) and the material of the cold reads described below. The README explains how to re-run the scripts and states which hash cannot be checked against the deposited copy.Notice (2026-10-__). Two sentences of the introduction will be revised in the next version; no computed value and no result changes.(1) The value H = 1.700979 of Ref. [Sayim, 5a] is described as "one of two long-range-nonstabilizerness certificates reported there". Following the correction notice of that work (2026-10-10), it is the long-range certificate reported there; the other item shows only that sampled ground-space states are not stabilizer states.(2) The sentence "the witness and its theorem are due to Ref. [Korbany et al., arXiv:2605.22424]" will also state that the idea of certifying long-range nonstabilizerness by non-integer mutual information goes back to D. A. Korbany, M. J. Gullans and L. Piroli, Phys. Rev. Lett. 135, 160404 (2025), and N. Parham, arXiv:2504.19966, as Sec. 3.1 of arXiv:2605.22424 states. Both works are already cited in the paper. Use of AI tools: In the research, processing, and writing of this paper and its results I worked together with generative AI tools, in practice a system of multiple coordinated AI instances that I set up and orchestrate (large language models, mainly Claude, by Anthropic, inside Claude Code and, for part of the work, in the Claude AI web interface). At their current context sizes I found it far more effective to work with several specialized instances, each with its own role and its own harness of rules and parameters that I designed and refined through feedback, than to load a single instance with all of the material; for my workflow that would have been inefficient, though this depends on the individual implementation. I lead this collaboration: I choose the research directions, set the goals, and make the final decisions in open exchange with the AI, learning actively as the work proceeds. The AI carries out the drafting, including the mathematical and technical parts, the numerical computation, and the literature search, under my direction. The AI works autonomously only task by task, within the structure I develop through feedback: it completes a task, and at open questions that need me it stops until the point is settled before the next step. Along the way I witness and take many of the decisions that shape the path, and it is common for me to spot things that need improvement. The work spans many separate runs, and a single simulation or build task alone can take up to an hour, so it could not happen all together in one autonomous run; and had I let the AI do all of it together alone, even if it is possible, it would no longer be my work but the AI's. I run multiple verifications at the different stages of the work and one before release, and all references are checked against the original sources. Before release, the statements and proofs of the main theorem were cold-read in one run, and two of its lemmas in a second run, each in a new Claude AI chat (Opus 5.5, high reasoning effort) given only extracted text: the first reader computed the list of blind states for k <= 16 itself, conjectured the rule of Theorem 1 and tested it at four larger k, and the readers reported imprecisions in the statements of two lemmas, which I corrected; the corrected text was not read again by a fresh reader. In the end what matters are human eyes, a principle that is itself written into the parameters of my system: I reach out to experts after publishing for review and feedback, so I learn what is solid and what must be corrected or falsified. My scripts for reproduction and review are released with this record. These tools are not authors; I am the author, and I take full responsibility for all scientific content and decisions leading to these results and their publication.
As a necessary resource for quantum computational advantage, quantum magic (nonstabilizerness) is of fundamental importance in the study of quantum computation and physics. We develop a systematic theory of \emph{long-range magic (LRM)} -- nonstabilizerness that cannot be erased by shallow unitary circuits -- and demonstrate its broad relevance. By bridging LRM with fault-tolerant logical gate theory, we show the emergence of LRM families from quantum error-correcting codes and devise a simple yet powerful method for testing transversal logical gates. Further, we introduce and characterize \emph{LRM phases} in which all ground states exhibit LRM, and identify certain non-Abelian topological orders as representative examples. Then, adopting a complexity theory perspective, we demonstrate the classicality of non-LRM systems in e.g.~preparation and learning settings, and present a ``no low-energy trivial magic'' (NLTM) conjecture with key motivation in the quantum PCP context, for which our LRM results suggest a promising route. Additionally, we demonstrate how to diagnose LRM with correlation functions. The concepts and results admit substantive extensions to approximate (robust) and nongeometric scenarios. Our LRM theory illuminates profound new connections among quantum resources, computational advantage, error correction and fault tolerance, many-body physics, and complexity theory.