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arXiv:1602.05035 · Progress in theoretical chemistry and physics

Topological Quantum Computation with Non-Abelian Anyons in Fractional Quantum Hall States

Lachezar S. Georgiev

Published Jan 1, 2017• 0 citations
arXivHTMLPDF
DOI
10.1007/978-3-319-50255-7_5

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•Sep 1, 2026•0 citations
We show that different classes of topological order can be distinguished by the dynamical symmetry algebra of edge excitations. A fundamental topological order is realized when this algebra is the largest possible, the algebra of quantum area-preserving diffeomorphisms, called W 1 + ∞ . We argue that this order is realized in the Jain hierarchy of fractional quantum Hall states and show that it is more robust than the standard Abelian Chern–Simons order since it has a lower entanglement entropy due to the non-Abelian character of the quasi-particle anyon excitations. These behave as SU( m ) quarks, where m is the number of components in the hierarchy. We propose the topological entanglement entropy as the experimental measure to detect the existence of these quantum Hall quarks. Non-Abelian anyons in the ν = 2/5 fractional quantum Hall states could be the primary candidates to realize qbits for topological quantum computation.
Topological quantum computation has emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations that are necessary for quantum computation are carried out by braiding quasiparticles and then measuring the multiquasiparticle states. The fault tolerance of a topological quantum computer arises from the nonlocal encoding of the quasiparticle states, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the $\ensuremath{\nu}=5∕2$ state, although several other prospective candidates have been proposed in systems as disparate as ultracold atoms in optical lattices and thin-film superconductors. In this review article, current research in this field is described, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. Both the mathematical underpinnings of topological quantum computation and the physics of the subject are addressed, using the $\ensuremath{\nu}=5∕2$ fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.
Topological quantum computation aims to encode quantum information in the nonlocal degrees of freedom of non-Abelian anyons and manipulate it through braiding and fusion operations， offering a promising pathway to fault-tolerant quantum computation. Non-Abelian anyons， with their nontrivial braiding statistics and intrinsic topological fault tolerance， are regarded as a key building block for realizing scalable and decoherence-resistant quantum computing. This review systematically surveys the key developments in anyon physics over the past four decades， from the early theoretical prediction of fractional statistics to direct experimental verification of anyon braiding in fractional quantum Hall states. The physical principles and key findings of essential experimental schemes， including Fabry-Pérot interferometers， Mach-Zehnder interferometers， and anyon colliders， are comprehensively covered. Based on these foundations， the characterization of topological order in even-denominator fractional quantum Hall states， exemplified by the ν=5/2 state， and transport and interferometric probes of non-Abelian statistics are discussed in detail. Furthermore， recent advances in topological quantum states in novel material platforms， including graphene and twisted transition metal dichalcogenides， are reviewed， together with artificial quantum platforms that simulate non-Abelian braiding using superconducting quantum processors. Finally， the key challenges and future directions on the path from anyon physics to scalable topological quantum computing prototype devices are outlined.