Topological Quantum Gates on FCIs with Bounded Quantum Geometry

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.