6,222 research outputs found
Reflections on the Role of Entanglement in the Explanation of Quantum Computational Speedup
Of the many and varied applications of quantum information theory, perhaps the most fascinating is the sub-field of quantum computation. In this sub-field, computational algorithms are designed which utilise the resources available in quantum systems in order to compute solutions to computational problems with, in some cases, exponentially fewer resources than any known classical algorithm. While the fact of quantum computational speedup is almost beyond doubt, the source of quantum speedup is still a matter of debate. In this paper I argue that entanglement is a necessary component for any explanation of quantum speedup and I address some purported counter-examples that some claim show that the contrary is true. In particular, I address Biham et al.'s mixed-state version of the Deutsch-Jozsa algorithm, and Knill \& Laflamme's deterministic quantum computation with one qubit (DQC1) model of quantum computation. I argue that these examples do not demonstrate that entanglement is unnecessary for the explanation of quantum speedup, but that they rather illuminate and clarify the role that entanglement does play
Portable random number generators
Computers are deterministic devices, and a computer-generated random number is a contradiction in terms. As a result, computer-generated pseudorandom numbers are fraught with peril for the unwary. We summarize much that is known about the most well-known pseudorandom number generators: congruential generators. We also provide machine-independent programs to implement the generators in any language that has 32-bit signed integers-for example C, C++, and FORTRAN. Based on an extensive search, we provide parameter values better than those previously available.Programming (Mathematics) ; Computers
Integrability of N=6 Chern-Simons Theory at Six Loops and Beyond
We study issues concerning perturbative integrability of N=6 Chern-Simons
theory at planar and weak `t Hooft coupling regime. By Feynman diagrammatics,
we derive so called maximal-ranged interactions in the quantum dilatation
generator, originating from homogeneous and inhomogeneous diagrams. These
diagrams require proper regularization of not only ultraviolet but also
infrared divergences. We first consider standard operator mixing method. We
show that homogeneous diagrams are obtainable by recursive method to all
orders. The method, however, is not easily extendable to inhomogeneous
diagrams. We thus consider two-point function method and study both operator
contents and spectrum of the quantum dilatation generator up to six loop
orders. We show that, of two possible classes of operators, only one linear
combination actually contributes. Curiously, this is exactly the same
combination as in N=4 super Yang-Mills theory. We then study spectrum of
anomalous dimension up to six loops. We find that the spectrum agrees perfectly
with the prediction based on quantum integrability. In evaluating the six loop
diagrams, we utilized remarkable integer-relation algorithm (PSLQ) developed by
Ferguson, Baily and Arno.Comment: 1+39 pages, 12 figures, references added, minor structural changes,
typos correcte
On the exact learnability of graph parameters: The case of partition functions
We study the exact learnability of real valued graph parameters which are
known to be representable as partition functions which count the number of
weighted homomorphisms into a graph with vertex weights and edge
weights . M. Freedman, L. Lov\'asz and A. Schrijver have given a
characterization of these graph parameters in terms of the -connection
matrices of . Our model of learnability is based on D. Angluin's
model of exact learning using membership and equivalence queries. Given such a
graph parameter , the learner can ask for the values of for graphs of
their choice, and they can formulate hypotheses in terms of the connection
matrices of . The teacher can accept the hypothesis as correct, or
provide a counterexample consisting of a graph. Our main result shows that in
this scenario, a very large class of partition functions, the rigid partition
functions, can be learned in time polynomial in the size of and the size of
the largest counterexample in the Blum-Shub-Smale model of computation over the
reals with unit cost.Comment: 14 pages, full version of the MFCS 2016 conference pape
Pseudo-random graphs
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science. Besides
being a fascinating study subject for their own sake, they serve as essential
instruments in proving an enormous number of combinatorial statements, making
their role quite hard to overestimate. Their tremendous success serves as a
natural motivation for the following very general and deep informal questions:
what are the essential properties of random graphs? How can one tell when a
given graph behaves like a random graph? How to create deterministically graphs
that look random-like? This leads us to a concept of pseudo-random graphs and
the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page
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