Character varieties and algebraic surfaces for the topology of quantum computing

TL;DR

This paper explores the relationship between the representation theory of finitely presented groups, their $SL_2(C)$ character varieties, and algebraic surfaces, proposing a topological quantum computing framework based on these mathematical structures.

Contribution

It introduces a novel connection between $SL_2(C)$ character varieties, algebraic surfaces, and topological quantum computing, providing explicit classifications and examples.

Findings

Character varieties relate to algebraic surfaces like Del Pezzo and $K_3$ surfaces.

Hopf link's character variety serves as a kernel for TQC models.

Connections between Bianchi groups, singular fibers, and quantum computing are established.

Abstract

It is shown that the representation theory of some finitely presented groups thanks to their $SL_2(\mathbb{C})$ 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 $SL_2(\mathbb{C})$ 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.