71,494 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
Minor stars in plane graphs with minimum degree five
The weight of a subgraph in is the sum of the degrees in of
vertices of . The {\em height} of a subgraph in is the maximum
degree of vertices of in . A star in a given graph is minor if its
center has degree at most five in the given graph. Lebesgue (1940) gave an
approximate description of minor -stars in the class of normal plane maps
with minimum degree five. In this paper, we give two descriptions of minor
-stars in plane graphs with minimum degree five. By these descriptions, we
can extend several results and give some new results on the weight and height
for some special plane graphs with minimum degree five.Comment: 11 pages, 3 figure
2 \pi-grafting and complex projective structures, I
Let be a closed oriented surface of genus at least two. Gallo, Kapovich,
and Marden asked if 2\pi-graftings produce all projective structures on
with arbitrarily fixed holonomy (Grafting Conjecture). In this paper, we show
that the conjecture holds true "locally" in the space of geodesic
laminations on via a natural projection of projective structures on
into in the Thurston coordinates. In the sequel paper, using this local
solution, we prove the conjecture for generic holonomy.Comment: 57 pages, 10 figures. To appear in Geometry & Topolog
Enumerative aspects of the Gross-Siebert program
We present enumerative aspects of the Gross-Siebert program in this
introductory survey. After sketching the program's main themes and goals, we
review the basic definitions and results of logarithmic and tropical geometry.
We give examples and a proof for counting algebraic curves via tropical curves.
To illustrate an application of tropical geometry and the Gross-Siebert program
to mirror symmetry, we discuss the mirror symmetry of the projective plane.Comment: A version of these notes will appear as a chapter in an upcoming
Fields Institute volume. 81 page
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