36 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
Coulomb Blockade Doppelgangers in Quantum Hall States
In this paper, we ask the question: How well can Coulomb blockade experiments
correctly identify and distinguish between different topological orders in
quantum Hall states? We definitively find the answer to be: Quite poorly. In
particular, we write the general expression for the spacing of resonance peaks
in a simple form that explicitly displays its dependence on the conformal
scaling dimensions of the systems' edge modes. This form makes transparent the
general argument that the Coulomb blockade peak spacings do not provide a
strongly indicative signature of the topological order of the system, since it
is only weakly related to the braiding statistics. We bolster this general
argument with examples for all the most physically relevant non-Abelian
candidate states, demonstrating that they have Coulomb blockade doppelgangers
-- candidate states at the same filling fraction with identical Coulomb
blockade signatures, but dramatically different topological orders and braiding
statistics.Comment: 12 pages, 1 figure; portions of this paper were formerly included in
Appendix C of arXiv:0903.3108; v2: examples added, minor corrections made;
v3: discussions of non-uniform filling and of hierarchical counterparts of
multi-component states added, minor corrections mad
Plasma Analogy and Non-Abelian Statistics for Ising-type Quantum Hall States
We study the non-Abelian statistics of quasiparticles in the Ising-type
quantum Hall states which are likely candidates to explain the observed Hall
conductivity plateaus in the second Landau level, most notably the one at
filling fraction nu=5/2. We complete the program started in Nucl. Phys. B 506,
685 (1997) and show that the degenerate four-quasihole and six-quasihole
wavefunctions of the Moore-Read Pfaffian state are orthogonal with equal
constant norms in the basis given by conformal blocks in a c=1+1/2 conformal
field theory. As a consequence, this proves that the non-Abelian statistics of
the excitations in this state are given by the explicit analytic continuation
of these wavefunctions. Our proof is based on a plasma analogy derived from the
Coulomb gas construction of Ising model correlation functions involving both
order and (at most two) disorder operators. We show how this computation also
determines the non-Abelian statistics of collections of more than six
quasiholes and give an explicit expression for the corresponding conformal
block-derived wavefunctions for an arbitrary number of quasiholes. Our method
also applies to the anti-Pfaffian wavefunction and to Bonderson-Slingerland
hierarchy states constructed over the Moore-Read and anti-Pfaffian states.Comment: 68 pages, 3 figures; v2: substantial revisions and additions for
clarity, minor correction
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
Instability and Degeneracy in the BMN Correspondence
Non-degenerate perturbation theory, which was used to calculate the scale
dimension of operators on the gauge theory side of the correspondence, breaks
down when effects of triple trace operators are included. We interpret this as
an instability of excited single-string states in the dual string theory for
decay into the continuum of degenerate 3-string states. We apply time-dependent
perturbation theory to calculate the decay widths from gauge theory. These
widths are new gauge theory data which can be compared with future calculations
in light cone string field theory.Comment: 23 pages, no figure
Kitaev's quantum double model from a local quantum physics point of view
A prominent example of a topologically ordered system is Kitaev's quantum
double model for finite groups (which in particular
includes , the toric code). We will look at these models from
the point of view of local quantum physics. In particular, we will review how
in the abelian case, one can do a Doplicher-Haag-Roberts analysis to study the
different superselection sectors of the model. In this way one finds that the
charges are in one-to-one correspondence with the representations of
, and that they are in fact anyons. Interchanging two of such
anyons gives a non-trivial phase, not just a possible sign change. The case of
non-abelian groups is more complicated. We outline how one could use
amplimorphisms, that is, morphisms to study the superselection
structure in that case. Finally, we give a brief overview of applications of
topologically ordered systems to the field of quantum computation.Comment: Chapter contributed to R. Brunetti, C. Dappiaggi, K. Fredenhagen, J.
Yngvason (eds), Advances in Algebraic Quantum Field Theory (Springer 2015).
Mainly revie
Anyons in a weakly interacting system
We describe a theoretical proposal for a system whose excitations are anyons
with the exchange phase pi/4 and charge -e/2, but, remarkably, can be built by
filling a set of single-particle states of essentially noninteracting
electrons. The system consists of an artificially structured type-II
superconducting film adjacent to a 2D electron gas in the integer quantum Hall
regime with unit filling fraction. The proposal rests on the observation that a
vacancy in an otherwise periodic vortex lattice in the superconductor creates a
bound state in the 2DEG with total charge -e/2. A composite of this
fractionally charged hole and the missing flux due to the vacancy behaves as an
anyon. The proposed setup allows for manipulation of these anyons and could
prove useful in various schemes for fault-tolerant topological quantum
computation.Comment: 7 pages with 3 figures. For related work and info visit
http://www.physics.ubc.ca/~fran
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
Disorder-assisted error correction in Majorana chains
It was recently realized that quenched disorder may enhance the reliability
of topological qubits by reducing the mobility of anyons at zero temperature.
Here we compute storage times with and without disorder for quantum chains with
unpaired Majorana fermions - the simplest toy model of a quantum memory.
Disorder takes the form of a random site-dependent chemical potential. The
corresponding one-particle problem is a one-dimensional Anderson model with
disorder in the hopping amplitudes. We focus on the zero-temperature storage of
a qubit encoded in the ground state of the Majorana chain. Storage and
retrieval are modeled by a unitary evolution under the memory Hamiltonian with
an unknown weak perturbation followed by an error-correction step. Assuming
dynamical localization of the one-particle problem, we show that the storage
time grows exponentially with the system size. We give supporting evidence for
the required localization property by estimating Lyapunov exponents of the
one-particle eigenfunctions. We also simulate the storage process for chains
with a few hundred sites. Our numerical results indicate that in the absence of
disorder, the storage time grows only as a logarithm of the system size. We
provide numerical evidence for the beneficial effect of disorder on storage
times and show that suitably chosen pseudorandom potentials can outperform
random ones.Comment: 50 pages, 7 figure
Interface engineering of quantum Hall effects in digital transition metal oxide heterostructures
Topological insulators are characterized by a nontrivial band topology driven
by the spin-orbit coupling. To fully explore the fundamental science and
application of topological insulators, material realization is indispensable.
Here we predict, based on tight-binding modeling and first-principles
calculations, that bilayers of perovskite-type transition-metal oxides grown
along the [111] crystallographic axis are potential candidates for
two-dimensional topological insulators. The topological band structure of these
materials can be fine-tuned by changing dopant ions, substrates, and external
gate voltages. We predict that LaAuO bilayers have a
topologically-nontrivial energy gap of about 0.15 eV, which is sufficiently
large to realize the quantum spin-Hall effect at room temperature. Intriguing
phenomena, such as fractional quantum Hall effect, associated with the
nearly-flat topologically-nontrivial bands found in systems are also
discussed.Comment: Main text 11 pages with 4 figures and 1 table. Supplementary
materials 4 pages with 2 figure