5 research outputs found
Topological Qubit Design and Leakage
We examine how best to design qubits for use in topological quantum
computation. These qubits are topological Hilbert spaces associated with small
groups of anyons. Op- erations are performed on these by exchanging the anyons.
One might argue that, in order to have as many simple single qubit operations
as possible, the number of anyons per group should be maximized. However, we
show that there is a maximal number of particles per qubit, namely 4, and more
generally a maximal number of particles for qudits of dimension d. We also look
at the possibility of having topological qubits for which one can perform
two-qubit gates without leakage into non-computational states. It turns out
that the requirement that all two-qubit gates are leakage free is very
restrictive and this property can only be realized for two-qubit systems
related to Ising-like anyon models, which do not allow for universal quantum
computation by braiding. Our results follow directly from the representation
theory of braid groups which means they are valid for all anyon models. We also
make some remarks on generalizations to other exchange groups.Comment: 13 pages, 3 figure
Clebsch-Gordan and 6j-coefficients for rank two quantum groups
We calculate (q-deformed) Clebsch-Gordan and 6j-coefficients for rank two
quantum groups. We explain in detail how such calculations are done, which
should allow the reader to perform similar calculations in other cases.
Moreover, we tabulate the q-Clebsch-Gordan and 6j-coefficients explicitly, as
well as some other topological data associated with theories corresponding to
rank-two quantum groups. Finally, we collect some useful properties of the
fusion rules of particular conformal field theories.Comment: 43 pages. v2: minor changes and added references. For mathematica
notebooks containing the various q-CG and 6j symbols, see
http://arxiv.org/src/1004.5456/an
The modular S-matrix as order parameter for topological phase transitions
We study topological phase transitions in discrete gauge theories in two
spatial dimensions induced by the formation of a Bose condensate. We analyse a
general class of euclidean lattice actions for these theories which contain one
coupling constant for each conjugacy class of the gauge group. To probe the
phase structure we use a complete set of open and closed anyonic string
operators. The open strings allow one to determine the particle content of the
condensate, whereas the closed strings enable us to determine the matrix
elements of the modular -matrix, also in the broken phase. From the measured
broken -matrix we may read off the sectors that split or get identified in
the broken phase, as well as the sectors that are confined. In this sense the
modular -matrix can be employed as a matrix valued non-local order parameter
from which the low-energy effective theories that occur in different regions of
parameter space can be fully determined.
To verify our predictions we studied a non-abelian anyon model based on the
quaternion group of order eight by Monte Carlo simulation. We
probe part of the phase diagram for the pure gauge theory and find a variety of
phases with magnetic condensates leading to various forms of (partial)
confinement in complete agreement with the algebraic breaking analysis. Also
the order of various transitions is established.Comment: 37 page
Non-locality of non-Abelian anyons
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.Comment: 7 pages, 3 figure
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