34 research outputs found
Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem
We study the approximation of the smallest eigenvalue of a Sturm-Liouville
problem in the classical and quantum settings. We consider a univariate
Sturm-Liouville eigenvalue problem with a nonnegative function from the
class and study the minimal number n(\e) of function evaluations
or queries that are necessary to compute an \e-approximation of the smallest
eigenvalue. We prove that n(\e)=\Theta(\e^{-1/2}) in the (deterministic)
worst case setting, and n(\e)=\Theta(\e^{-2/5}) in the randomized setting.
The quantum setting offers a polynomial speedup with {\it bit} queries and an
exponential speedup with {\it power} queries. Bit queries are similar to the
oracle calls used in Grover's algorithm appropriately extended to real valued
functions. Power queries are used for a number of problems including phase
estimation. They are obtained by considering the propagator of the discretized
system at a number of different time moments. They allow us to use powers of
the unitary matrix , where is an
matrix obtained from the standard discretization of the Sturm-Liouville
differential operator. The quantum implementation of power queries by a number
of elementary quantum gates that is polylog in is an open issue.Comment: 33 page
Constructive simulation and topological design of protocols
We give a topological simulation for tensor networks that we call the
two-string model. In this approach we give a new way to design protocols, and
we discover a new multipartite quantum communication protocol. We introduce the
notion of topologically-compressed transformations. Our new protocol can
implement multiple, non-local compressed transformations among multi-parties
using one multipartite resource state.Comment: 16 page
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For Which Error Criteria Can We Solve Nonlinear Equations?
For which error criteria can we solve a nonlinear scalar equation f(x) = 0 where f is a real function on the interval [a,b]
Holographic software for quantum networks
We introduce a pictorial approach to quantum information, called holographic software . Our software captures both algebraic and topological aspects of quantum networks. It yields a bi-directional dictionary to translate between a topological approach and an algebraic approach. Using our software, we give a topological simulation for quantum networks. The string Fourier transform (SFT) is our basic tool to transform product states into states with maximal entanglement entropy. We obtain a pictorial interpretation of Fourier transformation, of measurements, and of local transformations, including the n-qudit Pauli matrices and their representation by Jordan-Wigner transformations. We use our software to discover interesting new protocols for multipartite communication. In summary, we build a bridge linking the theory of planar para algebras with quantum information
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Is Gauss quadrature optimal for analytic functions?
We consider the problem of optimal quadratures for integrands f: [ -1 , 1 ] → R which have an analytic extension f to an open disk D, of radius r about the origin such that |f| ≤ 1 on Dr. If r = 1, we show that the penalty for sampling the integrand at zeros of the Legendre polynomial of degree n rather than at optimal points, tends to infinity with n. In particular there is an "infinite" penalty for using Gauss quadrature. On the other hand, if r > l, Gauss quadrature is almost optimal. These results hold for both the worst-case and asymptotic settings
In search of the authentic nation: landscape and national identity in Canada and Switzerland
While the study of nationalism and national identity has flourished in the last decade, little attention has been devoted to the conditions under which natural environments acquire significance in definitions of nationhood. This article examines the identity-forming role of landscape depictions in two polyethnic nation-states: Canada and Switzerland. Two types of geographical national identity are identified. The first – what we call the ‘nationalisation of nature’– portrays zarticular landscapes as expressions of national authenticity. The second pattern – what we refer to as the ‘naturalisation of the nation’– rests upon a notion of geographical determinism that depicts specific landscapes as forces capable of determining national identity. The authors offer two reasons why the second pattern came to prevail in the cases under consideration: (1) the affinity between wild landscape and the Romantic ideal of pure, rugged nature, and (2) a divergence between the nationalist ideal of ethnic homogeneity and the polyethnic composition of the two societies under consideration
E-learning in agri-business personnel training
Praca dotyczy zastosowania szkoleń e-learningowych dla kadr agrobiznesu. Specyfika nauk przyrodniczych, które stanowią główny nurt merytoryczny w tym rodzaju biznesu wymusza stosowanie specyficznych rozwiązań. Dotyczy to np. symulatorów procesów hodowlanych zawierających rozwiązania typu GIS.The work concerns the application of e-learning training for agribusiness staff. The natural sciences, which constitute the mainstream of business that typ, necessitates the use of specific solutions. This applies, for example, breeding process simulators incorporating GIS