2,637 research outputs found
Constructions and Noise Threshold of Hyperbolic Surface Codes
We show how to obtain concrete constructions of homological quantum codes
based on tilings of 2D surfaces with constant negative curvature (hyperbolic
surfaces). This construction results in two-dimensional quantum codes whose
tradeoff of encoding rate versus protection is more favorable than for the
surface code. These surface codes would require variable length connections
between qubits, as determined by the hyperbolic geometry. We provide numerical
estimates of the value of the noise threshold and logical error probability of
these codes against independent X or Z noise, assuming noise-free error
correction
Implementation of linear minimum area enclosing traingle algorithm
This article has been made available through the Brunel Open Access Publishing Fund.An algorithm which computes the minimum area triangle enclosing a convex polygon in linear time already exists in the literature. The paper describing the algorithm also proves that the provided solution is optimal and a lower complexity sequential algorithm cannot exist. However, only a high-level description of the algorithm was provided, making the implementation difficult to reproduce. The present note aims to contribute to the field by providing a detailed description of the algorithm which is easy to implement and reproduce, and a benchmark comprising 10,000 variable sized, randomly generated convex polygons for illustrating the linearity of the algorithm
Distance-Sensitive Planar Point Location
Let be a connected planar polygonal subdivision with edges
that we want to preprocess for point-location queries, and where we are given
the probability that the query point lies in a polygon of
. We show how to preprocess such that the query time
for a point~ depends on~ and, in addition, on the distance
from to the boundary of~---the further away from the boundary, the
faster the query. More precisely, we show that a point-location query can be
answered in time , where
is the shortest Euclidean distance of the query point~ to the
boundary of . Our structure uses space and
preprocessing time. It is based on a decomposition of the regions of
into convex quadrilaterals and triangles with the following
property: for any point , the quadrilateral or triangle
containing~ has area . For the special case where
is a subdivision of the unit square and
, we present a simpler solution that achieves a
query time of . The latter solution can be extended to
convex subdivisions in three dimensions
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