Experimental quantum deletion in an NMR quantum information processor

We report an NMR experimental realization of a rapid quantum deletion algorithm that deletes marked states in an unsorted database.Unlike classical deletion,where search and deletion are equivalent,quantum deletion can be implemented with only a single query,which achieves exponential speed-up compared to the optimal classical analog.In the experimental realization,the GRAPE algorithm was used to obtain an optimized NMR pulse sequence,and the efficient method of maximum-likelihood has been used to reconstruct the experimental output state.     

Maximal number of distinct H-eigenpairs for a two-dimensional real tensor

Based on the generalized characteristic polynomial introduced by J. Canny in Generalized characteristic polynomials [J. Symbolic Comput., 1990, 9(3): 241–250], it is immediate that for any m-order n-dimensional real tensor, the number of distinct H-eigenvalues is less than or equal to n(m?1) ^n?1. However, there is no known bounds on the maximal number of distinct Heigenvectors in general. We prove that for any m ? 2, an m-order 2-dimensional tensor A exists such that A has 2(m ? 1) distinct H-eigenpairs. We give examples of 4-order 2-dimensional tensors with six distinct H-eigenvalues as well as six distinct H-eigenvectors. We demonstrate the structure of eigenpairs for a higher order tensor is far more complicated than that of a matrix. Furthermore, we introduce a new class of weakly symmetric tensors, called p-symmetric tensors, and show under certain conditions, p-symmetry will effectively reduce the maximal number of distinct H-eigenvectors for a given two-dimensional tensor. Lastly, we provide a complete classification of the H-eigenvectors of a given 4-order 2-dimensional nonnegative p-symmetric tensor. Additionally, we give sufficient conditions which prevent a given 4-order 2-dimensional nonnegative irreducible weakly symmetric tensor from possessing six pairwise distinct H-eigenvectors.     

On Non-Linear Ordinary Differential Equations of Boundary Layer Type

A numerical method previously applied to linear two-point boundary value problems of boundary layer type is extended to some non-linear problems. Discretization of the differential equation leads to a set of non-linear algebraic equations, which is solved by a modified Newton's method; both the mesh spacing and the boundary layer parameter are iteratively adjusted during the solution process. Several examples are discussed; one of these concerns the problem of shock wave formation in a supersonic nozzle.