Fricke Topological Qubits
Résumé
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.