33 research outputs found
Non-Abelian Anyons and Topological Quantum Computation
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 {\it Non-Abelian anyons}, meaning that they obey {\it 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 \nu=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 \nu=5/2 fractional
quantum Hall state as the archetype of a non-Abelian topological state enabling
fault-tolerant quantum computation.Comment: Final Accepted form for RM
Braid Matrices and Quantum Gates for Ising Anyons Topological Quantum Computation
We study various aspects of the topological quantum computation scheme based
on the non-Abelian anyons corresponding to fractional quantum hall effect
states at filling fraction 5/2 using the Temperley-Lieb recoupling theory.
Unitary braiding matrices are obtained by a normalization of the degenerate
ground states of a system of anyons, which is equivalent to a modification of
the definition of the 3-vertices in the Temperley-Lieb recoupling theory as
proposed by Kauffman and Lomonaco. With the braid matrices available, we
discuss the problems of encoding of qubit states and construction of quantum
gates from the elementary braiding operation matrices for the Ising anyons
model. In the encoding scheme where 2 qubits are represented by 8 Ising anyons,
we give an alternative proof of the no-entanglement theorem given by Bravyi and
compare it to the case of Fibonacci anyons model. In the encoding scheme where
2 qubits are represented by 6 Ising anyons, we construct a set of quantum gates
which is equivalent to the construction of Georgiev.Comment: 25 pages, 13 figure
A magnetic model with a possible Chern-Simons phase
An elementary family of local Hamiltonians , is described for a dimensional quantum mechanical system of spin
particles. On the torus, the ground state space is
extensively degenerate but should collapse under \lperturbation" to
an anyonic system with a complete mathematical description: the quantum double
of the Chern-Simons modular functor at which
we call . The Hamiltonian defines a
\underline{quantum} \underline{loop}\underline{gas}. We argue that for and 2, is unstable and the collapse to can occur truly by perturbation. For ,
is stable and in this case finding must require either , help from finite
system size, surface roughening (see section 3), or some other trick, hence the
initial use of quotes {\l}\quad". A hypothetical phase diagram is included in
the introduction.Comment: Appendix by F. Goodman and H. Wenz
Anyonic statistics and large horizon diffeomorphisms for Loop Quantum Gravity Black Holes
We investigate the role played by large diffeomorphisms of quantum Isolated
Horizons for the statistics of LQG Black Holes by means of their relation to
the braid group. To this aim the symmetries of Chern-Simons theory are
recapitulated with particular regard to the aforementioned type of
diffeomorphisms. For the punctured spherical horizon, these are elements of the
mapping class group of , which is almost isomorphic to a corresponding
braid group on this particular manifold. The mutual exchange of quantum
entities in two dimensions is achieved by the braid group, rendering the
statistics anyonic. With this we argue that the quantum Isolated Horizon model
of LQG based on -Chern-Simons theory explicitly exhibits non-abelian
anyonic statistics. In this way a connection to the theory behind the
fractional quantum Hall effect and that of topological quantum computation is
established, where non-abelian anyons play a significant role.Comment: 20 pages, 8 figures, closest to published versio
Towards topological quantum computer
One of the principal obstacles on the way to quantum computers is the lack of
distinguished basis in the space of unitary evolutions and thus the lack of the
commonly accepted set of basic operations (universal gates). A natural choice,
however, is at hand: it is provided by the quantum R-matrices, the entangling
deformations of non-entangling (classical) permutations, distinguished from the
points of view of group theory, integrable systems and modern theory of
non-perturbative calculations in quantum field and string theory. Observables
in this case are (square modules of) the knot polynomials, and their pronounced
integrality properties could provide a key to error correction. We suggest to
use R-matrices acting in the space of irreducible representations, which are
unitary for the real-valued couplings in Chern-Simons theory, to build a
topological version of quantum computing.Comment: 14 page
A Class of -Invariant Topological Phases of Interacting Electrons
We describe a class of parity- and time-reversal-invariant topological states
of matter which can arise in correlated electron systems in 2+1-dimensions.
These states are characterized by particle-like excitations exhibiting exotic
braiding statistics. and invariance are maintained by a `doubling' of
the low-energy degrees of freedom which occurs naturally without doubling the
underlying microscopic degrees of freedom. The simplest examples have been the
subject of considerable interest as proposed mechanisms for high-
superconductivity. One is the `doubled' version (i.e. two opposite-chirality
copies) of the U(1) chiral spin liquid. The second example corresponds to
gauge theory, which describes a scenario for spin-charge separation. Our main
concern, with an eye towards applications to quantum computation, are richer
models which support non-Abelian statistics. All of these models, richer or
poorer, lie in a tightly-organized discrete family. The physical inference is
that a material manifesting the gauge theory or a doubled chiral spin
liquid might be easily altered to one capable of universal quantum computation.
These phases of matter have a field-theoretic description in terms of gauge
theories which, in their infrared limits, are topological field theories. We
motivate these gauge theories using a parton model or slave-fermion
construction and show how they can be solved exactly. The structure of the
resulting Hilbert spaces can be understood in purely combinatorial terms. The
highly-constrained nature of this combinatorial construction, phrased in the
language of the topology of curves on surfaces, lays the groundwork for a
strategy for constructing microscopic lattice models which give rise to these
phases.Comment: Typos fixed, references adde