1,080 research outputs found
Quantum computation with Turaev-Viro codes
The Turaev-Viro invariant for a closed 3-manifold is defined as the
contraction of a certain tensor network. The tensors correspond to tetrahedra
in a triangulation of the manifold, with values determined by a fixed spherical
category. For a manifold with boundary, the tensor network has free indices
that can be associated to qudits, and its contraction gives the coefficients of
a quantum error-correcting code. The code has local stabilizers determined by
Levin and Wen. For example, applied to the genus-one handlebody using the Z_2
category, this construction yields the well-known toric code.
For other categories, such as the Fibonacci category, the construction
realizes a non-abelian anyon model over a discrete lattice. By studying braid
group representations acting on equivalence classes of colored ribbon graphs
embedded in a punctured sphere, we identify the anyons, and give a simple
recipe for mapping fusion basis states of the doubled category to ribbon
graphs. We explain how suitable initial states can be prepared efficiently, how
to implement braids, by successively changing the triangulation using a fixed
five-qudit local unitary gate, and how to measure the topological charge.
Combined with known universality results for anyonic systems, this provides a
large family of schemes for quantum computation based on local deformations of
stabilizer codes. These schemes may serve as a starting point for developing
fault-tolerance schemes using continuous stabilizer measurements and active
error-correction.Comment: 53 pages, LaTeX + 199 eps figure
Parameterizing Quasiperiodicity: Generalized Poisson Summation and Its Application to Modified-Fibonacci Antenna Arrays
The fairly recent discovery of "quasicrystals", whose X-ray diffraction
patterns reveal certain peculiar features which do not conform with spatial
periodicity, has motivated studies of the wave-dynamical implications of
"aperiodic order". Within the context of the radiation properties of antenna
arrays, an instructive novel (canonical) example of wave interactions with
quasiperiodic order is illustrated here for one-dimensional (1-D) array
configurations based on the "modified-Fibonacci" sequence, with utilization of
a two-scale generalization of the standard Poisson summation formula for
periodic arrays. This allows for a "quasi-Floquet" analytic parameterization of
the radiated field, which provides instructive insights into some of the basic
wave mechanisms associated with quasiperiodic order, highlighting similarities
and differences with the periodic case. Examples are shown for quasiperiodic
infinite and spatially-truncated arrays, with brief discussion of computational
issues and potential applications.Comment: 29 pages, 10 figures. To be published in IEEE Trans. Antennas
Propagat., vol. 53, No. 6, June 200
Quadrature Points via Heat Kernel Repulsion
We discuss the classical problem of how to pick weighted points on a
dimensional manifold so as to obtain a reasonable quadrature rule
This problem, naturally, has a long history; the purpose of our paper is to
propose selecting points and weights so as to minimize the energy functional
\sum_{i,j =1}^{N}{ a_i a_j \exp\left(-\frac{d(x_i,x_j)^2}{4t}\right) }
\rightarrow \min, \quad \mbox{where}~t \sim N^{-2/d}, is the
geodesic distance and is the dimension of the manifold. This yields point
sets that are theoretically guaranteed, via spectral theoretic properties of
the Laplacian , to have good properties. One nice aspect is that the
energy functional is universal and independent of the underlying manifold; we
show several numerical examples
Competing states for the fractional quantum Hall effect in the 1/3-filled second Landau level
In this work, we investigate the nature of the fractional quantum Hall state
in the 1/3-filled second Landau level (SLL) at filling factor (and
8/3 in the presence of the particle-hole symmetry) via exact diagonalization in
both torus and spherical geometries. Specifically, we compute the overlap
between the exact 7/3 ground state and various competing states including (i)
the Laughlin state, (ii) the fermionic Haffnian state, (iii) the
antisymmetrized product state of two composite fermion seas at 1/6 filling, and
(iv) the particle-hole (PH) conjugate of the parafermion state. All these
trial states are constructed according to a guiding principle called the
bilayer mapping approach, where a trial state is obtained as the
antisymmetrized projection of a bilayer quantum Hall state with interlayer
distance as a variational parameter. Under the proper understanding of the
ground-state degeneracy in the torus geometry, the parafermion state can
be obtained as the antisymmetrized projection of the Halperin (330) state.
Similarly, it is proved in this work that the fermionic Haffnian state can be
obtained as the antisymmetrized projection of the Halperin (551) state. It is
shown that, while extremely accurate at sufficiently large positive Haldane
pseudopotential variation , the Laughlin state loses its
overlap with the exact 7/3 ground state significantly at . At slightly negative , it is shown that the
PH-conjugated parafermion state has a substantial overlap with the exact
7/3 ground state, which is the highest among the above four trial states.Comment: 22 pages, 5 figure
Tridiagonal substitution Hamiltonians
We consider a family of discrete Jacobi operators on the one-dimensional
integer lattice with Laplacian and potential terms modulated by a primitive
invertible two-letter substitution. We investigate the spectrum and the
spectral type, the fractal structure and fractal dimensions of the spectrum,
exact dimensionality of the integrated density of states, and the gap
structure. We present a review of previous results, some applications, and open
problems. Our investigation is based largely on the dynamics of trace maps.
This work is an extension of similar results on Schroedinger operators,
although some of the results that we obtain differ qualitatively and
quantitatively from those for the Schoedinger operators. The nontrivialities of
this extension lie in the dynamics of the associated trace map as one attempts
to extend the trace map formalism from the Schroedinger cocycle to the Jacobi
one. In fact, the Jacobi operators considered here are, in a sense, a test
item, as many other models can be attacked via the same techniques, and we
present an extensive discussion on this.Comment: 41 pages, 5 figures, 81 reference
Harmonic surface mapping algorithm for electrostatic potentials in an atomistic/continuum hybrid model for electrolyte solutions
Simulating charged many-body systems has been a computational demanding task
due to the long-range nature of electrostatic interaction. For the multi-scale
model of electrolytes which combines the strengths of atomistic/continuum
electrolyte representations, a harmonic surface mapping algorithm is developed
for fast and accurate evaluation of the electrostatic reaction potentials. Our
method reformulates the reaction potential into a sum of image charges for the
near-field, and a charge density on an auxiliary spherical surface for the
far-field, which can be further discretized into point charges. Fast multipole
method is used to accelerate the pairwise Coulomb summation. The accuracy and
efficiency of our algorithm, as well as the choice of relevant numerical
parameters are demonstrated in detail. As a concrete example, for charges close
to the dielectric interface, our method can improve the accuracy by two orders
of magnitudes compared to the Kirkwood series expansion method.Comment: 17 pages, 5 figure
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