Non-abelian braiding in abelian lattice models from lattice dislocations

Mattias Flygare

Published Jan 1, 2014• 0 citations

Abstract

Related papers

Icke-abelsk flätning i abelska gittermodeller genom dislokationer

Flygare, Mattias

Topological order is a new field of research involving exotic physics. Among other things it has been suggested as a means for realising fault-tolerant quantum computation. Topological degeneracy, i.e. the ground state degeneracy of a topologically ordered state, is one of the quantities that have been used to characterize such states. Topological order has also been suggested as a possible quantum information storage. We study two-dimensional lattice models defined on a closed manifold, specifically on a torus, and find that these systems exhibit topological degeneracy proportional to the genus of the manifold on which they are defined. We also find that the addition of lattice dislocations increases the ground state degeneracy, a behaviour that can be interpreted as artificially increasing the genus of the manifold. We derive the fusion and braiding rules of the model, which are then used to calculate the braiding properties of the dislocations themselves. These turn out to resemble non-abelian anyons, a property that is important for the possibility to achieve universal quantum computation. One can also emulate lattice dislocations synthetically, by adding an external field. This makes them more realistic for potential experimental realisations.

Jan 1, 20140 citations

Braid Group, Gauge Invariance and Topological Order(Fundamental Problems and Applications of Quantum Field Theory-Topological Aspects of Quantum Field Theory-)

Masatoshi Sato

素粒子論研究Apr 20, 20070 citations

Coupled wire models on compact manifolds and the entanglement entropy for Abelian and non-Abelian topological orders

Bo Han, Chi Yan Jeffrey Teo

Bulletin of the American Physical SocietyMar 4, 20190 citations

Non-Abelian anyons and topological quantum computation

Chetan Nayak, Steven H. Simon, Ady Stern +2 more

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.

arXiv:0707.1889Reviews of Modern PhysicsSep 12, 20086,846 citations

Topological Order in Condensed Matter Physics and a New View in Particle Physics(Current Topics)

Masatoshi Sato

ButsuriApr 5, 20080 citations

High spin correlations in topological and rainbow lattices

Lucy Sarah Byles

Quantum simulations are an invaluable tool in condensed matter physics, enabling the precise generation and control of physical phenomena abstracted from the complexity of the underlying many-body system. At the heart of such complexity lies entanglement, the intrinsically quantum correlations that have attracted renewed interest as a fundamental resource in quantum computation and information. In this thesis, we explore the use of one- and two-dimensional lattice models to simulate and harness such correlations for next-generation quantum technologies. We begin in one dimension with a spin-chain model that allows complete control over bipartite correlation structures. Specifically, we introduce a model of nearest-neighbour couplings and transverse field terms, whose exact solution yields a q-deformed rainbow ground state formed from a tensor product of q-deformed singlets, each with directly tunable entanglement. This framework acts as a universal simulator of free-fermion entanglement spectra, with numerical studies confirming high-fidelity realisations across a range of examples. We then turn to strong correlations in two dimensions, where topologically ordered systems host exotic quasiparticles known as anyons. Braiding non-Abelian anyons enacts unitary transformations on a logically encoded fusion space, offering a natural pathway to fault-tolerant quantum computation. Using Kitaev’s quantum double model D(G), we focus on the smallest non-Abelian instance, D(S3), realised on a lattice of d=6 qudits. We develop a minimal protocol in which braiding and fusion are simulated solely through operators that create and measure anyons, demonstrating that D(S3) anyons generate magic states, a vital resource for universal quantum computation. Finally, a dense encoding scheme is presented, rendering the minimal realisation of this protocol accessible with current quantum technologies and a circuit model for realisation within the framework of linear optics is presented. Notably, error analysis of this encoding scheme reveals an inherent passive protection provided by the topological encoding, extending beyond conventional notions of topological fault tolerance.

Sep 1, 20250 citations