897 research outputs found
Why should anyone care about computing with anyons?
In this article we present a pedagogical introduction of the main ideas and
recent advances in the area of topological quantum computation. We give an
overview of the concept of anyons and their exotic statistics, present various
models that exhibit topological behavior, and we establish their relation to
quantum computation. Possible directions for the physical realization of
topological systems and the detection of anyonic behavior are elaborated.Comment: 22 pages, 13 figures. Some changes to existing sections, several
references added, and a new section on criteria for TQO and TQC in lattice
system
Introduction to topological quantum computation with non-Abelian anyons
Topological quantum computers promise a fault tolerant means to perform
quantum computation. Topological quantum computers use particles with exotic
exchange statistics called non-Abelian anyons, and the simplest anyon model
which allows for universal quantum computation by particle exchange or braiding
alone is the Fibonacci anyon model. One classically hard problem that can be
solved efficiently using quantum computation is finding the value of the Jones
polynomial of knots at roots of unity. We aim to provide a pedagogical,
self-contained, review of topological quantum computation with Fibonacci
anyons, from the braiding statistics and matrices to the layout of such a
computer and the compiling of braids to perform specific operations. Then we
use a simulation of a topological quantum computer to explicitly demonstrate a
quantum computation using Fibonacci anyons, evaluating the Jones polynomial of
a selection of simple knots. In addition to simulating a modular circuit-style
quantum algorithm, we also show how the magnitude of the Jones polynomial at
specific points could be obtained exactly using Fibonacci or Ising anyons. Such
an exact algorithm seems ideally suited for a proof of concept demonstration of
a topological quantum computer.Comment: 51 pages, 51 figure
Topological Features in Ion Trap Holonomic Computation
Topological features in quantum computing provide controllability and noise
error avoidance in the performance of logical gates. While such resilience is
favored in the manipulation of quantum systems, it is very hard to identify
topological features in nature. This paper proposes a scheme where holonomic
quantum gates have intrinsic topological features. An ion trap is employed
where the vibrational modes of the ions are coherently manipulated with lasers
in an adiabatic cyclic way producing geometrical holonomic gates. A crucial
ingredient of the manipulation procedures is squeezing of the vibrational
modes, which effectively suppresses exponentially any undesired fluctuations of
the laser amplitudes, thus making the gates resilient to control errors.Comment: 9 pages, 4 figures, REVTE
Non-Abelian statistics as a Berry phase in exactly solvable models
We demonstrate how to directly study non-Abelian statistics for a wide class
of exactly solvable many-body quantum systems. By employing exact eigenstates
to simulate the adiabatic transport of a model's quasiparticles, the resulting
Berry phase provides a direct demonstration of their non-Abelian statistics. We
apply this technique to Kitaev's honeycomb lattice model and explicitly
demonstrate the existence of non-Abelian Ising anyons confirming the previous
conjectures. Finally, we present the manipulations needed to transport and
detect the statistics of these quasiparticles in the laboratory. Various
physically realistic system sizes are considered and exact predictions for such
experiments are provided.Comment: 10 pages, 3 figures. To appear in New Journal of Physic
Topological Quantum Gates with Quantum Dots
We present an idealized model involving interacting quantum dots that can
support both the dynamical and geometrical forms of quantum computation. We
show that by employing a structure similar to the one used in the Aharonov-Bohm
effect we can construct a topological two-qubit phase-gate that is to a large
degree independent of the exact values of the control parameters and therefore
resilient to control errors. The main components of the setup are realizable
with present technology.Comment: 8 pages, 3 figures, submitted to Jour. of Opt. B (special issue on
Quantum Computing
Parafermions in a Kagome lattice of qubits for topological quantum computation
Engineering complex non-Abelian anyon models with simple physical systems is
crucial for topological quantum computation. Unfortunately, the simplest
systems are typically restricted to Majorana zero modes (Ising anyons). Here we
go beyond this barrier, showing that the parafermion model of
non-Abelian anyons can be realized on a qubit lattice. Our system additionally
contains the Abelian anyons as low-energetic excitations. We
show that braiding of these parafermions with each other and with the
anyons allows the entire Clifford group to be
generated. The error correction problem for our model is also studied in
detail, guaranteeing fault-tolerance of the topological operations. Crucially,
since the non-Abelian anyons are engineered through defect lines rather than as
excitations, non-Abelian error correction is not required. Instead the error
correction problem is performed on the underlying Abelian model, allowing high
noise thresholds to be realized.Comment: 11+10 pages, 14 figures; v2: accepted for publication in Phys. Rev.
X; 4 new figures, performance of phase-gate explained in more detai
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