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Universal Quantum Computing and Three-Manifolds

Institut FEMTO-ST CNRS UMR 6174, Université de Bourgogne/Franche-Comté, 15 B Avenue des Montboucons, F-25044 Besançon, France
Quantum Gravity Research, Los Angeles, CA 90290, USA
Author to whom correspondence should be addressed.
Symmetry 2018, 10(12), 773;
Received: 23 November 2018 / Revised: 12 December 2018 / Accepted: 14 December 2018 / Published: 19 December 2018
(This article belongs to the Special Issue Number Theory and Symmetry)
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 positive operator-valued measure (POVM) that one recognizes to be a 3-manifold M 3 . More precisely, the d-dimensional POVMs defined from subgroups of finite index of the modular group P S L ( 2 , Z ) correspond to d-fold M 3 - coverings over the trefoil knot. In this paper, we also investigate quantum information on a few ‘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 the software SnapPy. Further connections between POVMs based UQC and M 3 ’s obtained from Dehn fillings are explored. View Full-Text
Keywords: quantum computation; IC-POVMs; knot theory; three-manifolds; branch coverings; Dehn surgeries quantum computation; IC-POVMs; knot theory; three-manifolds; branch coverings; Dehn surgeries
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MDPI and ACS Style

Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Universal Quantum Computing and Three-Manifolds. Symmetry 2018, 10, 773.

AMA Style

Planat M, Aschheim R, Amaral MM, Irwin K. Universal Quantum Computing and Three-Manifolds. Symmetry. 2018; 10(12):773.

Chicago/Turabian Style

Planat, Michel, Raymond Aschheim, Marcelo M. Amaral, and Klee Irwin. 2018. "Universal Quantum Computing and Three-Manifolds" Symmetry 10, no. 12: 773.

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