Schreiber Type IIA Anyons (Rev #4, changes)

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An article that we are writing at CQTS:

• Layered Topological Hopf Anyons in FCIs with Bounded Quantum Geometry Gates on
  FCIs with Bounded Quantum Geometry

  manuscript: [to appear here]

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Two-dimensional, Two-band two-band 2Dfractional Chern insulators (FCI) have recently been shown to in principle admit unusual abeliananyon statistics 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 commuting braidingobservables, cyclically permuted by lattice translations, for Chern numbers $|C| \gt 1$ . The cyclic structure entails mildly nonabelian braiding phases and otherwise matches thespectral flow organizing exhibited hybrid by theFCI’s hybrid Wannier states into through the effectivequantum Hall layers of Qi and Barkeshli , with the metric bound being the states’ real-space spread. We observe that, for$|C| = 2$states , ’ real-space spread. We give a complete classification of the resulting topological quantum gate gates set implementable may on contain this thetopological order, among them we find the non-Clifford T-gate . This and result the points to a novel candidate realization of nonabelian non-Clifford topological order within the known orbital structure ofentangling 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 processors computers.

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• Based on:

  • Fractional Quantum Hall Anyons via the Algebraic Topology of Surplus Flux Quanta

  • Identifying Anyonic Topological Order in Fractional Quantum Anomalous Hall Systems

  • On the Derivation of Twisted K-Theory from M-Theory

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