13,546 research outputs found
Temperley-Lieb R-matrices from generalized Hadamard matrices
New sets of rank n-representations of Temperley-Lieb algebra TL_N(q) are
constructed. They are characterized by two matrices obeying a generalization of
the complex Hadamard property. Partial classifications for the two matrices are
given, in particular when they reduce to Fourier or Butson matrices.Comment: 17 page
Approximating Subadditive Hadamard Functions on Implicit Matrices
An important challenge in the streaming model is to maintain small-space
approximations of entrywise functions performed on a matrix that is generated
by the outer product of two vectors given as a stream. In other works, streams
typically define matrices in a standard way via a sequence of updates, as in
the work of Woodruff (2014) and others. We describe the matrix formed by the
outer product, and other matrices that do not fall into this category, as
implicit matrices. As such, we consider the general problem of computing over
such implicit matrices with Hadamard functions, which are functions applied
entrywise on a matrix. In this paper, we apply this generalization to provide
new techniques for identifying independence between two vectors in the
streaming model. The previous state of the art algorithm of Braverman and
Ostrovsky (2010) gave a -approximation for the distance
between the product and joint distributions, using space , where is the length of the stream and denotes the
size of the universe from which stream elements are drawn. Our general
techniques include the distance as a special case, and we give an
improved space bound of
Fast Hadamard transforms for compressive sensing of joint systems: measurement of a 3.2 million-dimensional bi-photon probability distribution
We demonstrate how to efficiently implement extremely high-dimensional
compressive imaging of a bi-photon probability distribution. Our method uses
fast-Hadamard-transform Kronecker-based compressive sensing to acquire the
joint space distribution. We list, in detail, the operations necessary to
enable fast-transform-based matrix-vector operations in the joint space to
reconstruct a 16.8 million-dimensional image in less than 10 minutes. Within a
subspace of that image exists a 3.2 million-dimensional bi-photon probability
distribution. In addition, we demonstrate how the marginal distributions can
aid in the accuracy of joint space distribution reconstructions
Explicit Inversion for Two Brownian-Type Matrices
We present explicit inverses of two Brownian--type matrices, which are
defined as Hadamard products of certain already known matrices. The matrices
under consideration are defined by parameters and their lower Hessenberg
form inverses are expressed analytically in terms of these parameters. Such
matrices are useful in the theory of digital signal processing and in testing
matrix inversion algorithms.Comment: v3 has been submitted to Applied Mathematics
(http://www.SciRP.org/journal/am) and accepted after revision; v4, i.e. the
present version, is the revised E-print (title modified; some remarks and
Eqs. (6)-(7) added in Sec. 1; Secs. 2 and 3 reformed; Sec. 5 added;
References [6]-[7] added); 10 page
Quantum Algorithms for Weighing Matrices and Quadratic Residues
In this article we investigate how we can employ the structure of
combinatorial objects like Hadamard matrices and weighing matrices to device
new quantum algorithms. We show how the properties of a weighing matrix can be
used to construct a problem for which the quantum query complexity is
ignificantly lower than the classical one. It is pointed out that this scheme
captures both Bernstein & Vazirani's inner-product protocol, as well as
Grover's search algorithm.
In the second part of the article we consider Paley's construction of
Hadamard matrices, which relies on the properties of quadratic characters over
finite fields. We design a query problem that uses the Legendre symbol chi
(which indicates if an element of a finite field F_q is a quadratic residue or
not). It is shown how for a shifted Legendre function f_s(i)=chi(i+s), the
unknown s in F_q can be obtained exactly with only two quantum calls to f_s.
This is in sharp contrast with the observation that any classical,
probabilistic procedure requires more than log(q) + log((1-e)/2) queries to
solve the same problem.Comment: 18 pages, no figures, LaTeX2e, uses packages {amssymb,amsmath};
classical upper bounds added, presentation improve
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