Fricke Topological Qubits

We recently proposed that topological quantum computing might be based on $SL(2,\mathbb{C})$ representations of the fundamental group \(\pi_1(S^3\setminus K)\) for the complement of a link \(K\) in the $3$-sphere. The restriction to links whose associated $SL(2,\mathbb{C})$ character variety \(\mathcal{V}\) contains a Fricke surface \(\kappa_d=xyz -x^2-y^2-z^2+d\) 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 \(5_1^2\), the Berge link \(6_2^2\), the double-eight link \(6_3^2\)] or the link \(7_3^2\), the \(\mathcal{V}\) for those links contains the reducible component \(\kappa_4\), the so-called Cayley cubic. In addition, the \(\mathcal{V}\) for the later two links contains the irreducible component \(\kappa_3\), or \(\kappa_2\), respectively. Taking \(\rho\) to be a representation with character \(\kappa_d\) (\(d<4\)), with $|x|,|y|,|z| \le 2$, then \(\rho(\pi_1)\) fixes a unique point in the hyperbolic space \(\mathcal{H}_3\) and is 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 $4$-punctured sphere and Painlev\'e VI equation. The $0$-surgery on the $3$ circles of the Borromean rings L6a4 is taken as an example.