Preprint Article Version 1 This version is not peer-reviewed

Universal Quantum Computing and Three-Manifolds

Version 1 : Received: 23 November 2018 / Approved: 26 November 2018 / Online: 26 November 2018 (10:11:29 CET)

A peer-reviewed article of this Preprint also exists.

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

Journal reference: Symmetry 2018, 10, 773
DOI: 10.3390/sym10120773

Abstract

A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S3. 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 M3. More precisely, the d-dimensional POVMs defined from subgroups of finite index of the modular group PSL(2, Z) correspond to d-fold M3- coverings over the trefoil knot. In this paper, one also investigates 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 M3’s obtained from Dehn fillings are explored.

Subject Areas

quantum computation, IC-POVMs, knot theory, three-manifolds, branch coverings, Dehn surgeries.

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