Braiding Anyons in the ν = 5/2 Fractional Quantum Hall Effect State

We study the problem of anyons with statistics \ensuremath{\nu} in a strong magnetic field by means of a similarity transformation. The ground state of this problem exhibits an ``integer'' quantum Hall effect at filling factor 1/\ensuremath{\nu} with quasiparticle excitations of charge q/\ensuremath{\nu} and statistics -1/\ensuremath{\nu}. Certain electron fractional quantum Hall states can be considered as a realization of this, for example, the 2/5, 3/7,. . . hierarchy of the 1/3 state. This may explain the observed quasiparticle-quasihole asymmetry in the fractional quantum hall hierarchy.We study the problem of anyons with statistics \ensuremath{\nu} in a strong magnetic field by means of a similarity transformation. The ground state of this problem exhibits an ``integer'' quantum Hall effect at filling factor 1/\ensuremath{\nu} with quasiparticle excitations of charge q/\ensuremath{\nu} and statistics -1/\ensuremath{\nu}. Certain electron fractional quantum Hall states can be considered as a realization of this, for example, the 2/5, 3/7,. . . hierarchy of the 1/3 state. This may explain the observed quasiparticle-quasihole asymmetry in the fractional quantum hall hierarchy.

Physical Review LettersApr 1, 199119 citations

Anyons and the Fractional Quantum Hall Effect

Lecture notes in physics. M, MonographsJan 1, 19920 citations

Universal quantum computation with the ν=5∕2 fractional quantum Hall state

Sergey B. Bravyi

We consider topological quantum computation (TQC) with a particular class of anyons that are believed to exist in the fractional quantum Hall effect state at Landau-level filling fraction $\ensuremath{\nu}=5∕2$. Since the braid group representation describing the statistics of these anyons is not computationally universal, one cannot directly apply the standard TQC technique. We propose to use very noisy nontopological operations such as direct short-range interactions between anyons to simulate a universal set of gates. Assuming that all TQC operations are implemented perfectly, we prove that the threshold error rate for nontopological operations is above $14%$ . The total number of nontopological computational elements that one needs to simulate a quantum circuit with $L$ gates scales as $L(\mathrm{ln}\phantom{\rule{0.2em}{0ex}}L{)}^{3}$ .We consider topological quantum computation (TQC) with a particular class of anyons that are believed to exist in the fractional quantum Hall effect state at Landau-level filling fraction $\ensuremath{\nu}=5∕2$. Since the braid group representation describing the statistics of these anyons is not computationally universal, one cannot directly apply the standard TQC technique. We propose to use very noisy nontopological operations such as direct short-range interactions between anyons to simulate a universal set of gates. Assuming that all TQC operations are implemented perfectly, we prove that the threshold error rate for nontopological operations is above $14%$. The total number of nontopological computational elements that one needs to simulate a quantum circuit with $L$ gates scales as $L(\mathrm{ln}\phantom{\rule{0.2em}{0ex}}L{)}^{3}$.

arXiv:quant-ph/0511178Physical Review AApr 12, 2006251 citations

Spinning braid-group representation and the fractional quantum Hall effect

Christopher Ting, C.H. Lai

arXiv:hep-th/9202024Nuclear Physics BMay 1, 19930 citations

Many-Anyon Braiding in Non-Abelian Fractional Quantum Hall Effect with Hybrid Monte Carlo Simulation

Ting-Tung Wang, Ha Quang Trung, Qianhui Xu +3 more

We employ the hybrid Monte Carlo method to efficiently compute the many-anyon non-Abelian braiding matrices associated with different braiding schemes of the Moore-Read quasiholes. A novel proposal in this work is that anyon braiding schemes based on a global rotation are robust against finite-size effects, as demonstrated by benchmarking their errors in the braiding matrix against those of a simple two-anyon exchange. Moreover, we investigate how electron-electron interactions and local electrostatic trapping potentials influence the energetic preference of different fusion channels. Their effect on the non-Abelian braiding matrices has been verified, a surprising phenomenon that demonstrates long-range entanglement of non-Abelian states. Our results are relevant to the experimental realization of non-Abelian physics in fractional quantum Hall and other analogous systems, including the fast-growing field of fractional quantum anomalous Hall states in moiré materials.We employ the hybrid Monte Carlo method to efficiently compute the many-anyon non-Abelian braiding matrices associated with different braiding schemes of the Moore-Read quasiholes. A novel proposal in this work is that anyon braiding schemes based on a global rotation are robust against finite-size effects, as demonstrated by benchmarking their errors in the braiding matrix against those of a simple two-anyon exchange. Moreover, we investigate how electron-electron interactions and local electrostatic trapping potentials influence the energetic preference of different fusion channels. Their effect on the non-Abelian braiding matrices has been verified, a surprising phenomenon that demonstrates long-range entanglement of non-Abelian states. Our results are relevant to the experimental realization of non-Abelian physics in fractional quantum Hall and other analogous systems, including the fast-growing field of fractional quantum anomalous Hall states in moiré materials.

arXiv:2608.17424Aug 18, 20260 citations

Ground State for Fractional Quantum Hall Effect

Ahmed Jellal

International Journal of Theoretical PhysicsFeb 1, 20000 citations

14%

. The total number of nontopological computational elements that one needs to simulate a quantum circuit with

L

gates scales as

L(\mathrm{ln}\phantom{\rule{0.2em}{0ex}}L{)}^{3}