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arXiv:2610.25472

Topological Quantum Gates on FCIs with Bounded Quantum Geometry

Hisham Sati, Urs Schreiber

quant-phcond-mat.str-elmath-phSubmitted Oct 18, 2026• Announced Oct 20, 2026• 0 citations
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Abstract

Two-band 2D fractional Chern insulators (FCIs) have recently been shown, in principle, to admit unusual anyonic order, associated with Bloch Hopfions localized in momentum space. Here we show that bounding the integrated quantum metric along one primitive Brillouin-zone cycle may refine this topological order to $|C|$ distinct braiding observables, cyclically permuted by lattice translations, for Chern numbers $|C| > 1$. The cyclic structure entails mildly non-abelian braiding phases and otherwise matches the spectral flow exhibited by the FCI's hybrid Wannier states, with the metric bound equivalently a bound on the states' real-space spread. We give a classification of quantum gates in principle implementable on this topological order; among them we find the non-Clifford T-gate and the non-Clifford entangling CCZ-gate. This points to novel pathways to fault-tolerant quantum processes within the known orbital structure of FCIs, potentially informing experimental searches towards hardware for future topological quantum computers.

Comments
19 pages + references, 4 figures

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Two-band 2D fractional Chern insulators (FCIs) have recently been shown, in principle, to admit unusual anyonic order, associated with Bloch Hopfions localized in momentum space. Here we show that bounding the integrated quantum metric along one primitive Brillouin-zone cycle may refine this topological order to ∣C∣|C|∣C∣ distinct braiding observables, cyclically permuted by lattice translations, for Chern numbers ∣C∣>1|C| > 1∣C∣>1. The cyclic structure entails mildly non-abelian braiding phases and otherwise matches the spectral flow exhibited by the FCI's hybrid Wannier states, with the metric bound equivalently a bound on the states' real-space spread.
We give a classification of quantum gates in principle implementable on this topological order; among them we find the non-Clifford T-gate and the non-Clifford entangling CCZ-gate. This points to novel pathways to fault-tolerant quantum processes within the known orbital structure of FCIs, potentially informing experimental searches towards hardware for future topological quantum computers.
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The recent theoretical discovery of fractional Chern insulators (FCIs) has provided an important new way to realize topologically ordered states in lattice models. In earlier works, on-site and nearest-neighbor Hubbard-like interactions have been used extensively to stabilize Abelian FCIs in systems with nearly flat, topologically nontrivial bands. However, attempts to use two-body interactions to stabilize non-Abelian FCIs, where the ground state in the presence of impurities can be massively degenerate and manipulated through anyon braiding, have proven very difficult in uniform lattice systems. Here, we study the remarkable effect of long-range interactions in a lattice model that possesses an exactly flat lowest band with a unit Chern number. When spinless bosons with two-body long-range interactions partially fill the lowest Chern band, we find convincing evidence of gapped, bosonic Read-Rezayi (RR) phases with non-Abelian anyon statistics. We characterize these states through studying topological degeneracies, the overlap between the ground states of two-body interactions and the exact RR ground states of three- and four-body interactions, and state counting in the particle-cut entanglement spectrum. Moreover, we demonstrate how an approximate lattice form of Haldane's pseudopotentials, analogous to that in the continuum, can be used as an efficient guiding principle in the search for lattice models with stable non-Abelian phases.
A flat band with Chern number C = 0 , well isolated from the rest of Hilbert space by a gap much larger than interaction strength, is an unexpected context for fractional quantum Hall physics. In this work, we demonstrate the emergence of the fractional Chern insulator (FCI) states in such a trivial flat band using large-scale exact diagonalization and infinite density matrix renormalization group simulations. Hosted by an anisotropic lattice, this band has a sharp peak for the quantum geometry. At 1/3 hole filling, we find the unexpected FCI with 3-fold ground-state degeneracy and σ H = − 1 / 3 ( e 2 / h ) . The momentum-space carrier distribution shows that the quantum geometry peak tends to push the interacting holes away, toward other regions of the Brillouin zone with the nearly "ideal" quantum geometry, underlying the formation of the FCI in the C = 0 flat band. In comparison, we find only trivial ground states when the quantum geometry peak becomes less avoidable. Our work not only shows that FCl states could exist without the single-particle topology, relying on the more intrinsic quantum geometry, but also suggests a possibly generic scenario to explore FCIs in otherwise trivial flat bands.
Fractional Chern insulators (FCIs) are lattice analogues of fractional quantum Hall states, hosting topological order and fractionalized anyonic excitations. Their realization poses a demanding target for quantum simulation. Here, we realize a bosonic FCI on an IBM Nighthawk superconducting quantum processor using classically optimized variational circuits. We target a Laughlin-like state that is an exact zero-energy ground state of a long-range parent Hamiltonian on a torus, eliminating physical edges and providing a favorable geometry for accessing fractional quantum Hall physics in a small system. On a 4×44\times44×4 qubit lattice, we prepare both the four-boson FCI ground state and a three-boson state hosting two quasiholes. Measurements of particle densities and correlations reproduce the characteristic structure of the target states, while reconstructed chiral magnetization currents resolve the circulating current pattern associated with the quasiholes. The preparation of these strongly correlated topological states with shallow circuits establishes a route toward the digital simulation and manipulation of fractionalized excitations, including measurements of topological invariants and anyonic braiding.
Abstract: Fractional Chern insulators (FCIs) in moiré materials present a unique platform for exploring strongly correlated topological phases beyond the paradigm of ideal quantum geometry. While analytical approaches to FCIs and fractional quantum Hall states (FQHS) often rely on idealized Bloch wavefunctions, realistic moiré models lack direct tunability of quantum metric and Berry curvature, limiting theoretical and numerical exploration. Here, we introduce an unsupervised machine learning framework to model interacting Hamiltonians directly through the distribution of single-particle form factors. Using a variational autoencoder (VAE), we show that unsupervised learning can not only distinguish FCI and non-FCI states, but also generate new form factors with distinct topological character, not present in the training set. This latent space enables the generation and interpolation of form factors for topological flatbands with Chern number ∣ C ∣ = 1, enabling the discovery of unobserved many-body states such as charge density waves. Principal component analysis (PCA) further reveals that the dominant patterns in the form factors—reflecting correlations across the Brillouin zone—can be decomposed into components with approximately quantized Chern numbers, providing new insights into the global and topological structure of quantum geometry. Our results highlight the ability of machine learning to generalize and model topological quantum systems, paving the way for the inverse design of form factors with tailored quantum geometry and many-body phases in flatband materials.