Version 1
: Received: 8 October 2018 / Approved: 8 October 2018 / Online: 8 October 2018 (16:45:17 CEST)
How to cite:
Planat, M.; Aschheim, R.; Amaral, M.; Irwin, K. Universal Quantum Computing and Three-Manifolds. Preprints2018, 2018100161. https://doi.org/10.20944/preprints201810.0161.v1
Planat, M.; Aschheim, R.; Amaral, M.; Irwin, K. Universal Quantum Computing and Three-Manifolds. Preprints 2018, 2018100161. https://doi.org/10.20944/preprints201810.0161.v1
Planat, M.; Aschheim, R.; Amaral, M.; Irwin, K. Universal Quantum Computing and Three-Manifolds. Preprints2018, 2018100161. https://doi.org/10.20944/preprints201810.0161.v1
APA Style
Planat, M., Aschheim, R., Amaral, M., & Irwin, K. (2018). Universal Quantum Computing and Three-Manifolds. Preprints. https://doi.org/10.20944/preprints201810.0161.v1
Chicago/Turabian Style
Planat, M., Marcelo Amaral and Klee Irwin. 2018 "Universal Quantum Computing and Three-Manifolds" Preprints. https://doi.org/10.20944/preprints201810.0161.v1
Abstract
A single qubit may be represented on the Bloch sphere or similarly on the $3$-sphere $S^3$. Our goal is to dress this correspondence by converting the language of universal quantum computing (uqc) to that of $3$-manifolds. A magic state and the Pauli group acting on it define a model of uqc as a POVM that one recognizes to be a $3$-manifold $M^3$. E. g., the $d$-dimensional POVMs defined from subgroups of finite index of the modular group $PSL(2,\mathbb{Z})$ correspond to $d$-fold $M^3$- coverings over the trefoil knot. In this paper, one also investigates quantum information on a few \lq universal' knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on SnapPy. Further connections between POVMs based uqc and $M^3$'s obtained from Dehn fillings are explored.
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.
