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arXiv:2509.08053 · Physical review. B./Physical review. B

Resource complexity of symmetry-protected topological phases

Alberto Giuseppe Catalano, Sven Benjamin Kožić, Gianpaolo Torre, Carola Ciaramelletti, Simone Paganelli, Fabio Franchini, S. M. Giampaolo

Published Mar 12, 2026• 1 citation
arXivHTMLPDF

Abstract

We pursue the identification of quantum resources carried by topological order, by evaluating quantum magic, quantified through the rank-$2$ Stabilizer Rényi entropy $\mathcal{M}_2$, in one-dimensional systems hosting symmetry-protected topological phases (SPTP). Focusing on models with an exact duality between an SPTP and a trivial one, namely the dimerized XX and the Cluster-Ising chains, we show that dual points exhibit identical amounts of magic, even thought they belong to distinct topological sectors. A subextensive asymmetry arises only under open boundary conditions, where edge effects break the duality, but this correction is non-topological and depends on microscopic parameters. These results stand in contrast to the case of topological frustration, where delocalized excitations enhance the magic logarithmically with system size. They also complement recent analyses in the literature, showing that the total magic is largely insensitive to the presence of topological order, hence suggesting that topological order is not necessarily a genuine computational resource.

DOI
10.1103/t4fm-t7tc

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We pursue the identification of quantum resources carried by topological order, by evaluating quantum magic, quantified through the rank-222 Stabilizer Rényi entropy M2\mathcal{M}_2M2​, in one-dimensional systems hosting symmetry-protected topological phases (SPTP). Focusing on models with an exact duality between an SPTP and a trivial one, namely the dimerized XX and the Cluster-Ising chains, we show that dual points exhibit identical amounts of magic, even thought they belong to distinct topological sectors. A subextensive asymmetry arises only under open boundary conditions, where edge effects break the duality, but this correction is non-topological and depends on microscopic parameters. These results stand in contrast to the case of topological frustration, where delocalized excitations enhance the magic logarithmically with system size. They also complement recent analyses in the literature, showing that the total magic is largely insensitive to the presence of topological order, hence suggesting that topological order is not necessarily a genuine computational resource.
Topological matter provides natural platforms for robust, non-local information storage, central to quantum error correction. Yet, while the relation between entanglement and topology is well established, little is known about the role of nonstabilizerness (or magic), a pivotal concept in fault-tolerant quantum computation, in topological phases. We introduce the concept of topological magic response, the ability of a state to spread over stabilizer space when perturbed by finite-depth non-Clifford circuits. Unlike a topological invariant or order parameter, this response function probes how a phase reacts to non-Clifford perturbations, revealing the presence of non-local quantum correlations. In Ising-type spin chains, we show that symmetry-broken and paramagnetic phases lack such a response, whereas symmetry-protected topological (SPT) phases always display it. To capture this, we utilize a combination of stabilizer Rényi entropies that, in analogy with topological entanglement entropy, isolates non-locally stored information. Using exact analytic computations and matrix product states simulations based on an algorithmic technique we introduce, we show that SPT phases doped with TTT gates support robust topological magic response, while trivial phases remain featureless.
Quantum entanglement is widely used as a diagnostic of topological phases of matter. Beyond entanglement, non-stabilizerness captures a distinct aspect of quantum many-body states by quantifying their distance from the manifold of stabilizer states. In this work, we study the stabilizer Rényi entropy in symmetry protected topological (SPT) phases, including both gapped SPT, non-intrinsically gapless SPT, and intrinsically gapless SPT phases. Under symmetry preserving perturbations, we find numerically that the stabilizer Rényi entropy exhibits an extremum near the phase transition. However, the stabilizer Rényi entropy alone cannot distinguish different SPT phases. In contrast, the Pauli spectrum reveals a characteristic crossing structure at the transition point. This crossing reflects the exchange of dominant Pauli-string correlations associated with the non-local string order parameters of the two topological distinct phases. For gapped SPT and non-intrinsically gapless SPT phases, the crossing structure can be understood from a local-unitary duality that maps the Pauli spectrum between the two phases. For intrinsically gapless SPT phases, such a local-unitary mapping is absent. Instead, we find that the Pauli spectrum mapping is generated by a non-invertible duality transformation. These results show that although the stabilizer Rényi entropy provides only a coarse diagnostic of phase transitions, the Pauli spectrum contains finer information about the exchange of string order sectors. Our findings demonstrate that quantum magic offers a complementary perspective for characterizing both gapped and gapless SPT phases.
Although in general boundary conditions do not affect the bulk properties of a system, some of them are special and defy such expectation. This is the case, for instance, of those inducing geometrical frustration in a classical magnet. Recently, the study of such settings in quantum systems (dubbed topological frustration) has uncovered peculiar features, interesting both from a fundamental and technological point of view. In this work, we present and discuss the behavior of several quantum resources in presence of TF, namely the (disconnected) entanglement entropy and the non-stabilizerness Renyi entropy. We will show that, compared to their non-frustrated counterparts, TF adds a distinct contribution to these resources, due to a stable, delocalized, topological excitation. Remarkably, this contribution can be calculated analytically, due to its similarities with that of a W-state.
We introduce the concepts of a symmetry-protected sign problem and symmetry-protected magic to study the complexity of symmetry-protected topological (SPT) phases of matter. In particular, we say a state has a symmetry-protected sign problem or symmetry-protected magic, if finite-depth quantum circuits composed of symmetric gates are unable to transform the state into a non-negative real wave function or stabilizer state, respectively. We prove that states belonging to certain SPT phases have these properties, as a result of their anomalous symmetry action at a boundary. For example, we find that one-dimensional Z2×Z2 SPT states (e.g. cluster state) have a symmetry-protected sign problem, and two-dimensional Z2 SPT states (e.g. Levin-Gu state) have symmetry-protected magic. Furthermore, we comment on the relation between a symmetry-protected sign problem and the computational wire property of one-dimensional SPT states. In an appendix, we also introduce explicit decorated domain wall models of SPT phases, which may be of independent interest.
Despite growing interest in beyond-group symmetries in quantum condensed matter systems, there are relatively few microscopic lattice models explicitly realizing these symmetries, and many phenomena have yet to be studied at the microscopic level. We introduce a one-dimensional stabilizer Hamiltonian composed of group-based Pauli operators whose ground state is a G × Rep ( G ) -symmetric state: the G -cluster state introduced by Brell []. We show that this state lies in a symmetry-protected topological (SPT) phase protected by G × Rep ( G ) symmetry, distinct from the symmetric product state by a duality argument. We identify several signatures of SPT order, namely, protected edge modes, string order parameters, and topological response. We discuss how G -cluster states may be used as a universal resource for measurement-based quantum computation, explicitly working out the case where G is a semidirect product of Abelian groups.
We discuss physical constructions and the boundary properties of various symmetry-protected topological phases that involve 1-form symmetries from one spatial dimension to four spatial dimensions (4d4d4d). For example, the prototype three-dimensional (3d3d3d) boundary state of 4d4d4d SPT states involving 1-form symmetries can be either a gapless photon phase (quantum electrodynamics) or gapped topological order enriched by 1-form symmetries; that is, the loop excitations of these topological orders carry nontrivial 1-form symmetry charges. This study also serves the purpose of diagnosing anomalies of 3d3d3d states of matter. Connection between SPT states with 1-form symmetries and condensed-matter systems such as quantum dimer models at one lower dimension will also be discussed. Whether a quantum dimer model can have a trivial gapped phase depends on the nature of its corresponding bulk state in one higher dimension.