1,147 research outputs found
On finding dense sub-lattices as low energy states of a quantum Hamiltonian
Lattice-based cryptography has emerged as one of the most prominent
candidates for post-quantum cryptography, projected to be secure against the
imminent threat of large-scale fault-tolerant quantum computers. The Shortest
Vector Problem (SVP) is to find the shortest non-zero vector in a given
lattice. It is fundamental to lattice-based cryptography and believed to be
hard even for quantum computers. We study a natural generalization of the SVP
known as the -Densest Sub-lattice Problem (-DSP): to find the densest
-dimensional sub-lattice of a given lattice. We formulate -DSP as finding
the first excited state of a Z-basis Hamiltonian, making -DSP amenable to
investigation via an array of quantum algorithms, including Grover search,
quantum Gibbs sampling, adiabatic, and Variational Quantum Algorithms. The
complexity of the algorithms depends on the basis through which the input
lattice is presented. We present a classical polynomial-time algorithm that
takes an arbitrary input basis and preprocesses it into inputs suited to
quantum algorithms. With preprocessing, we prove that qubits suffice
for solving -DSP for dimensional input lattices. We empirically
demonstrate the performance of a Quantum Approximate Optimization Algorithm
-DSP solver for low dimensions, highlighting the influence of a good
preprocessed input basis. We then discuss the hardness of -DSP in relation
to the SVP, to see if there is reason to build post-quantum cryptography on
-DSP. We devise a quantum algorithm that solves -DSP with run-time
exponent . Therefore, for fixed , -DSP is no more than
polynomially harder than the SVP
Core Decomposition in Multilayer Networks: Theory, Algorithms, and Applications
Multilayer networks are a powerful paradigm to model complex systems, where
multiple relations occur between the same entities. Despite the keen interest
in a variety of tasks, algorithms, and analyses in this type of network, the
problem of extracting dense subgraphs has remained largely unexplored so far.
In this work we study the problem of core decomposition of a multilayer
network. The multilayer context is much challenging as no total order exists
among multilayer cores; rather, they form a lattice whose size is exponential
in the number of layers. In this setting we devise three algorithms which
differ in the way they visit the core lattice and in their pruning techniques.
We then move a step forward and study the problem of extracting the
inner-most (also known as maximal) cores, i.e., the cores that are not
dominated by any other core in terms of their core index in all the layers.
Inner-most cores are typically orders of magnitude less than all the cores.
Motivated by this, we devise an algorithm that effectively exploits the
maximality property and extracts inner-most cores directly, without first
computing a complete decomposition.
Finally, we showcase the multilayer core-decomposition tool in a variety of
scenarios and problems. We start by considering the problem of densest-subgraph
extraction in multilayer networks. We introduce a definition of multilayer
densest subgraph that trades-off between high density and number of layers in
which the high density holds, and exploit multilayer core decomposition to
approximate this problem with quality guarantees. As further applications, we
show how to utilize multilayer core decomposition to speed-up the extraction of
frequent cross-graph quasi-cliques and to generalize the community-search
problem to the multilayer setting
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
Random perfect lattices and the sphere packing problem
Motivated by the search for best lattice sphere packings in Euclidean spaces
of large dimensions we study randomly generated perfect lattices in moderately
large dimensions (up to d=19 included). Perfect lattices are relevant in the
solution of the problem of lattice sphere packing, because the best lattice
packing is a perfect lattice and because they can be generated easily by an
algorithm. Their number however grows super-exponentially with the dimension so
to get an idea of their properties we propose to study a randomized version of
the algorithm and to define a random ensemble with an effective temperature in
a way reminiscent of a Monte-Carlo simulation. We therefore study the
distribution of packing fractions and kissing numbers of these ensembles and
show how as the temperature is decreased the best know packers are easily
recovered. We find that, even at infinite temperature, the typical perfect
lattices are considerably denser than known families (like A_d and D_d) and we
propose two hypotheses between which we cannot distinguish in this paper: one
in which they improve Minkowsky's bound phi\sim 2^{-(0.84+-0.06) d}, and a
competitor, in which their packing fraction decreases super-exponentially,
namely phi\sim d^{-a d} but with a very small coefficient a=0.06+-0.04. We also
find properties of the random walk which are suggestive of a glassy system
already for moderately small dimensions. We also analyze local structure of
network of perfect lattices conjecturing that this is a scale-free network in
all dimensions with constant scaling exponent 2.6+-0.1.Comment: 19 pages, 22 figure
Densest Lattice Packings of 3-Polytopes
Based on Minkowski's work on critical lattices of 3-dimensional convex bodies
we present an efficient algorithm for computing the density of a densest
lattice packing of an arbitrary 3-polytope. As an application we calculate
densest lattice packings of all regular and Archimedean polytopes.Comment: 37 page
Precise Algorithm to Generate Random Sequential Addition of Hard Hyperspheres at Saturation
Random sequential addition (RSA) time-dependent packing process, in which
congruent hard hyperspheres are randomly and sequentially placed into a system
without interparticle overlap, is a useful packing model to study disorder in
high dimensions. Of particular interest is the infinite-time {\it saturation}
limit in which the available space for another sphere tends to zero. However,
the associated saturation density has been determined in all previous
investigations by extrapolating the density results for near-saturation
configurations to the saturation limit, which necessarily introduces numerical
uncertainties. We have refined an algorithm devised by us [S. Torquato, O.
Uche, and F.~H. Stillinger, Phys. Rev. E {\bf 74}, 061308 (2006)] to generate
RSA packings of identical hyperspheres. The improved algorithm produce such
packings that are guaranteed to contain no available space using finite
computational time with heretofore unattained precision and across the widest
range of dimensions (). We have also calculated the packing and
covering densities, pair correlation function and structure factor
of the saturated RSA configurations. As the space dimension increases,
we find that pair correlations markedly diminish, consistent with a recently
proposed "decorrelation" principle, and the degree of "hyperuniformity"
(suppression of infinite-wavelength density fluctuations) increases. We have
also calculated the void exclusion probability in order to compute the
so-called quantizer error of the RSA packings, which is related to the second
moment of inertia of the average Voronoi cell. Our algorithm is easily
generalizable to generate saturated RSA packings of nonspherical particles
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