Planat, M.; Amaral, M.M.; Fang, F.; Chester, D.; Aschheim, R.; Irwin, K. Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing. Symmetry2022, 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.
Planat, M.; Amaral, M.M.; Fang, F.; Chester, D.; Aschheim, R.; Irwin, K. Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing. Symmetry2022, 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.
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
Computer Science and Mathematics, Geometry and Topology
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
