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Pré-Publication, Document De Travail Année : 2022

Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing

Marcelo Amaral
Fang Fang
David Chester
  • Fonction : Auteur
Raymond Aschheim
Klee Irwin

Résumé

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.

Dates et versions

hal-03641420 , version 1 (14-04-2022)

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Citer

Marcelo Amaral, Fang Fang, David Chester, Raymond Aschheim, Klee Irwin, et al.. Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing. 2022. ⟨hal-03641420⟩
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