5,610 research outputs found
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
A digital computer is generally believed to be an efficient universal
computing device; that is, it is believed able to simulate any physical
computing device with an increase in computation time of at most a polynomial
factor. This may not be true when quantum mechanics is taken into
consideration. This paper considers factoring integers and finding discrete
logarithms, two problems which are generally thought to be hard on a classical
computer and have been used as the basis of several proposed cryptosystems.
Efficient randomized algorithms are given for these two problems on a
hypothetical quantum computer. These algorithms take a number of steps
polynomial in the input size, e.g., the number of digits of the integer to be
factored.Comment: 28 pages, LaTeX. This is an expanded version of a paper that appeared
in the Proceedings of the 35th Annual Symposium on Foundations of Computer
Science, Santa Fe, NM, Nov. 20--22, 1994. Minor revisions made January, 199
The Complexity of Change
Many combinatorial problems can be formulated as "Can I transform
configuration 1 into configuration 2, if certain transformations only are
allowed?". An example of such a question is: given two k-colourings of a graph,
can I transform the first k-colouring into the second one, by recolouring one
vertex at a time, and always maintaining a proper k-colouring? Another example
is: given two solutions of a SAT-instance, can I transform the first solution
into the second one, by changing the truth value one variable at a time, and
always maintaining a solution of the SAT-instance? Other examples can be found
in many classical puzzles, such as the 15-Puzzle and Rubik's Cube.
In this survey we shall give an overview of some older and more recent work
on this type of problem. The emphasis will be on the computational complexity
of the problems: how hard is it to decide if a certain transformation is
possible or not?Comment: 28 pages, 6 figure
Poloidal-toroidal decomposition in a finite cylinder. II. Discretization, regularization and validation
The Navier-Stokes equations in a finite cylinder are written in terms of
poloidal and toroidal potentials in order to impose incompressibility.
Regularity of the solutions is ensured in several ways: First, the potentials
are represented using a spectral basis which is analytic at the cylindrical
axis. Second, the non-physical discontinuous boundary conditions at the
cylindrical corners are smoothed using a polynomial approximation to a steep
exponential profile. Third, the nonlinear term is evaluated in such a way as to
eliminate singularities. The resulting pseudo-spectral code is tested using
exact polynomial solutions and the spectral convergence of the coefficients is
demonstrated. Our solutions are shown to agree with exact polynomial solutions
and with previous axisymmetric calculations of vortex breakdown and of
nonaxisymmetric calculations of onset of helical spirals. Parallelization by
azimuthal wavenumber is shown to be highly effective
Quantum information and statistical mechanics: an introduction to frontier
This is a short review on an interdisciplinary field of quantum information
science and statistical mechanics. We first give a pedagogical introduction to
the stabilizer formalism, which is an efficient way to describe an important
class of quantum states, the so-called stabilizer states, and quantum
operations on them. Furthermore, graph states, which are a class of stabilizer
states associated with graphs, and their applications for measurement-based
quantum computation are also mentioned. Based on the stabilizer formalism, we
review two interdisciplinary topics. One is the relation between quantum error
correction codes and spin glass models, which allows us to analyze the
performances of quantum error correction codes by using the knowledge about
phases in statistical models. The other is the relation between the stabilizer
formalism and partition functions of classical spin models, which provides new
quantum and classical algorithms to evaluate partition functions of classical
spin models.Comment: 15pages, 4 figures, to appear in Proceedings of 4th YSM-SPIP (Sendai,
14-16 December 2012
An efficient high-order algorithm for acoustic scattering from penetrable thin structures in three dimensions
This paper presents a high-order accelerated algorithm for the solution of the integral-equation formulation of volumetric scattering problems. The scheme is particularly well suited to the analysis of “thin” structures as they arise in certain applications (e.g., material coatings); in addition, it is also designed to be used in conjunction with existing low-order FFT-based codes to upgrade their order of accuracy through a suitable treatment of material interfaces. The high-order convergence of the new procedure is attained through a combination of changes of parametric variables (to resolve the singularities of the Green function) and “partitions of unity” (to allow for a simple implementation of spectrally accurate quadratures away from singular points). Accelerated evaluations of the interaction between degrees of freedom, on the other hand, are accomplished by incorporating (two-face) equivalent source approximations on Cartesian grids. A detailed account of the main algorithmic components of the scheme are presented, together with a brief review of the corresponding error and performance analyses which are exemplified with a variety of numerical results
- …