327 research outputs found
Homological Error Correction: Classical and Quantum Codes
We prove several theorems characterizing the existence of homological error
correction codes both classically and quantumly. Not every classical code is
homological, but we find a family of classical homological codes saturating the
Hamming bound. In the quantum case, we show that for non-orientable surfaces it
is impossible to construct homological codes based on qudits of dimension
, while for orientable surfaces with boundaries it is possible to
construct them for arbitrary dimension . We give a method to obtain planar
homological codes based on the construction of quantum codes on compact
surfaces without boundaries. We show how the original Shor's 9-qubit code can
be visualized as a homological quantum code. We study the problem of
constructing quantum codes with optimal encoding rate. In the particular case
of toric codes we construct an optimal family and give an explicit proof of its
optimality. For homological quantum codes on surfaces of arbitrary genus we
also construct a family of codes asymptotically attaining the maximum possible
encoding rate. We provide the tools of homology group theory for graphs
embedded on surfaces in a self-contained manner.Comment: Revtex4 fil
Open String Diagrams I: Topological Type
An arbitrary Feynman graph for string field theory interactions is analysed
and the homeomorphism type of the corresponding world sheet surface is
completely determined even in the non-orientable cases. Algorithms are found to
mechanically compute the topological characteristics of the resulting surface
from the structure of the signed oriented graph. Whitney's
permutation-theoretic coding of graphs is utilized
Asymmetric quantum codes on non-orientable surfaces
In this paper, we construct new families of asymmetric quantum surface codes
(AQSCs) over non-orientable surfaces of genus by applying tools of
hyperbolic geometry. More precisely, we prove that if the genus of a
non-orientable surface is even , then the parameters of the
corresponding AQSC are equal to the parameters of a surface code obtained from
an orientable surface of genus . Additionally, if is a non-orientable
surface of genus , we show that the new surface code constructed on a tessellation over has the ratio better than the ratio of an AQSC
constructed on the same tessellation over an orientable surface of
the same genus
Virtual Knot Theory --Unsolved Problems
This paper is an introduction to the theory of virtual knots and links and it
gives a list of unsolved problems in this subject.Comment: 33 pages, 7 figures, LaTeX documen
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