Transport properties of anyons in random topological environments

2010

We study the single particle dynamics of a mobile non-Abelian anyon hopping around many pinned anyons on a surface. The dynamics is modelled by a discrete time quantum walk and the spatial degree of freedom of the mobile anyon becomes entangled with the fusion degrees of freedom of the collective system. Each quantum trajectory makes a closed braid on the world lines of the particles establishing a direct connection between statistical dynamics and quantum link invariants. We find that asymptotically a mobile Ising anyon becomes so entangled with its environment that its statistical dynamics reduces to a classical random walk with linear dispersion in contrast to particles with Abelian statistics which have quadratic dispersion.

Physical Review Letters, 2011

We study the single particle dynamics of a mobile non-Abelian anyon hopping around many pinned anyons on a surface. The dynamics is modelled by a discrete time quantum walk and the spatial degree of freedom of the mobile anyon becomes entangled with the fusion degrees of freedom of the collective system. Each quantum trajectory makes a closed braid on the world lines of the particles establishing a direct connection between statistical dynamics and quantum link invariants. We find that asymptotically a mobile Ising anyon becomes so entangled with its environment that its statistical dynamics reduces to a classical random walk with linear dispersion in contrast to particles with Abelian statistics which have quadratic dispersion. PACS numbers: 05.30.Pr, 05.40.Fb, 03.65.Vf Anyons are point like particles with more general statistics than bosons or fermions. They were shown to exist in systems where the physics is constrained to two dimensions [1]. Beyond mere possible existence they where found to be a good description for low lying quasi-particle excitations of fractional quantum Hall systems [2, 3] and they exactly describe excitations in various strongly correlated two dimensional spin lattice models . Recently there has been tremendous experimental progress in preparation and control of systems capable of exhibiting topological order with the goal to observe anyonic statistics. This is further motivated by the discovery that braiding some types of non-Abelian anyons can be used for naturally fault tolerant quantum computing . The quantum physics of anyonic systems is very rich but is only beginning to be explored in its own right. For example, there have been investigations of the equilibrium properties of dynamically interacting, but static, non-Abelian anyons in chains [9] and two dimensional lattices .

2010

The one dimensional quantum walk of anyonic systems is presented. The anyonic walker performs braiding operations with stationary anyons of the same type ordered canonically on the line of the walk. Abelian as well as non-Abelian anyons are studied and it is shown that they have very different properties. Abelian anyonic walks demonstrate the expected quadratic quantum speedup. Non-Abelian anyonic walks are much more subtle. The exponential increase of the system's Hilbert space and the particular statistical evolution of non-Abelian anyons give a variety of new behaviors. The position distribution of the walker is related to Jones polynomials, topological invariants of the links created by the anyonic world-lines during the walk. Several examples such as the SU(2) k and the quantum double models are considered that provide insight to the rich diffusion properties of anyons.

Reviews of Modern Physics, 2008

Topological quantum computation has recently 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 which are necessary for quantum computation are carried out by braiding quasiparticles, and then measuring the multi-quasiparticle states. The fault-tolerance of a topological quantum computer arises from the non-local encoding of the states of the quasiparticles, 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 ν = 5/2 state, although several other prospective candidates have been proposed in systems as disparate as ultra-cold atoms in optical lattices and thin film superconductors. In this review article, we describe current research in this field, 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. We address both the mathematical underpinnings of topological quantum computation and the physics of the subject using the ν = 5/2 fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

arXiv: Quantum Physics, 2020

We investigate numerically and theoretically the effect of spatial disorder on two-dimensional split-step discrete-time quantum walks with two internal "coin" states. Spatial disorder can lead to Anderson localization, inhibiting the spread of quantum walks, putting them at a disadvantage against their diffusively spreading classical counterparts. We find that spatial disorder of the most general type, i.e., position-dependent Haar random coin operators, does not lead to Anderson localization, but to a diffusive spread instead. This is a delocalization, which happens because disorder places the quantum walk to a critical point between different anomalous Floquet-Anderson insulating topological phases. We base this explanation on the relationship of this general quantum walk to a simpler case more studied in the literature, and for which disorder-induced delocalization of a topological origin has been observed. We review topological delocalization for the simpler quantum wa...

Physical Review Letters, 2007

We discuss generalizations of quantum spin Hamiltonians using anyonic degrees of freedom. The simplest model for interacting anyons energetically favors neighboring anyons to fuse into the trivial (''identity'') channel, similar to the quantum Heisenberg model favoring neighboring spins to form spin singlets. Numerical simulations of a chain of Fibonacci anyons show that the model is critical with a dynamical critical exponent z 1, and described by a two-dimensional (2D) conformal field theory with central charge c 7

New Journal of Physics, 2009

Topological systems, such as fractional quantum Hall liquids, promise to successfully combat environmental decoherence while performing quantum computation. These highly correlated systems can support non-Abelian anyonic quasiparticles that can encode exotic entangled states. To reveal the non-local character of these encoded states we demonstrate the violation of suitable Bell inequalities. We provide an explicit recipe for the preparation, manipulation and measurement of the desired correlations for a large class of topological models. This proposal gives an operational measure of non-locality for anyonic states and it opens up the possibility to violate the Bell inequalities in quantum Hall liquids or spin lattices.

Nature Communications, 2016

Physical Review B, 2013

Topological degeneracy is the degeneracy of the ground states in a many-body system in the large-system-size limit. Topological degeneracy cannot be lifted by any local perturbation of the Hamiltonian. The topological degeneracies on closed manifolds have been used to discover/define topological order in many-body systems, which contain excitations with fractional statistics. In this paper, we study a new type of topological degeneracy induced by condensing anyons along a line in two-dimensional topological ordered states. Such topological degeneracy can be viewed as carried by each end of the line defect, which is a generalization of Majorana zero modes. The topological degeneracy can be used as a quantum memory. The ends of line defects carry projective non-Abelian statistics even though they are produced by the condensation of Abelian anyons, and braiding them allows us to perform fault tolerant quantum computations.

2016

The aim of this text is to provide an introduction to the theory of topological quantum computation. We give an introduction to the theory of anyons (two-dimensional quasi-particle excitations that have exotic statistics) and how we can use these to perform fault-tolerant quantum computation. Additionally, we give a complete description of an exactly solvable spin lattice model whose local low-energy excitations of the Hamiltonian behave as anyons. We conclude by indicating how this model can be generalized so as to perform universal quantum computation.