410 research outputs found
On the Approximation of the Quantum Gates using Lattices
A central question in Quantum Computing is how matrices in can be
approximated by products over a small set of "generators". A topology will be
defined on so as to introduce the notion of a covering exponent
\cite{letter}, which compares the length of products required to covering
with balls against the Haar measure of
balls. An efficient universal set over will be constructed using the
Pauli matrices, using the metric of the covering exponent. Then, the
relationship between and will be manipulated to correlate angles
between points on and give a conjecture on the maximum of angles between
points on a lattice. It will be shown how this conjecture can be used to
compute the covering exponent, and how it can be generalized to universal sets
in .Comment: This is an updated version of arxiv.org:1506.0578
On the Whitney distortion extension problem for and and its applications to interpolation and alignment of data in
Let , open. In this paper we provide a sharp
solution to the following Whitney distortion extension problems: (a) Let
be a map. If is compact (with some
geometry) and the restriction of to is an almost isometry with small
distortion, how to decide when there exists a one-to-one and
onto almost isometry with small distortion
which agrees with in a neighborhood of and a Euclidean motion
away from . (b) Let
be map. If is compact (with some geometry) and the
restriction of to is an almost isometry with small distortion, how
to decide when there exists a one-to-one and onto
almost isometry with small distortion which
agrees with in a neighborhood of and a Euclidean motion away from . Our results complement those of [14,15,20]
where there, is a finite set. In this case, the problem above is also a
problem of interpolation and alignment of data in .Comment: This is part three of four papers with C. Fefferman (arXiv:1411.2451,
arXiv:1411.2468, involve-v5-n2-p03-s.pdf) dealing with the problem of Whitney
type extensions of distortions from certain compact sets to distorted diffeomorphisms on $\Bbb R^n
On surface completion and image inpainting by biharmonic functions: Numerical aspects
Numerical experiments with smooth surface extension and image inpainting
using harmonic and biharmonic functions are carried out. The boundary data used
for constructing biharmonic functions are the values of the Laplacian and
normal derivatives of the functions on the boundary. Finite difference schemes
for solving these harmonic functions are discussed in detail.Comment: Revised 21 July, 2017. Revised 12 January, 2018. To appear in
International Journal of Mathematics and Mathematical Science
Isometries and Equivalences Between Point Configurations, Extended To -diffeomorphisms
In this paper, we deal with the Orthogonal Procrustes Problem, in which two
point configurations are compared in order to construct a map to optimally
align the two sets. This extends this to -diffeomorphisms,
introduced by [1] Damelin and Fefferman. Examples will be given for when
complete maps can not be constructed, for if the distributions do match, and
finally an algorithm for partitioning the configurations into polygons for
convenient construction of the maps.Comment: version: 11.26.1
An Algebraic-Coding Equivalence to the Maximum Distance Separable Conjecture
We formulate an Algebraic-Coding Equivalence to the Maximum Distance
Separable Conjecture. Specifically, we present novel proofs of the following
equivalent statements. Let be a fixed pair of integers satisfying
is a prime power and . We denote by the vector
space of functions from a finite field to itself, which can be
represented as the space of
polynomial functions. We denote by the
set of polynomials that are either the zero polynomial, or have at most
distinct roots in . Given two subspaces of ,
we denote by their span. We prove that the following are
equivalent.
[A] Suppose that either: 1. is odd 2. is even and .
Then there do not exist distinct subspaces and of
such that:
3. 4. . 5. 6. 7.
.
[B] Suppose is odd, or, if is even, . There is
no integer with such that the Reed-Solomon code
over of dimension can have columns
added to it, such that:
8. Any submatrix of containing
the first columns of is independent. 9. is independent.
[C] The MDS conjecture is true for the given .Comment: This is version: 5.6.18. arXiv admin note: substantial text overlap
with arXiv:1611.0235
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