5 research outputs found
Euclidean algorithms are Gaussian
This study provides new results about the probabilistic behaviour of a class
of Euclidean algorithms: the asymptotic distribution of a whole class of
cost-parameters associated to these algorithms is normal. For the cost
corresponding to the number of steps Hensley already has proved a Local Limit
Theorem; we give a new proof, and extend his result to other euclidean
algorithms and to a large class of digit costs, obtaining a faster, optimal,
rate of convergence. The paper is based on the dynamical systems methodology,
and the main tool is the transfer operator. In particular, we use recent
results of Dolgopyat.Comment: fourth revised version - 2 figures - the strict convexity condition
used has been clarifie
Quantum binary quadratic form reduction
Quadratic form reduction enjoys broad uses both in classical and quantum algorithms
such as in the celebrated LLL algorithm for lattice reduction. In this paper, we propose the first quantum
circuit for definite binary quadratic form reduction that achieves O(n log n) depth, O(n^2)
width and O(n^2 log(n)) quantum gates. The proposed work is based on a
binary variant of the reduction algorithm of the definite quadratic form. As
side results, we show a quantum circuit performing bit rotation
with O(log n) depth, O(n) width, and O(n log n) gates, in addition to a circuit performing
integer logarithm computation with O(log n) depth, O(n) width, and O(n) gates
Algorithms Seminar, 2001-2002
These seminar notes constitute the proceedings of a seminar devoted to the analysis of algorithms and related topics. The subjects covered include combinatorics, symbolic computation, asymptotic analysis, number theory, as well as the analysis of algorithms, data structures, and network protocols