Topological phase transitions challenge conventional paradigms in many-body physics by separating phases that are locally indistinguishable yet globally distinct. Using a quantum simulator of interacting erbium atoms in an optical lattice, we observe such a transition between one-dimensional crystalline symmetry-protected topological phases (CSPTs). We detect the critical point through non-local string order parameters and reveal its connection to the transition predicted between the Mott and Haldane insulators. Moreover, we demonstrate a striking property: stacking two identical systems eliminates the transition, confirming the predicted group structure and invertibility of SPTs. Finally, while introducing symmetry-breaking disorder also removes the transition, disorder averaging restores it. Consequently, the adjacent phases realize a form of mixed-state quantum order wherein the criticality between them depends on the observer's information. Our results demonstrate how topology and information influence quantum phase transitions, opening the doors to probing novel critical phenomena in programmable quantum matter.

Topological order (TO) provides a natural platform for storing and manipulating quantum information. However, its stability to noise has only been systematically understood for Abelian TOs. In this Letter, we exploit the nondeterministic fusion of non-Abelian anyons to inform active error-correction and design decoders where the fusion products, instead of flag qubits, herald the noise. This intrinsic heralding offers improved thresholds over those of Abelian counterparts. Furthermore, we use Bayesian inference to obtain a statistical mechanics model for fixed-point non-Abelian TOs with perfect measurements under any noise model, which yields the optimal threshold conditioned on measuring anyon syndromes. We numerically illustrate these results for D_{4}≅Z_{4}⋊Z_{2} TO. In particular, for non-Abelian charge noise and perfect syndrome measurement, we find a conditioned optimal threshold p_{c}=0.218(1), whereas an intrinsically heralded minimum-weight perfect matching (MWPM) decoder already gives p_{c}=0.208 42(2), outperforming standard honeycomb-lattice MWPM with p_{c}=0.158 60(1). Our Letter highlights how non-Abelian properties can enhance stability, rather than reduce it, and discusses potential generalizations for achieving fault tolerance.

Correlation functions ⟨ O_1(x) O_2(0) ⟩ reveal scaling dimensions through spatial decay. We instead consider static susceptibility, the change in ⟨ O_1(x) ⟩ from perturbing the Hamiltonian by O_2(0), which we term causation for short. In a conformal field theory (CFT), dimensional analysis predicts decay of |x|^-2Δ for correlation and |x|^-2Δ+1 for causation. Yet we find causation can decay up to fifteen additional orders in x through a general mechanism, which we trace to time-derivative fields being unable to contribute to static response. In higher-dimensional CFTs, this mechanism ensures leading causation arises from primaries, even when descendants dominate correlation, which we leverage with DMRG to identify a previously unresolved corner primary of Δ≈ 8.8 and a heavy magnetic line defect primary of Δ≈ 4.6 in the (2+1)D critical Ising model. Moreover, the same mechanism governs edge-mode localization in (1+1)D gapless symmetry-protected topological phases, explaining previously observed anomalously small edge-mode splittings and guiding our construction of spin chains with splittings as small as 1/L^18 and 1/L^25.

We argue that the presence of any exact U(1) higher-form symmetry, under mild assumptions, presents a fundamental obstruction to ergodicity under unitary dynamics in lattice systems with local interactions and finite on-site Hilbert space dimension. Focusing on the two-dimensional case, we show that such systems necessarily exhibit Hilbert space fragmentation and explicitly construct Krylov sectors whose number scales exponentially with system size. While these sectors cannot be distinguished by symmetry quantum numbers, we identify the emergent integrals of motion that characterize them. Our symmetry-based approach is insensitive to details of the Hamiltonian and the lattice, providing a systematic explanation for ergodicity breaking in a range of systems, including quantum link models.

Understanding emergent phenomena in out-of-equilibrium interacting many-body systems is an exciting frontier in physical science. While quantum simulators represent a promising approach to this long-standing problem, in practice it can be challenging to directly realize the required interactions, measure arbitrary observables, and mitigate errors. Here we use coherent mapping between the Rydberg and hyperfine qubits in a neutral atom array simulator to engineer and probe complex quantum dynamics. We combine efficient analog dynamics with fully programmable state preparation and measurement, leverage non-destructive readout for loss information and atomic qubit reuse, and use an atom reservoir for replacing lost atoms. With this analog-digital approach, we first demonstrate dynamical engineering of ring-exchange and particle hopping dynamics via Floquet driving and measure the spectral function of single excitations by evolving initial superposition states. Extending these techniques to a 271-site kagome lattice, we employ closed-loop optimization to target an out-of-equilibrium critical quantum spin liquid of the Rokhsar-Kivelson type. We observe the key features of such a state, including the absence of local order, many-body coherences between nearly equal-amplitude dimer configurations over up to 18 sites, and universal correlations consistent with predictions from field theory. Together, these results pave the way for using dynamical control in analog-digital quantum simulators to study complex quantum many-body systems.

Topological quantum computation relies on braiding non-Abelian anyons, but requires the underlying topological order to survive imperfect state preparation and environmental noise. We show that the instability of topological order to wavefunction deformations and to decoherence, with the latter probed by syndrome distributions, are generically captured by stat-mech models whose symmetries naturally expose the corrupting anyonic excitations. As an example, we combine this framework with Monte-Carlo simulations to resolve the stability of D_4 topological order under deformations and quantum channels that proliferate multiple non-Abelian anyon species that individually are unable to condense. We show that beyond a finite threshold, proliferation of two non-Abelian anyon species parasitically condenses a shared Abelian-anyon fusion outcome, destroying the topological order. Our symmetry-based approach sharply differentiates the resulting trivial phase from that obtained by condensing all Abelian charges; in other words, the trivial phase "remembers" which anyons condensed. This framework provides a first step into identifying the relevant symmetry for optimal decoders, conditioned on syndrome measurements, of non-Abelian topological order.

Although symmetry-protected topological phases (SPTs) can be distinguished by their entanglement properties, it has been unclear how to extract this information directly from expectation values beyond the 1D case. Here, we close this gap and propose a pair of nonlocal order parameters that can detect and distinguish all bosonic SPTs in 2D protected by internal discrete, Abelian unitary symmetries. The desired topological invariants are extracted by these quantities by effectively simulating the SPT path integral on topologically non-trivial spacetime manifolds. Our order parameters are defined in terms of expectation values of partial symmetry and permutation operations acting on fixed numbers of replicas of the system in finite spatial regions. These expectation values correspond to symmetry-twisted versions of multipartite entanglement quantities known as multi-entropies. We show explicitly that our two order parameters detect symmetry-protected four-party and six-party entanglement, respectively, and we constrain possible "spurious" contributions. We analytically test our proposal in fixed-point lattice models. Our results suggest multipartite entanglement to be a defining feature of SPTs; indeed, we expect our methods to generalize to fermionic and higher-dimensional systems.

A quantum computer requires the ability to store and manipulate information globally to protect against local noise. Topologically ordered phases1,2 offer two routes: encoding information in the ground-state subspace3 or in anyonic excitations1,4,5. The toric code1 exemplifies the first approach but does not intrinsically support a universal gate set. The latter-topological quantum computation-implements gates by braiding non-Abelian anyons6 around each other. However, the simplest non-Abelian generalizations of the toric code cannot achieve universality by braiding alone7-9. Here we demonstrate that anyon fusion, used as a computational primitive, renders these minimally non-Abelian topologically ordered states universal. We prepare a 54-qubit ground state of the quantum double of S3, the smallest non-Abelian group, on the H2 processor of Quantinuum. We encode logical information in the global fusion space of non-Abelian anyons, and by combining braiding with fusion, we realize a universal topological gate set and read-out, which we demonstrate by topologically preparing a magic state. This demonstrates that the S3 topologically ordered state is scalably preparable, yet rich enough to support a universal gate set. More broadly, this work opens up new pathways for harnessing the intrinsic properties of quantum matter to manipulate quantum information.

Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, S_3, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes S_3 TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.

Quantum many-body scars are nonthermal states exhibiting persistent revivals in an otherwise ergodic, nonintegrable quantum system. Here we leverage the phenomenon of kinetic frustration – the destructive interference of multiple quantum paths – to create exact scars. The simplicity makes these models directly suitable for implementation on multiple existing quantum simulation platforms. In particular, we show how frustrated hardcore bosons in cold atom Bose-Hubbard simulators and polar molecule or Rydberg atom tweezer arrays have persistent oscillations whose lifetimes can be tuned with experimentally accessible parameters, like the Hubbard interaction or a Floquet drive. Second, we propose an experimentally realizable scar within a non-integrable Fermi-Hubbard model where the frustration arises from the fermionic exchange statistics, which admits a one-to-one mapping with the bosonic model in the scar subspace. Finally, we introduce a practical heuristic based on the energy distribution of eigenstates for systematically predicting and optimizing quantum many-body scar lifetimes. Their cross-platform realizability and long lifetimes make them well-suited for benchmarking coherence and exploring nonergodic dynamics in current and near-term quantum devices.

Gapless quantum phases can become distinct when internal symmetries are enforced, in analogy with gapped symmetry-protected topological (SPT) phases. However, this distinction does not always lead to protected edge modes, raising the question of how the bulk-boundary correspondence is generalized to gapless cases. We propose that the spatial interface between gapless phases -- rather than their boundaries -- provides a more robust fingerprint. We show that whenever two 1+1d conformal field theories (CFTs) differ in symmetry charge assignments of local operators or twisted sectors, any symmetry-preserving spatial interface between the theories must flow to a non-invertible defect. We illustrate this general result for different versions of the Ising CFT with $\mathbb{Z}_2 \times \mathbb{Z}_2^T$ symmetry, obtaining a complete classification of allowed conformal interfaces. When the Ising CFTs differ by nonlocal operator charges, the interface hosts 0+1d symmetry-breaking phases with finite-size splittings scaling as $1/L^3$, as well as continuous phase transitions between them. For general gapless phases differing by an SPT entangler, the interfaces between them can be mapped to conformal defects with a certain defect 't Hooft anomaly. This classification also gives implications for higher-dimensional examples, including symmetry-enriched variants of the 2+1d Ising CFT. Our results establish a physical indicator for symmetry-enriched criticality through symmetry-protected interfaces, giving a new handle on the interplay between topology and gapless phases.

The development of programmable quantum devices can be measured by the complexity of many-body states that they are able to prepare. Among the most significant are topologically ordered states of matter, which enable robust quantum information storage and processing. While topological orders are more readily accessible with qudits, experimental realizations have thus far been limited to lattice models of qubits. Here, we prepare and measure a ground state of the Z 3 toric code state on 24 qutrits (obtained by encoding one qutrit into two qubits) in a trapped ion quantum processor with fidelity per qutrit exceeding 96.5(3)%. We manipulate two types of defects which go beyond the conventional qubit toric code: a parafermion, and its bound state which is related to charge conjugation symmetry. We further demonstrate defect fusion and the transfer of entanglement between anyons and defects, which we use to control topological qutrits. Our work opens up the space of long-range entangled states with qudit degrees of freedom for use in quantum simulation and universal error-correcting codes.

It is well known that an exponentially localized Hamiltonian must be gapless if its ground state has algebraic correlations. We show that even certain exponentially decaying correlations can imply gaplessness. This is exemplified by the deformed toric code ∝exp(β∑_ℓ Z_ℓ) |𝖳𝖢⟩, where |𝖳𝖢⟩ is a fixed-point toric code wavefunction. Although it has a confined regime for β > β_c, recent work has drawn attention to its perimeter law loop correlations. Here, we show that these unusual loop correlations – namely, perimeter law coexisting with a 1-form symmetry whose disorder operator has long-range order – imply that any local parent Hamiltonian must either be gapless or have a degeneracy scaling with system size. Moreover, we construct a variational low-energy state for arbitrary local frustration-free Hamiltonians, upper bounding the finite-size gap by O(1/L^3) on periodic boundary conditions. Strikingly, these variational states look like loop waves – non-quasiparticle analogs of spin waves – generated from the ground state by non-local loop operators. Our findings have implications for identifying the subset of Hilbert space to which gapped ground states belong, and the techniques have wide applicability. For instance, a corollary of our first result is that Glauber dynamics for the ordered phase of the two-dimensional classical Ising model on the torus must have a gapless Markov transition matrix, with our second result bounding its gap.

The effect of decoherence on topological order (TO) has been most deeply understood for the toric code, the paragon of Abelian TOs. We show that certain non-Abelian TOs can be analyzed and understood to a similar degree, despite being significantly richer. We consider both wavefunction deformations and quantum channels acting on D_4 TO, which has recently been realized on a quantum processor. By identifying the corresponding local statistical mechanical spin or rotor model with D_4 symmetry, we find a remarkable stability against proliferating non-Abelian anyons. This is shown by leveraging a reformulation in terms of the tractable O(2) loop model in the pure state case, and n coupled O(2) loop models for Rényi-n quantities in the decoherence case – corresponding to worldlines of the proliferating anyon with quantum dimension 2. In particular, we find that the purity (n=2) remains deep in the D_4 TO for any decoherence strength, while the n →∞ limit becomes critical upon maximally decohering a particular anyon type, similar to our wavefunction deformation result. The information-theoretic threshold (n→ 1) appears to be controlled by a disordered version of these stat-mech models, akin to the toric code case although significantly more robust. We furthermore use Monte Carlo simulations to explore the phase diagrams when multiple anyon types proliferate at the same time, leading to a continued stability of the D_4 TO in addition to critical phases with emergent U(1) symmetry. Instead of loop models, these are now described by net models corresponding to different anyon types coupled together according to fusion rules.This opens up the exploration of statistical mechanical models for decohered non-Abelian TO, which can inform optimal decoders, and which in an ungauged formulation examples of non-Abelian strong-to-weak symmetry breaking.

Measurements and feedback have emerged as powerful resources for creating many-body quantum states. However, a detailed understanding has been restricted to fixed-point representatives of phases of matter. Here, we go beyond this and characterize the patterns of many-body entanglement that can be deterministically created from measurement. Focusing on one spatial dimension, a framework is developed for the case where a single round of measurements is the only entangling operation. We show this creates matrix-product states and identify necessary and sufficient tensor conditions for preparability, which uniquely determine the preparation protocol. We use these conditions to both classify preparable quantum states and characterize their physical constraints. In particular, we find a trade-off between the richness of the preparable entanglement spectrum and correlation functions, which leads to a no-go theorem for preparing certain quantum states. More broadly, we connect properties of the preparation protocol to the resulting phase of matter, including trivial, symmetry-breaking, and symmetry-protected topological phases—for both uniform and modulated symmetries. This work offers a resource-theoretic perspective on preparable quantum entanglement and shows how to systematically create states of matter, away from their fixed points, in quantum devices.

Creating and manipulating anyons and symmetry defects in topological phases, especially those with a non-Abelian character, constitutes a primitive for topological quantum computation. We provide a physical protocol for implementing the ribbon operators of non-Abelian anyons and symmetry defects. We utilize dualities, in particular, the Kramers-Wannier or gauging map, which have previously been used to construct topologically ordered ground states by relating them to simpler states. In this Letter, ribbon operators are implemented by applying a gauging procedure to a lower-dimensional region of such states. This protocol uses sequential unitary circuits or, in certain cases, constant-depth adaptive circuits. We showcase this for anyons and defects in the Z_{3} toric code and S_{3} quantum double. The general applicability of our method is demonstrated by deriving unitary expressions for ribbon operators of various (twisted) quantum doubles.

Decohering topological order (TO) is central to the many-body physics of open quantum matter and decoding transitions. We identify statistical mechanical models for decohering non-Abelian TOs, which have been crucial for understanding the error threshold of Abelian stabilizer codes. The decohered density matrix can be described by loop models, whose topological loop weight N is the quantum dimension of the decohering anyon-reducing to the Ising model if N = 1. In particular, the Renyi-n moments correspond to n coupled O(N) loop models. Moreover, by diagonalizing the density matrix at maximal error rate, we connect the fidelity between two logically distinct ground states to random O(N) loop and spin models. We find a remarkable stability to quantum channels which proliferate non-Abelian anyons with large quantum dimension, with the possibility of critical phases for smaller dimensions. Intuitively, this stability is due to non-Abelian anyons not admitting finite-depth string operators. We confirm our framework with exact results for Kitaev quantum double models, and with numerical simulations for the non-Abelian phase of the Kitaev honeycomb model. Our work opens up the possibility of non-Abelian TO being robust against maximally proliferating certain anyons, which can inform error-correction studies of these topological memories.

Symmetry-protected topological (SPT) phases are commonly required to have an energy gap, but recent work has extended the concept to gapless settings. This raises a natural question: what happens at transitions between inequivalent gapless SPTs? We address this for the simplest known case among gapless SPTs protected by a unitary symmetry group acting faithfully on the low-energy theory. To this end, we consider a qutrit version of the nearest-neighbor XX chain. Trimerizing the chain explicitly breaks an anomalous symmetry and produces three distinct gapped SPT phases protected by a unitary ℤ_3 ×ℤ_3 symmetry. Their phase boundaries are given by three inequivalent gapless SPTs without any gapped symmetry sectors, each described by a symmetry-enriched version of an orbifolded Potts^2 conformal field theory with central charge c=8/5. We provide an analytic derivation of this critical theory in a particular regime and confirm its stability using tensor network simulations. Remarkably, the three gapless SPTs meet at a c = 2 multicritical point, where the protecting ℤ_3 ×ℤ_3 symmetry exhibits a mixed anomaly with the ℤ_3 entangler symmetry that permutes the SPT classes. We further explore how discrete gauging gives dipole-symmetric models, offering insights into dipole symmetry-breaking and SPTs, as well as symmetry-enriched multiversality. Altogether, this work uncovers a rich phase diagram of a minimal qutrit chain, whose purely nearest-neighbor interactions make it a promising candidate for experimental realization, including the prospect of critical phases with stable edge modes.

The effect of decoherence on topological order (TO) has been most deeply understood for the toric code, the paragon of Abelian TOs. We show that certain non-Abelian TOs can be analyzed and understood to a similar degree, despite being significantly richer. We consider both wave-function deformations and quantum channels acting on D4 TO, which has recently been realized on a quantum processor. By identifying the corresponding local statistical mechanical spin or rotor model with D4 symmetry, we find a remarkable stability against proliferating non-Abelian anyons. This is shown by leveraging a reformulation in terms of the tractable O(2) loop model in the pure state case and n coupled O(2) loop models for Renyi-n quantities in the decoherence case-corresponding to worldlines of the proliferating anyon with quantum dimension 2. In particular, we find that the purity (n 1/4 2) remains deep in the D4 TO for any decoherence strength, while the n-* infinity limit becomes critical upon maximally decohering a particular anyon type, similar to our wave-function deformation result. The information-theoretic threshold (n-* 1) appears to be controlled by a disordered version of these statistical mechanical models, akin to the toric code case although significantly more robust. We furthermore use Monte Carlo simulations to explore the phase diagrams when multiple anyon types proliferate at the same time, leading to a continued stability of the D4 TO in addition to critical phases with emergent U(1) symmetry. Instead of loop models, these are now described by net models corresponding to different anyon types coupled together according to fusion rules. This opens up the exploration of statistical mechanical models for decohered non-Abelian TO, which can inform optimal decoders and which in an ungauged formulation provides examples of non-Abelian strong-to-weak symmetry breaking.

Non-Abelian topological order (TO) enables topologically protected quantum computation with its anyonic quasiparticles. Recently, TO with S_3 gauge symmetry was identified as a sweet spot – simple enough to emerge from finite-depth adaptive circuits yet powerful enough to support a universal topological gate-set. In these notes, we review how anyon braiding and measurement in S_3 TO are primitives for topological quantum computation and we explicitly demonstrate universality. These topological operations are made concrete in the S_3 quantum double lattice model, aided by the introduction of a generalized ribbon operator. This provides a roadmap for near-term quantum platforms.