213 research outputs found
Generalized Finite Element Systems for smooth differential forms and Stokes problem
We provide both a general framework for discretizing de Rham sequences of
differential forms of high regularity, and some examples of finite element
spaces that fit in the framework. The general framework is an extension of the
previously introduced notion of Finite Element Systems, and the examples
include conforming mixed finite elements for Stokes' equation. In dimension 2
we detail four low order finite element complexes and one infinite family of
highorder finite element complexes. In dimension 3 we define one low order
complex, which may be branched into Whitney forms at a chosen index. Stokes
pairs with continuous or discontinuous pressure are provided in arbitrary
dimension. The finite element spaces all consist of composite polynomials. The
framework guarantees some nice properties of the spaces, in particular the
existence of commuting interpolators. It also shows that some of the examples
are minimal spaces.Comment: v1: 27 pages. v2: 34 pages. Numerous details added. v3: 44 pages. 8
figures and several comments adde
COMPARISON ON DIFFERENT DISCRETE FRACTIONAL FOURIER TRANSFORM (DFRFT) APPROACHES
As an extension of conventional Fourier transform and a time-frequency signal analysis tool, the fractional Fourier transforms (FRFT) are suitable for dealing with various types of non-stationary signals. Taking advantage of the properties and non-stationary features of linear chirp signals in the Fourier transform domain, several methods of extraction and parameter estimation for chirp signals are proposed, and a comparative study has been done on chirp signal estimation. Computation of the discrete fractional Fourier transform (DFRFT) and its chirp concentration properties are dependent on the basis of DFT eigenvectors used in the computation. Several DFT-eigenvector bases have been proposed for the transform, and there is no common framework for comparing them. In this thesis, we compare several different approaches from a conceptual viewpoint and point out the differences between them. We discuss five different approaches, namely: (1) the bilinear transformation method, (2) the Grunbaum method, (3) the Dickenson-Steiglitz method, also known as the S-matrix method, (4) the quantum mechanics in finite dimension( QMFD) method, and (5) the higher order S-matrix method, to find centered DFT (CDFT) commuting matrices and the various properties of these commuting matrices. We study the nature of eigenvalues and eigenvectors of these commuting matrices to determine whether they resemble those of corresponding continuous Gauss-Hermite operator. We also measure the performance of these five approaches in terms of mailobe-to-sidelobe ratio, 10-dB bandwidth, quality factor, linearity of eigenvalues, parameter estimation error, and, finally peak-to-parameter mapping regions. We compare the five approaches using these several parameters and point out the best approach for chirp signal applications
One-body reduced density-matrix functional theory in finite basis sets at elevated temperatures
In this review we provide a rigorous and self-contained presentation of
one-body reduced density-matrix (1RDM) functional theory. We do so for the case
of a finite basis set, where density-functional theory (DFT) implicitly becomes
a 1RDM functional theory. To avoid non-uniqueness issues we consider the case
of fermionic and bosonic systems at elevated temperature and variable particle
number, i.e, a grand-canonical ensemble. For the fermionic case the Fock space
is finite-dimensional due to the Pauli principle and we can provide a rigorous
1RDM functional theory relatively straightforwardly. For the bosonic case,
where arbitrarily many particles can occupy a single state, the Fock space is
infinite-dimensional and mathematical subtleties (not every hermitian
Hamiltonian is self-adjoint, expectation values can become infinite, and not
every self-adjoint Hamiltonian has a Gibbs state) make it necessary to impose
restrictions on the allowed Hamiltonians and external non-local potentials. For
simple conditions on the interaction of the bosons a rigorous 1RDM functional
theory can be established, where we exploit the fact that due to the finite
one-particle space all 1RDMs are finite-dimensional. We also discuss the
problems arising from 1RDM functional theory as well as DFT formulated for an
infinite-dimensional one-particle space.Comment: 55 pages, 7 figure
Eigenfunction Statistics on Quantum Graphs
We investigate the spatial statistics of the energy eigenfunctions on large
quantum graphs. It has previously been conjectured that these should be
described by a Gaussian Random Wave Model, by analogy with quantum chaotic
systems, for which such a model was proposed by Berry in 1977. The
autocorrelation functions we calculate for an individual quantum graph exhibit
a universal component, which completely determines a Gaussian Random Wave
Model, and a system-dependent deviation. This deviation depends on the graph
only through its underlying classical dynamics. Classical criteria for quantum
universality to be met asymptotically in the large graph limit (i.e. for the
non-universal deviation to vanish) are then extracted. We use an exact field
theoretic expression in terms of a variant of a supersymmetric sigma model. A
saddle-point analysis of this expression leads to the estimates. In particular,
intensity correlations are used to discuss the possible equidistribution of the
energy eigenfunctions in the large graph limit. When equidistribution is
asymptotically realized, our theory predicts a rate of convergence that is a
significant refinement of previous estimates. The universal and
system-dependent components of intensity correlation functions are recovered by
means of an exact trace formula which we analyse in the diagonal approximation,
drawing in this way a parallel between the field theory and semiclassics. Our
results provide the first instance where an asymptotic Gaussian Random Wave
Model has been established microscopically for eigenfunctions in a system with
no disorder.Comment: 59 pages, 3 figure
Practical Quantum Chemistry on Near Term Quantum Computers
Solutions to the time-independent Schrödinger equation for molecular systems allow chemical properties to be studied without the direct need for the material. However, the dimension of this problem grows exponentially with the size of the quantum system under consideration making conventional treatment intractable. Quantum computers can efficiently represent and evolve quantum states. Their use offers a possible way to perform simulations on molecules previously impossible to model. However, given the constraints of current quantum computers even studying small systems is limited by the number of qubits, circuit depth and runtime of a chosen quantum algorithm. The work in this thesis is to explore and provide new tools to make chemical simulation more practical on near-term devices. First, the unitary partitioning measurement reduction strategy is explored. This reduces the runtime of the variational quantum eigensolver algorithm (VQE). We then apply this reduction technique to the contextual subspace method, which approximates a problem by introducing artificial symmetries based on the solution of noncontextual version of the problem that reduces the number of qubits required for simulation. We provide a modification to the original algorithm that makes an exponentially scaling part of the technique quadratic. Finally, we develop the projection-based embedding (PBE) technique to allow chemical systems to be studied using state-of-the-art classical methods in conjuncture with quantum computing protocols in a multiscale hierarchy. This allows molecular problems much larger than conventionally studied on quantum hardware to be approached
Comparative study of semiclassical approaches to quantum dynamics
Quantum states can be described equivalently by density matrices, Wigner
functions or quantum tomograms. We analyze the accuracy and performance of
three related semiclassical approaches to quantum dynamics, in particular with
respect to their numerical implementation. As test cases, we consider the time
evolution of Gaussian wave packets in different one-dimensional geometries,
whereby tunneling, resonance and anharmonicity effects are taken into account.
The results and methods are benchmarked against an exact quantum mechanical
treatment of the system, which is based on a highly efficient Chebyshev
expansion technique of the time evolution operator.Comment: 32 pages, 8 figures, corrected typos and added references; version as
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