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arXiv · 2204.06872

Character varieties and algebraic surfaces for the topology of quantum computing

Michel Planat·Marcelo M. Amaral·Fang Fang·David Chester·Raymond Aschheim·Klee Irwin
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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.

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Michel Planat, Marcelo M. Amaral, Fang Fang, David Chester, Raymond Aschheim, Klee Irwin. 2022-04-14. Character varieties and algebraic surfaces for the topology of quantum computing. https://doi.org/10.20944/preprints202204.0126.v1

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