1,607 research outputs found
A note on the computation of the fraction of smallest denominator in between two irreducible fractions
International audienceGiven two irreducible fractions f and g, with f < g, we characterize the fraction h such that f < h < g and the denominator of h is as small as possible. An output-sensitive algorithm of time complexity O(d), where d is the depth of h is derived from this characterization
Constructing minimal telescopers for rational functions in three discrete variables
We present a new algorithm for constructing minimal telescopers for rational
functions in three discrete variables. This is the first discrete
reduction-based algorithm that goes beyond the bivariate case. The termination
of the algorithm is guaranteed by a known existence criterion of telescopers.
Our approach has the important feature that it avoids the potentially costly
computation of certificates. Computational experiments are also provided so as
to illustrate the efficiency of our approach
Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets
We generalize an efficient exact synthesis algorithm for single-qubit
unitaries over the Clifford+T gate set which was presented by Kliuchnikov,
Maslov and Mosca. Their algorithm takes as input an exactly synthesizable
single-qubit unitary--one which can be expressed without error as a product of
Clifford and T gates--and outputs a sequence of gates which implements it. The
algorithm is optimal in the sense that the length of the sequence, measured by
the number of T gates, is smallest possible. In this paper, for each positive
even integer we consider the "Clifford-cyclotomic" gate set consisting of
the Clifford group plus a z-rotation by . We present an
efficient exact synthesis algorithm which outputs a decomposition using the
minimum number of z-rotations. For the Clifford+T case
the group of exactly synthesizable unitaries was shown to be equal to the group
of unitaries with entries over the ring .
We prove that this characterization holds for a handful of other small values
of but the fraction of positive even integers for which it fails to hold is
100%.Comment: v2: published versio
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