1,841 research outputs found
Towards topological quantum computer
One of the principal obstacles on the way to quantum computers is the lack of
distinguished basis in the space of unitary evolutions and thus the lack of the
commonly accepted set of basic operations (universal gates). A natural choice,
however, is at hand: it is provided by the quantum R-matrices, the entangling
deformations of non-entangling (classical) permutations, distinguished from the
points of view of group theory, integrable systems and modern theory of
non-perturbative calculations in quantum field and string theory. Observables
in this case are (square modules of) the knot polynomials, and their pronounced
integrality properties could provide a key to error correction. We suggest to
use R-matrices acting in the space of irreducible representations, which are
unitary for the real-valued couplings in Chern-Simons theory, to build a
topological version of quantum computing.Comment: 14 page
Evolution method and "differential hierarchy" of colored knot polynomials
We consider braids with repeating patterns inside arbitrary knots which
provides a multi-parametric family of knots, depending on the "evolution"
parameter, which controls the number of repetitions. The dependence of knot
(super)polynomials on such evolution parameters is very easy to find. We apply
this evolution method to study of the families of knots and links which include
the cases with just two parallel and anti-parallel strands in the braid, like
the ordinary twist and 2-strand torus knots/links and counter-oriented 2-strand
links. When the answers were available before, they are immediately reproduced,
and an essentially new example is added of the "double braid", which is a
combination of parallel and anti-parallel 2-strand braids. This study helps us
to reveal with the full clarity and partly investigate a mysterious
hierarchical structure of the colored HOMFLY polynomials, at least, in
(anti)symmetric representations, which extends the original observation for the
figure-eight knot to many (presumably all) knots. We demonstrate that this
structure is typically respected by the t-deformation to the superpolynomials.Comment: 31 page
Racah matrices and hidden integrability in evolution of knots
We construct a general procedure to extract the exclusive Racah matrices S
and \bar S from the inclusive 3-strand mixing matrices by the evolution method
and apply it to the first simple representations R =[1], [2], [3] and [2,2].
The matrices S and \bar S relate respectively the maps (R\otimes R)\otimes \bar
R\longrightarrow R with R\otimes (R \otimes \bar R) \longrightarrow R and
(R\otimes \bar R) \otimes R \longrightarrow R with R\otimes (\bar R \otimes R)
\longrightarrow R. They are building blocks for the colored HOMFLY polynomials
of arbitrary arborescent (double fat) knots. Remarkably, the calculation
realizes an unexpected integrability property underlying the evolution
matrices.Comment: 16 page
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