Search results for "Linear"
showing 10 items of 7165 documents
Spatial Search on Grids with Minimum Memory
2015
We study quantum algorithms for spatial search on finite dimensional grids. Patel et al. and Falk have proposed algorithms based on a quantum walk without a coin, with different operators applied at even and odd steps. Until now, such algorithms have been studied only using numerical simulations. In this paper, we present the first rigorous analysis for an algorithm of this type, showing that the optimal number of steps is $O(\sqrt{N\log N})$ and the success probability is $O(1/\log N)$, where $N$ is the number of vertices. This matches the performance achieved by algorithms that use other forms of quantum walks.
Exceptional Quantum Walk Search on the Cycle
2016
Quantum walks are standard tools for searching graphs for marked vertices, and they often yield quadratic speedups over a classical random walk's hitting time. In some exceptional cases, however, the system only evolves by sign flips, staying in a uniform probability distribution for all time. We prove that the one-dimensional periodic lattice or cycle with any arrangement of marked vertices is such an exceptional configuration. Using this discovery, we construct a search problem where the quantum walk's random sampling yields an arbitrary speedup in query complexity over the classical random walk's hitting time. In this context, however, the mixing time to prepare the initial uniform state…
Nonmalleable encryption of quantum information
2008
We introduce the notion of "non-malleability" of a quantum state encryption scheme (in dimension d): in addition to the requirement that an adversary cannot learn information about the state, here we demand that no controlled modification of the encrypted state can be effected. We show that such a scheme is equivalent to a "unitary 2-design" [Dankert et al.], as opposed to normal encryption which is a unitary 1-design. Our other main results include a new proof of the lower bound of (d^2-1)^2+1 on the number of unitaries in a 2-design [Gross et al.], which lends itself to a generalization to approximate 2-design. Furthermore, while in prime power dimension there is a unitary 2-design with =…
QCD sum rule calculation ofK ℓ3 form factors
1992
We present a combined finite energy sum rule (FESR) and analytic continuation by duality (ACD) calculation of the (neutral)K l3 decay. We confirm the Callan-Treiman relation and investigate the validity of a linear fit for the form factors. Furthermore, we obtain ζ=−0.1...−0.3, consistent with the mean experimental value ζ=−0.1±0.09.
On the conical density properties of measures on $\mathbb{R}^n$
2005
We compare conical density properties and spherical density properties for general Borel measures on $\mathbb{R}^n$ . As a consequence, we obtain results for packing and Hausdorff measures $\mathcal{P}_h$ and $\mathcal{H}_h$ provided that the gauge function $h$ satisfies certain conditions. One consequence of our general results is the following: let $m, n\,{\in}\,\mathbb{N}, 0\,{\lt}\,s\,{\lt}\,m\,{\leq}\,n$ , $0\,{\lt}\,\eta\,{\lt}\,1$ , and suppose that $V$ is an $m$ -dimensional linear subspace of $\mathbb{R}^n$ . Let $\mu$ be either the $s$ -dimensional Hausdorff measure or the $s$ -dimensional packing measure restricted to a set $A$ with $\mu(A)\,{\lt}\,\infty$ . Then for $\mu$ -almos…
Stochastic factorizations, sandwiched simplices and the topology of the space of explanations
2003
We study the space of stochastic factorizations of a stochastic matrix V, motivated by the statistical problem of hidden random variables. We show that this space is homeomorphic to the space of simplices sandwiched between two nested convex polyhedra, and use this geometrical model to gain some insight into its structure and topology. We prove theorems describing its homotopy type, and, in the case where the rank of V is 2, we give a complete description, including bounds on the number of connected components, and examples in which these bounds are attained. We attempt to make the notions of topology accessible and relevant to statisticians.
Characterization and Extraction of Irredundant Tandem Motifs
2012
We address the problem of extracting pairs of subwords (m1,m2) from a text string s of length n, such that, given also an integer constant d in input, m1 and m2 occur in tandem within a maximum distance of d symbols in s. The main effort of this work is to eliminate the possible redundancy from the candidate set of the so found tandem motifs. To this aim, we first introduce the concept of maximality, characterized by four specific conditions, that we show to be not deducible by the corresponding notion of maximality already defined for "simple" (i.e., non tandem) motifs. Then, we further eliminate the remaining redundancy by defining the concept of irredundancy for tandem motifs. We prove t…
On operator valued sequences of multipliers and R-boundedness
2007
AbstractIn recent papers (cf. [J.L. Arregui, O. Blasco, (p,q)-Summing sequences, J. Math. Anal. Appl. 274 (2002) 812–827; J.L. Arregui, O. Blasco, (p,q)-Summing sequences of operators, Quaest. Math. 26 (2003) 441–452; S. Aywa, J.H. Fourie, On summing multipliers and applications, J. Math. Anal. Appl. 253 (2001) 166–186; J.H. Fourie, I. Röntgen, Banach space sequences and projective tensor products, J. Math. Anal. Appl. 277 (2) (2003) 629–644]) the concept of (p,q)-summing multiplier was considered in both general and special context. It has been shown that some geometric properties of Banach spaces and some classical theorems can be described using spaces of (p,q)-summing multipliers. The p…
Elementary (-1)-curves of P-3
2006
In this note we deal with rational curves in P^3 which are images of a line by means of a finite sequence of cubo-cubic Cremona transformations. We prove that these curves can always be obtained applying to the line a sequence of such transformations increasing at each step the degree of the curve. As a corollary we get a result about curves that can give speciality for linear systems of P^3.
Complemented Subspaces and Interpolation Properties in Spaces of Polynomials
1997
LetXbe a Banach space whose dualX* has typep ∈ (1, 2]. Ifmis an integer greater thanp/(p − 1) and (xn) is a seminormalized sequence weakly convergent to zero, there is a subsequence (yn) of (xn) such that, for each element (an) ofl∞, there is anm-homogeneous continuous polynomialPonXwithP(yn) = an,n = 1, 2,… . Some interpolation and complementation properties are also given in P(mlp), form < p, as well as in other spaces of polynomials and multilinear functionals.