221 research outputs found
Constructive spherical codes on layers of flat tori
A new class of spherical codes is constructed by selecting a finite subset of
flat tori from a foliation of the unit sphere S^{2L-1} of R^{2L} and designing
a structured codebook on each torus layer. The resulting spherical code can be
the image of a lattice restricted to a specific hyperbox in R^L in each layer.
Group structure and homogeneity, useful for efficient storage and decoding, are
inherited from the underlying lattice codebook. A systematic method for
constructing such codes are presented and, as an example, the Leech lattice is
used to construct a spherical code in R^{48}. Upper and lower bounds on the
performance, the asymptotic packing density and a method for decoding are
derived.Comment: 9 pages, 5 figures, submitted to IEEE Transactions on Information
Theor
Curves on torus layers and coding for continuous alphabet sources
In this paper we consider the problem of transmitting a continuous alphabet
discrete-time source over an AWGN channel. The design of good curves for this
purpose relies on geometrical properties of spherical codes and projections of
-dimensional lattices. We propose a constructive scheme based on a set of
curves on the surface of a 2N-dimensional sphere and present comparisons with
some previous works.Comment: 5 pages, 4 figures. Accepted for presentation at 2012 IEEE
International Symposium on Information Theory (ISIT). 2th version: typos
corrected. 3rd version: some typos corrected, a footnote added in Section III
B, a comment added in the beggining of Section V and Theorem I adde
Spherical Designs and Heights of Euclidean Lattices
We study the connection between the theory of spherical designs and the
question of extrema of the height function of lattices. More precisely, we show
that a full-rank n-dimensional Euclidean lattice, all layers of which hold a
spherical 2-design, realises a stationary point for the height function, which
is defined as the first derivative at 0 of the spectral zeta function of the
associated flat torus. Moreover, in order to find out the lattices for which
this 2-design property holds, a strategy is described which makes use of theta
functions with spherical coefficients, viewed as elements of some space of
modular forms. Explicit computations in dimension up to 7, performed with
Pari/GP and Magma, are reported.Comment: 22 page
Basic Understanding of Condensed Phases of Matter via Packing Models
Packing problems have been a source of fascination for millenia and their
study has produced a rich literature that spans numerous disciplines.
Investigations of hard-particle packing models have provided basic insights
into the structure and bulk properties of condensed phases of matter, including
low-temperature states (e.g., molecular and colloidal liquids, crystals and
glasses), multiphase heterogeneous media, granular media, and biological
systems. The densest packings are of great interest in pure mathematics,
including discrete geometry and number theory. This perspective reviews
pertinent theoretical and computational literature concerning the equilibrium,
metastable and nonequilibrium packings of hard-particle packings in various
Euclidean space dimensions. In the case of jammed packings, emphasis will be
placed on the "geometric-structure" approach, which provides a powerful and
unified means to quantitatively characterize individual packings via jamming
categories and "order" maps. It incorporates extremal jammed states, including
the densest packings, maximally random jammed states, and lowest-density jammed
structures. Packings of identical spheres, spheres with a size distribution,
and nonspherical particles are also surveyed. We close this review by
identifying challenges and open questions for future research.Comment: 33 pages, 20 figures, Invited "Perspective" submitted to the Journal
of Chemical Physics. arXiv admin note: text overlap with arXiv:1008.298
Energy minimization, periodic sets and spherical designs
We study energy minimization for pair potentials among periodic sets in
Euclidean spaces. We derive some sufficient conditions under which a point
lattice locally minimizes the energy associated to a large class of potential
functions. This allows in particular to prove a local version of Cohn and
Kumar's conjecture that , , and the
Leech lattice are globally universally optimal, regarding energy minimization,
and among periodic sets of fixed point density.Comment: 16 pages; incorporated referee comment
Geometric optimization problems in quantum computation and discrete mathematics: Stabilizer states and lattices
This thesis consists of two parts:
Part I deals with properties of stabilizer states and their convex
hull, the stabilizer polytope. Stabilizer states, Pauli measurements
and Clifford unitaries are the three building blocks of the stabilizer
formalism whose computational power is limited by the Gottesman-
Knill theorem. This model is usually enriched by a magic state to get
a universal model for quantum computation, referred to as quantum
computation with magic states (QCM). The first part of this thesis
will investigate the role of stabilizer states within QCM from three
different angles.
The first considered quantity is the stabilizer extent, which provides
a tool to measure the non-stabilizerness or magic of a quantum state.
It assigns a quantity to each state roughly measuring how many stabilizer
states are required to approximate the state. It has been shown
that the extent is multiplicative under taking tensor products when
the considered state is a product state whose components are composed
of maximally three qubits. In Chapter 2, we will prove that
this property does not hold in general, more precisely, that the stabilizer
extent is strictly submultiplicative. We obtain this result as
a consequence of rather general properties of stabilizer states. Informally
our result implies that one should not expect a dictionary to be
multiplicative under taking tensor products whenever the dictionary
size grows subexponentially in the dimension.
In Chapter 3, we consider QCM from a resource theoretic perspective.
The resource theory of magic is based on two types of quantum
channels, completely stabilizer preserving maps and stabilizer operations.
Both classes have the property that they cannot generate additional
magic resources. We will show that these two classes of quantum
channels do not coincide, specifically, that stabilizer operations are a
strict subset of the set of completely stabilizer preserving channels.
This might have the consequence that certain tasks which are usually
realized by stabilizer operations could in principle be performed better
by completely stabilizer preserving maps.
In Chapter 4, the last one of Part I, we consider QCM via the polar
dual stabilizer polytope (also called the Lambda-polytope). This polytope
is a superset of the quantum state space and every quantum state
can be written as a convex combination of its vertices. A way to
classically simulate quantum computing with magic states is based on
simulating Pauli measurements and Clifford unitaries on the vertices
of the Lambda-polytope. The complexity of classical simulation with respect
to the polytope is determined by classically simulating the updates
of vertices under Clifford unitaries and Pauli measurements. However,
a complete description of this polytope as a convex hull of its vertices is
only known in low dimensions (for up to two qubits or one qudit when
odd dimensional systems are considered). We make progress on this
question by characterizing a certain class of operators that live on the
boundary of the Lambda-polytope when the underlying dimension is an odd
prime. This class encompasses for instance Wigner operators, which
have been shown to be vertices of Lambda. We conjecture that this class
contains even more vertices of Lambda. Eventually, we will shortly sketch
why applying Clifford unitaries and Pauli measurements to this class
of operators can be efficiently classically simulated.
Part II of this thesis deals with lattices. Lattices are discrete subgroups
of the Euclidean space. They occur in various different areas of
mathematics, physics and computer science. We will investigate two
types of optimization problems related to lattices.
In Chapter 6 we are concerned with optimization within the space of
lattices. That is, we want to compare the Gaussian potential energy
of different lattices. To make the energy of lattices comparable we
focus on lattices with point density one. In particular, we focus on
even unimodular lattices and show that, up to dimension 24, they are
all critical for the Gaussian potential energy. Furthermore, we find
that all n-dimensional even unimodular lattices with n 24 are local
minima or saddle points. In contrast in dimension 32, there are even
unimodular lattices which are local maxima and others which are not
even critical.
In Chapter 7 we consider flat tori R^n/L, where L is an n-dimensional
lattice. A flat torus comes with a metric and our goal is to approximate
this metric with a Hilbert space metric. To achieve this, we
derive an infinite-dimensional semidefinite optimization program that
computes the least distortion embedding of the metric space R^n/L into
a Hilbert space. This program allows us to make several interesting
statements about the nature of least distortion embeddings of flat tori.
In particular, we give a simple proof for a lower bound which gives
a constant factor improvement over the previously best lower bound
on the minimal distortion of an embedding of an n-dimensional flat
torus. Furthermore, we show that there is always an optimal embedding
into a finite-dimensional Hilbert space. Finally, we construct
optimal least distortion embeddings for the standard torus R^n/Z^n and
all 2-dimensional flat tori
Minimization of energy per particle among Bravais lattices in R^2 : Lennard-Jones and Thomas-Fermi cases
We study the two dimensional Lennard-Jones energy per particle of lattices
and we prove that the minimizer among Bravais lattices with sufficiently large
density is triangular and that is not the case for sufficiently small density.
We give other results about the global minimizer of this energy. Moreover we
study the energy per particle stemming from Thomas-Fermi model in two
dimensions and we prove that the minimizer among Bravais lattices with fixed
density is triangular. We use a result of Montgomery from Number Theory about
the minimization of Theta functions in the plane.Comment: 15 pages, 4 figures. Final Versio
Dimension reduction techniques for the minimization of theta functions on lattices
We consider the minimization of theta functions
amongst
lattices , by reducing the dimension of the
problem, following as a motivation the case , where minimizers are
supposed to be either the BCC or the FCC lattices. A first way to reduce
dimension is by considering layered lattices, and minimize either among
competitors presenting different sequences of repetitions of the layers, or
among competitors presenting different shifts of the layers with respect to
each other. The second case presents the problem of minimizing theta functions
also on translated lattices, namely minimizing . Another way to reduce dimension is by considering
lattices with a product structure or by successively minimizing over concentric
layers. The first direction leads to the question of minimization amongst
orthorhombic lattices, whereas the second is relevant for asymptotics
questions, which we study in detail in two dimensions.Comment: 45 pages. 7 figure
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