1,605 research outputs found
Quantum algorithms for problems in number theory, algebraic geometry, and group theory
Quantum computers can execute algorithms that sometimes dramatically
outperform classical computation. Undoubtedly the best-known example of this is
Shor's discovery of an efficient quantum algorithm for factoring integers,
whereas the same problem appears to be intractable on classical computers.
Understanding what other computational problems can be solved significantly
faster using quantum algorithms is one of the major challenges in the theory of
quantum computation, and such algorithms motivate the formidable task of
building a large-scale quantum computer. This article will review the current
state of quantum algorithms, focusing on algorithms for problems with an
algebraic flavor that achieve an apparent superpolynomial speedup over
classical computation.Comment: 20 pages, lecture notes for 2010 Summer School on Diversities in
Quantum Computation/Information at Kinki Universit
The Unification and Decomposition of Processing Structures Using Lattice Theoretic Methods
The purpose of this dissertation is to demonstrate that lattice theoretic methods can be used to decompose and unify computational structures over a variety of processing systems. The unification arguments provide a better understanding of the intricacies of the development of processing system decomposition. Since abstract algebraic techniques are used, the decomposition process is systematized which makes it conducive to the use of computers as tools for decomposition. A general algorithm using the lattice theoretic method is developed to examine the structures and therefore decomposition properties of integer and polynomial rings. Two fundamental representations, the Sino-correspondence and the weighted radix representation, are derived for integer and polynomial structures and are shown to be a natural result of the decomposition process. They are used in developing systematic methods for decomposing discrete Fourier transforms and discrete linear systems. That is, fast Fourier transforms and partial fraction expansions of linear systems are a result of the natural representation derived using the lattice theoretic method. The discrete Fourier transform is derived from a lattice theoretic base demonstrating its independence of the continuous form and of the field over which it is computed. The same properties are demonstrated for error control codes based on polynomials. Partial fraction expansions are shown to be independent of the concept of a derivative for repeated roots and the field used to implement them
Quantum phase uncertainty in mutually unbiased measurements and Gauss sums
Mutually unbiased bases (MUBs), which are such that the inner product between
two vectors in different orthogonal bases is constant equal to the inverse
, with the dimension of the finite Hilbert space, are becoming
more and more studied for applications such as quantum tomography and
cryptography, and in relation to entangled states and to the Heisenberg-Weil
group of quantum optics. Complete sets of MUBs of cardinality have been
derived for prime power dimensions using the tools of abstract algebra
(Wootters in 1989, Klappenecker in 2003). Presumably, for non prime dimensions
the cardinality is much less. The bases can be reinterpreted as quantum phase
states, i.e. as eigenvectors of Hermitean phase operators generalizing those
introduced by Pegg & Barnett in 1989. The MUB states are related to additive
characters of Galois fields (in odd characteristic p) and of Galois rings (in
characteristic 2). Quantum Fourier transforms of the components in vectors of
the bases define a more general class of MUBs with multiplicative characters
and additive ones altogether. We investigate the complementary properties of
the above phase operator with respect to the number operator. We also study the
phase probability distribution and variance for physical states and find them
related to the Gauss sums, which are sums over all elements of the field (or of
the ring) of the product of multiplicative and additive characters. Finally we
relate the concepts of mutual unbiasedness and maximal entanglement. This
allows to use well studied algebraic concepts as efficient tools in our quest
of minimal uncertainty in quantum information primitives.Comment: 11 page
Notes on the Riemann Hypothesis
These notes were written from a series of lectures given in March 2010 at the
Universidad Complutense of Madrid and then in Barcelona for the centennial
anniversary of the Spanish Mathematical Society (RSME). Our aim is to give an
introduction to the Riemann Hypothesis and a panoramic view of the world of
zeta and L-functions. We first review Riemann's foundational article and
discuss the mathematical background of the time and his possible motivations
for making his famous conjecture. We discuss some of the most relevant
developments after Riemann that have contributed to a better understanding of
the conjecture.Comment: 2 sections added, 55 pages, 6 figure
Multiplierless DCT Algorithm for Image Compression Applications
This paper presents a novel error-free (infinite-precision) architecture for the fast implementation of 8x8
2-D Discrete Cosine Transform. The architecture uses a new algebraic integer encoding of a 1-D radix-8 DCT
that allows the separable computation of a 2-D 8x8 DCT without any intermediate number representation
conversions. This is a considerable improvement on previously introduced algebraic integer encoding techniques
to compute both DCT and IDCT which eliminates the requirements to approximate the transformation matrix ele-
ments by obtaining their exact representations and hence mapping the transcendental functions without any
errors. Apart from the multiplication-free nature, this new mapping scheme fits to this algorithm, eliminating any
computational or quantization errors and resulting short-word-length and high-speed-design
Diffraction from visible lattice points and k-th power free integers
We prove that the set of visible points of any lattice of dimension at least
2 has pure point diffraction spectrum, and we determine the diffraction
spectrum explicitly. This settles previous speculation on the exact nature of
the diffraction in this situation, see math-ph/9903046 and references therein.
Using similar methods we show the same result for the 1-dimensional set of k-th
power free integers with k at least 2. Of special interest is the fact that
neither of these sets is a Delone set --- each has holes of unbounded inradius.
We provide a careful formulation of the mathematical ideas underlying the study
of diffraction from infinite point sets.Comment: 45 pages, with minor corrections and improvements; dedicated to
Ludwig Danzer on the occasion of his 70th birthda
Rational series and asymptotic expansion for linear homogeneous divide-and-conquer recurrences
Among all sequences that satisfy a divide-and-conquer recurrence, the
sequences that are rational with respect to a numeration system are certainly
the most immediate and most essential. Nevertheless, until recently they have
not been studied from the asymptotic standpoint. We show how a mechanical
process permits to compute their asymptotic expansion. It is based on linear
algebra, with Jordan normal form, joint spectral radius, and dilation
equations. The method is compared with the analytic number theory approach,
based on Dirichlet series and residues, and new ways to compute the Fourier
series of the periodic functions involved in the expansion are developed. The
article comes with an extended bibliography
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