13,836 research outputs found
A digital analogue of the Jordan curve theorem
AbstractWe study certain closure operations on Z2, with the aim of showing that they can provide a suitable framework for solving problems of digital topology. The Khalimsky topology on Z2, which is commonly used as a basic structure in digital topology nowadays, can be obtained as a special case of the closure operations studied. By proving an analogy of the Jordan curve theorem for these closure operations, we show that they provide a convenient model of the real plane and can therefore be used for studying topological and geometric properties of digital images. We also discuss some advantages of the closure operations investigated over the Khalimsky topology
Introduction to topological quantum computation with non-Abelian anyons
Topological quantum computers promise a fault tolerant means to perform
quantum computation. Topological quantum computers use particles with exotic
exchange statistics called non-Abelian anyons, and the simplest anyon model
which allows for universal quantum computation by particle exchange or braiding
alone is the Fibonacci anyon model. One classically hard problem that can be
solved efficiently using quantum computation is finding the value of the Jones
polynomial of knots at roots of unity. We aim to provide a pedagogical,
self-contained, review of topological quantum computation with Fibonacci
anyons, from the braiding statistics and matrices to the layout of such a
computer and the compiling of braids to perform specific operations. Then we
use a simulation of a topological quantum computer to explicitly demonstrate a
quantum computation using Fibonacci anyons, evaluating the Jones polynomial of
a selection of simple knots. In addition to simulating a modular circuit-style
quantum algorithm, we also show how the magnitude of the Jones polynomial at
specific points could be obtained exactly using Fibonacci or Ising anyons. Such
an exact algorithm seems ideally suited for a proof of concept demonstration of
a topological quantum computer.Comment: 51 pages, 51 figure
Quantum finite automata and linear context-free languages: a decidable problem
We consider the so-called measure once finite quantum automata model introduced by Moore and Crutchfield in 2000. We show that given a language recognized by such a device and a linear context-free language, it is recursively decidable whether or not they have a nonempty intersection. This extends a result of Blondel et al. which can be interpreted as solving the problem with the free monoid in place of the family of linear context-free languages. © 2013 Springer-Verlag
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