243,592 research outputs found
Configuration Complexities of Hydrogenic Atoms
The Fisher-Shannon and Cramer-Rao information measures, and the LMC-like or
shape complexity (i.e., the disequilibrium times the Shannon entropic power) of
hydrogenic stationary states are investigated in both position and momentum
spaces. First, it is shown that not only the Fisher information and the
variance (then, the Cramer-Rao measure) but also the disequilibrium associated
to the quantum-mechanical probability density can be explicitly expressed in
terms of the three quantum numbers (n, l, m) of the corresponding state.
Second, the three composite measures mentioned above are analytically,
numerically and physically discussed for both ground and excited states. It is
observed, in particular, that these configuration complexities do not depend on
the nuclear charge Z. Moreover, the Fisher-Shannon measure is shown to
quadratically depend on the principal quantum number n. Finally, sharp upper
bounds to the Fisher-Shannon measure and the shape complexity of a general
hydrogenic orbital are given in terms of the quantum numbers.Comment: 22 pages, 7 figures, accepted i
Three-point functions in the SU(2) sector at strong coupling
Extending the methods developed in our previous works (arXiv:1110.3949,
arXiv:1205.6060), we compute the three-point functions at strong coupling of
the non-BPS states with large quantum numbers corresponding to the composite
operators belonging to the so-called SU(2) sector in the
super-Yang-Mills theory in four dimensions. This is achieved by the
semi-classical evaluation of the three-point functions in the dual string
theory in the spacetime, using the general one-cut finite
gap solutions as the external states. In spite of the complexity of the
contributions from various parts in the intermediate stages, the final answer
for the three-point function takes a remarkably simple form, exhibiting the
structure reminiscent of the one obtained at weak coupling. In particular, in
the Frolov-Tseytlin limit the result is expressed in terms of markedly similar
integrals, however with different contours of integration. We discuss a natural
mechanism for introducing additional singularities on the worldsheet without
affecting the infinite number of conserved charges, which can modify the
contours of integration.Comment: 128 pages (A summary is given in section 1); v2 minor improvement
Algorithmic information and incompressibility of families of multidimensional networks
This article presents a theoretical investigation of string-based generalized
representations of families of finite networks in a multidimensional space.
First, we study the recursive labeling of networks with (finite) arbitrary node
dimensions (or aspects), such as time instants or layers. In particular, we
study these networks that are formalized in the form of multiaspect graphs. We
show that, unlike classical graphs, the algorithmic information of a
multidimensional network is not in general dominated by the algorithmic
information of the binary sequence that determines the presence or absence of
edges. This universal algorithmic approach sets limitations and conditions for
irreducible information content analysis in comparing networks with a large
number of dimensions, such as multilayer networks. Nevertheless, we show that
there are particular cases of infinite nesting families of finite
multidimensional networks with a unified recursive labeling such that each
member of these families is incompressible. From these results, we study
network topological properties and equivalences in irreducible information
content of multidimensional networks in comparison to their isomorphic
classical graph.Comment: Extended preprint version of the pape
Shor’s Algorithm: How Quantum Computing Affects Cybersecurity
Honorable Mention Winner
Almost all of today’s computer security relies on something known as the RSA cryptosystem. This system relies on a mathematical, specifically number theory, problem known as prime factorization, where a composite number is broken down into its two prime number factors. This in an ideal method for encryption because it is easy to multiply two numbers, encoding the data, but it much harder to determine which numbers were originally multiplied together, thus hard to decode the data. If this composite number is sufficiently large, there is no known algorithm for efficiently breaking it down – at least not in classical computation. Peter Shor developed an algorithm in 1994, however, which can factor integers very efficiently and thus break down RSA encryption by employing some mathematical principles of quantum mechanics, specifically quantum parallelism, which allows for an exponential speedup with some quantum algorithms. The main goal of this research is to implement these quantum principles, as well as some necessary classical components, to demonstrate Shor’s algorithm and its superior time complexity. To do this, we needed to build Shor’s algorithm in the form of a quantum circuit, which can be done using python and employing the libraries of Qiskit, a quantum computing simulation program developed by IBM. The goal of this program is to show that Shor’s algorithm successfully returns the factors of some integer and compare it to classical computation
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