5 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
Quantum Algebras Associated With Bell States
The antisymmetric solution of the braided Yang--Baxter equation called the
Bell matrix becomes interesting in quantum information theory because it can
generate all Bell states from product states. In this paper, we study the
quantum algebra through the FRT construction of the Bell matrix. In its four
dimensional representations via the coproduct of its two dimensional
representations, we find algebraic structures including a composition series
and a direct sum of its two dimensional representations to characterize this
quantum algebra. We also present the quantum algebra using the FRT construction
of Yang--Baxterization of the Bell matrix.Comment: v1: 15 pages, 2 figures, latex; v2: 18 pages, 2 figures, latex,
references and notes adde
Degeneracy of non-abelian quantum Hall states on the torus: domain walls and conformal field theory
We analyze the non-abelian Read-Rezayi quantum Hall states on the torus,
where it is natural to employ a mapping of the many-body problem onto a
one-dimensional lattice model. On the thin torus--the Tao-Thouless (TT)
limit--the interacting many-body problem is exactly solvable. The Read-Rezayi
states at filling are known to be exact ground states of a
local repulsive -body interaction, and in the TT limit this is manifested
in that all states in the ground state manifold have exactly particles on
any consecutive sites. For the two-body correlations of these
states also imply that there is no more than one particle on adjacent
sites. The fractionally charged quasiparticles and quasiholes appear as domain
walls between the ground states, and we show that the number of distinct domain
wall patterns gives rise to the nontrivial degeneracies, required by the
non-abelian statistics of these states. In the second part of the paper we
consider the quasihole degeneracies from a conformal field theory (CFT)
perspective, and show that the counting of the domain wall patterns maps one to
one on the CFT counting via the fusion rules. Moreover we extend the CFT
analysis to topologies of higher genus.Comment: 15 page
One-particle density matrix and momentum distribution function of one-dimensional anyon gases
We present a systematic study of the Green functions of a one-dimensional gas
of impenetrable anyons. We show that the one-particle density matrix is the
determinant of a Toeplitz matrix whose large N asymptotic is given by the
Fisher-Hartwig conjecture. We provide a careful numerical analysis of this
determinant for general values of the anyonic parameter, showing in full
details the crossover between bosons and fermions and the reorganization of the
singularities of the momentum distribution function.
We show that the one-particle density matrix satisfies a Painleve VI
differential equation, that is then used to derive the small distance and large
momentum expansions. We find that the first non-vanishing term in this
expansion is always k^{-4}, that is proved to be true for all couplings in the
Lieb-Liniger anyonic gas and that can be traced back to the presence of a delta
function interaction in the Hamiltonian.Comment: 21 pages, 4 figure
Architectural design for a topological cluster state quantum computer
The development of a large scale quantum computer is a highly sought after
goal of fundamental research and consequently a highly non-trivial problem.
Scalability in quantum information processing is not just a problem of qubit
manufacturing and control but it crucially depends on the ability to adapt
advanced techniques in quantum information theory, such as error correction, to
the experimental restrictions of assembling qubit arrays into the millions. In
this paper we introduce a feasible architectural design for large scale quantum
computation in optical systems. We combine the recent developments in
topological cluster state computation with the photonic module, a simple chip
based device which can be used as a fundamental building block for a large
scale computer. The integration of the topological cluster model with this
comparatively simple operational element addresses many significant issues in
scalable computing and leads to a promising modular architecture with complete
integration of active error correction exhibiting high fault-tolerant
thresholds.Comment: 14 Pages, 8 Figures, changes to the main text, new appendix adde