It is shown that the representation theory of some finitely presented groups thanks to their SL2(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 SL2(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 fH (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 E~6 and D~4 contain fH. A surface birationally equivalent to a K3 surface is another compound of their character varieties.
Citation: 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. https://doi.org/
10.3390/sym14050915
Academic Editors: Stefan Heusler
and Ivan Arraut
Received: 19 April 2022
Accepted: 28 April 2022
Published: 30 April 2022
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symmetry
S S
Article
Character Varieties and Algebraic Surfaces for the Topology of
Quantum Computing
Michel Planat 1,*,†
, Marcelo M. Amaral 2,†
, Fang Fang 2,†
, David Chester 2,†
, Raymond Aschheim 2,†
and Klee Irwin 2,†
1
Institut FEMTO-ST CNRS UMR 6174, Université de Bourgogne-Franche-Comté, F-25044 Besançon, France
2 Quantum Gravity Research, Los Angeles, CA 90290, USA; marcelo@quantumgravityresearch.org (M.M.A.);
fang@quantumgravityresearch.org (F.F.); davidc@quantumgravityresearch.org (D.C.);
raymond@quantumgravityresearch.org (R.A.); klee@quantumgravityresearch.org (K.I.)
* Correspondence: michel.planat@femto-st.fr
†
These authors contributed equally to this work.
Abstract: It is shown that the representation theory of some finitely presented groups thanks to
their SL2(C) character variety is related to algebraic surfaces. We make use of the Enriques–Kodaira
classification of algebraic surfaces and related topological tools to make such surfaces explicit. We
study the connection of SL2(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
fH (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