6,272 research outputs found
Average case quantum lower bounds for computing the boolean mean
We study the average case approximation of the Boolean mean by quantum
algorithms. We prove general query lower bounds for classes of probability
measures on the set of inputs. We pay special attention to two probabilities,
where we show specific query and error lower bounds and the algorithms that
achieve them. We also study the worst expected error and the average expected
error of quantum algorithms and show the respective query lower bounds. Our
results extend the optimality of the algorithm of Brassard et al.Comment: 18 page
Qubit Complexity of Continuous Problems
The number of qubits used by a quantum algorithm will be a crucial
computational resource for the foreseeable future. We show how to obtain the
classical query complexity for continuous problems. We then establish a simple
formula for a lower bound on the qubit complexity in terms of the classical
query complexityComment: 6 pages, 2 figure
Average Case Tractability of Non-homogeneous Tensor Product Problems
We study d-variate approximation problems in the average case setting with
respect to a zero-mean Gaussian measure. Our interest is focused on measures
having a structure of non-homogeneous linear tensor product, where covariance
kernel is a product of univariate kernels. We consider the normalized average
error of algorithms that use finitely many evaluations of arbitrary linear
functionals. The information complexity is defined as the minimal number n(h,d)
of such evaluations for error in the d-variate case to be at most h. The growth
of n(h,d) as a function of h^{-1} and d depends on the eigenvalues of the
covariance operator and determines whether a problem is tractable or not. Four
types of tractability are studied and for each of them we find the necessary
and sufficient conditions in terms of the eigenvalues of univariate kernels. We
illustrate our results by considering approximation problems related to the
product of Korobov kernels characterized by a weights g_k and smoothnesses r_k.
We assume that weights are non-increasing and smoothness parameters are
non-decreasing. Furthermore they may be related, for instance g_k=g(r_k) for
some non-increasing function g. In particular, we show that approximation
problem is strongly polynomially tractable, i.e., n(h,d)\le C h^{-p} for all d
and 0<h<1, where C and p are independent of h and d, iff liminf |ln g_k|/ln k
>1. For other types of tractability we also show necessary and sufficient
conditions in terms of the sequences g_k and r_k
Neural Network Methods for Boundary Value Problems Defined in Arbitrarily Shaped Domains
Partial differential equations (PDEs) with Dirichlet boundary conditions
defined on boundaries with simple geometry have been succesfuly treated using
sigmoidal multilayer perceptrons in previous works. This article deals with the
case of complex boundary geometry, where the boundary is determined by a number
of points that belong to it and are closely located, so as to offer a
reasonable representation. Two networks are employed: a multilayer perceptron
and a radial basis function network. The later is used to account for the
satisfaction of the boundary conditions. The method has been successfuly tested
on two-dimensional and three-dimensional PDEs and has yielded accurate
solutions
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