We recently proposed that topological quantum computing might be based on
representations of the fundamental group
for the complement of a link
K in the three-sphere. The restriction
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We recently proposed that topological quantum computing might be based on
representations of the fundamental group
for the complement of a link
K in the three-sphere. The restriction to links whose associated
character variety
contains a Fricke surface
is desirable due to the connection of Fricke spaces to elementary topology. Taking
K as the Hopf link
, one of the three arithmetic two-bridge links (the Whitehead link
, the Berge link
or the double-eight link
) or the link
, the
for those links contains the reducible component
, the so-called Cayley cubic. In addition, the
for the latter two links contains the irreducible component
, or
, respectively. Taking
to be a representation with character
(
), with
, then
fixes a unique point in the hyperbolic space
and is a conjugate to a
representation (a qubit). Even though details on the physical implementation remain open, more generally, we show that topological quantum computing may be developed from the point of view of three-bridge links, the topology of the four-punctured sphere and Painlevé VI equation. The 0-surgery on the three circles of the Borromean rings L6a4 is taken as an example.
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