Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing

Version 1 : Received: 12 April 2022 / Approved: 13 April 2022 / Online: 13 April 2022 (10:37:41 CEST)

A peer-reviewed article of this Preprint also exists.

Planat, M.; Amaral, M.M.; Fang, F.; Chester, D.; Aschheim, R.; Irwin, K. Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing. Symmetry 2022, 14, 915. Planat, M.; Amaral, M.M.; Fang, F.; Chester, D.; Aschheim, R.; Irwin, K. Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing. Symmetry 2022, 14, 915.

Journal reference: Symmetry 2022, 14, 915
DOI: 10.3390/sym14050915

## 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.

## Keywords

$SL_2(\mathbb{C})$ character varieties; algebraic surfaces; magic state quantum computing; topological quantum computing; aperiodicity

## Subject

MATHEMATICS & COMPUTER SCIENCE, Geometry & Topology

Views 0