Fricke Topological Qubits

We recently proposed that topological quantum computing might be based on SL(2,C) representations of the fundamental group π1(S3\K) for the complement of a link K in the three-sphere. The restriction to links whose associated SL(2,C) character variety V contains a Fricke surface κd=xyz−x2−y2−z2+d is desirable due to the connection of Fricke spaces to elementary topology. Taking K as the Hopf link L2a1, one of the three arithmetic two-bridge links (the Whitehead link 521, the Berge link 622 or the double-eight link 623) or the link 723, the V for those links contains the reducible component κ4, the so-called Cayley cubic. In addition, the V for the latter two links contains the irreducible component κ3, or κ2, respectively. Taking ρ to be a representation with character κd (d<4), with |x|,|y|,|z|≤2, then ρ(π1) fixes a unique point in the hyperbolic space H3 and is a conjugate to a SU(2) 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.