2,954 research outputs found
Wannier Function Approach to Realistic Coulomb Interactions in Layered Materials and Heterostructures
We introduce an approach to derive realistic Coulomb interaction terms in
free standing layered materials and vertical heterostructures from ab-initio
modelling of the corresponding bulk materials. To this end, we establish a
combination of calculations within the framework of the constrained random
phase approximation, Wannier function representation of Coulomb matrix elements
within some low energy Hilbert space and continuum medium electrostatics, which
we call Wannier function continuum electrostatics (WFCE). For monolayer and
bilayer graphene we reproduce full ab-initio calculations of the Coulomb matrix
elements within an accuracy of eV or better. We show that realistic
Coulomb interactions in bilayer graphene can be manipulated on the eV scale by
different dielectric and metallic environments. A comparison to electronic
phase diagrams derived in [M. M. Scherer et al., Phys. Rev. B 85, 235408
(2012)] suggests that the electronic ground state of bilayer graphene is a
layered antiferromagnet and remains surprisingly unaffected by these strong
changes in the Coulomb interaction.Comment: 12 pages, 8 figure
Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method
Given the matrix polynomial , we
consider the associated polynomial eigenvalue problem. This problem, viewed in
terms of computing the roots of the scalar polynomial , is treated
in polynomial form rather than in matrix form by means of the Ehrlich-Aberth
iteration. The main computational issues are discussed, namely, the choice of
the starting approximations needed to start the Ehrlich-Aberth iteration, the
computation of the Newton correction, the halting criterion, and the treatment
of eigenvalues at infinity. We arrive at an effective implementation which
provides more accurate approximations to the eigenvalues with respect to the
methods based on the QZ algorithm. The case of polynomials having special
structures, like palindromic, Hamiltonian, symplectic, etc., where the
eigenvalues have special symmetries in the complex plane, is considered. A
general way to adapt the Ehrlich-Aberth iteration to structured matrix
polynomial is introduced. Numerical experiments which confirm the effectiveness
of this approach are reported.Comment: Submitted to Linear Algebra App
High-Performance Solvers for Dense Hermitian Eigenproblems
We introduce a new collection of solvers - subsequently called EleMRRR - for
large-scale dense Hermitian eigenproblems. EleMRRR solves various types of
problems: generalized, standard, and tridiagonal eigenproblems. Among these,
the last is of particular importance as it is a solver on its own right, as
well as the computational kernel for the first two; we present a fast and
scalable tridiagonal solver based on the Algorithm of Multiple Relatively
Robust Representations - referred to as PMRRR. Like the other EleMRRR solvers,
PMRRR is part of the freely available Elemental library, and is designed to
fully support both message-passing (MPI) and multithreading parallelism (SMP).
As a result, the solvers can equally be used in pure MPI or in hybrid MPI-SMP
fashion. We conducted a thorough performance study of EleMRRR and ScaLAPACK's
solvers on two supercomputers. Such a study, performed with up to 8,192 cores,
provides precise guidelines to assemble the fastest solver within the ScaLAPACK
framework; it also indicates that EleMRRR outperforms even the fastest solvers
built from ScaLAPACK's components
On Ergodic Secrecy Capacity for Gaussian MISO Wiretap Channels
A Gaussian multiple-input single-output (MISO) wiretap channel model is
considered, where there exists a transmitter equipped with multiple antennas, a
legitimate receiver and an eavesdropper each equipped with a single antenna. We
study the problem of finding the optimal input covariance that achieves ergodic
secrecy capacity subject to a power constraint where only statistical
information about the eavesdropper channel is available at the transmitter.
This is a non-convex optimization problem that is in general difficult to
solve. Existing results address the case in which the eavesdropper or/and
legitimate channels have independent and identically distributed Gaussian
entries with zero-mean and unit-variance, i.e., the channels have trivial
covariances. This paper addresses the general case where eavesdropper and
legitimate channels have nontrivial covariances. A set of equations describing
the optimal input covariance matrix are proposed along with an algorithm to
obtain the solution. Based on this framework, we show that when full
information on the legitimate channel is available to the transmitter, the
optimal input covariance has always rank one. We also show that when only
statistical information on the legitimate channel is available to the
transmitter, the legitimate channel has some general non-trivial covariance,
and the eavesdropper channel has trivial covariance, the optimal input
covariance has the same eigenvectors as the legitimate channel covariance.
Numerical results are presented to illustrate the algorithm.Comment: 27 pages, 10 figure
A Pedagogical Intrinsic Approach to Relative Entropies as Potential Functions of Quantum Metrics: the - Family
The so-called -z-\textit{R\'enyi Relative Entropies} provide a huge
two-parameter family of relative entropies which includes almost all well-known
examples of quantum relative entropies for suitable values of the parameters.
In this paper we consider a log-regularized version of this family and use it
as a family of potential functions to generate covariant symmetric
tensors on the space of invertible quantum states in finite dimensions. The
geometric formalism developed here allows us to obtain the explicit expressions
of such tensor fields in terms of a basis of globally defined differential
forms on a suitable unfolding space without the need to introduce a specific
set of coordinates. To make the reader acquainted with the intrinsic formalism
introduced, we first perform the computation for the qubit case, and then, we
extend the computation of the metric-like tensors to a generic -level
system. By suitably varying the parameters and , we are able to recover
well-known examples of quantum metric tensors that, in our treatment, appear
written in terms of globally defined geometrical objects that do not depend on
the coordinates system used. In particular, we obtain a coordinate-free
expression for the von Neumann-Umegaki metric, for the Bures metric and for the
Wigner-Yanase metric in the arbitrary -level case.Comment: 50 pages, 1 figur
Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that
We report on some recent developments in the search for optimal network
topologies. First we review some basic concepts on spectral graph theory,
including adjacency and Laplacian matrices, and paying special attention to the
topological implications of having large spectral gaps. We also introduce
related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we
discuss two different dynamical feautures of networks: synchronizability and
flow of random walkers and so that they are optimized if the corresponding
Laplacian matrix have a large spectral gap. From this, we show, by developing a
numerical optimization algorithm that maximum synchronizability and fast random
walk spreading are obtained for a particular type of extremely homogeneous
regular networks, with long loops and poor modular structure, that we call
entangled networks. These turn out to be related to Ramanujan and Cage graphs.
We argue also that these graphs are very good finite-size approximations to
Bethe lattices, and provide almost or almost optimal solutions to many other
problems as, for instance, searchability in the presence of congestion or
performance of neural networks. Finally, we study how these results are
modified when studying dynamical processes controlled by a normalized (weighted
and directed) dynamics; much more heterogeneous graphs are optimal in this
case. Finally, a critical discussion of the limitations and possible extensions
of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted
for pub. in JSTA
On pole-swapping algorithms for the eigenvalue problem
Pole-swapping algorithms, which are generalizations of the QZ algorithm for
the generalized eigenvalue problem, are studied. A new modular (and therefore
more flexible) convergence theory that applies to all pole-swapping algorithms
is developed. A key component of all such algorithms is a procedure that swaps
two adjacent eigenvalues in a triangular pencil. An improved swapping routine
is developed, and its superiority over existing methods is demonstrated by a
backward error analysis and numerical tests. The modularity of the new
convergence theory and the generality of the pole-swapping approach shed new
light on bi-directional chasing algorithms, optimally packed shifts, and bulge
pencils, and allow the design of novel algorithms
Measure of decoherence in quantum error correction for solid-state quantum computing
We considered the interaction of semiconductor quantum register with noisy
environment leading to various types of qubit errors. We analysed both phase
and amplitude decays during the process of electron-phonon interaction. The
performance of quantum error correction codes (QECC) which will be inevitably
used in full scale quantum information processors was studied in realistic
conditions in semiconductor nanostructures. As a hardware basis for quantum bit
we chose the quantum spatial states of single electron in semiconductor coupled
double quantum dot system. The modified 5- and 9-qubit quantum error correction
(QEC) algorithms by Shor and DiVincenzo without error syndrome extraction were
applied to quantum register. 5-qubit error correction procedures were
implemented for Si charge double dot qubits in the presence of acoustic phonon
environment. Chi-matrix, Choi-Jamiolkowski state and measure of decoherence
techniques were used to quantify qubit fault-tolerance. Our results showed that
the introduction of above quantum error correction techniques at small phonon
noise levels provided quadratic improvement of output error rates. The
efficiency of 5-qubits quantum error correction algorithm in semiconductor
quantum information processors was demonstrated
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