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Symmetry 2018, 10(12), 773; https://doi.org/10.3390/sym10120773

Universal Quantum Computing and Three-Manifolds

1
Institut FEMTO-ST CNRS UMR 6174, Université de Bourgogne/Franche-Comté, 15 B Avenue des Montboucons, F-25044 Besançon, France
2
Quantum Gravity Research, Los Angeles, CA 90290, USA
*
Author to whom correspondence should be addressed.
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)
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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 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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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 (CC BY 4.0).
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Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Universal Quantum Computing and Three-Manifolds. Symmetry 2018, 10, 773.

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