Character varieties and algebraic surfaces for the topology of quantum computing

Abstract

It is shown that the representation theory of some finitely presented groups thanks to their $S{L}_2(\mathbb{C})$ character variety is related to algebraic surfaces. We make use of the Enriques-Kodaira classification of algebraic surfaces and the related topological tools to make such surfaces explicit. We study the connection of $S{L}_2(\mathbb{C})$ character varieties to topological quantum computing (TQC) as an alternative to the concept of anyons. The Hopf link $H$, whose character variety is a Del Pezzo surface $f_H$ (the trace of the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic state computing, derived from the trefoil knot in our previous work, may be seen as TQC from the Hopf link. The character variety of some two-generator Bianchi groups as well as that of the fundamental group for the singular fibers ${\tilde{E}}_6$ and ${\tilde{D}}_4$ contain $f_H$. A surface birationally equivalent to a $K_3$ surface is another compound of their character varieties.