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Quantum Computation and Measurements from an Exotic Space-Time R^{4}

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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

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Author to whom correspondence should be addressed.

Received: 17 March 2020 / Revised: 3 April 2020 / Accepted: 16 April 2020 / Published: 5 May 2020

(This article belongs to the Special Issue Symmetry in Quantum Systems)

The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a ‘magic’ state $\left|\psi \right.\u27e9$ in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite ‘contextual’ geometry. In the present work, we choose G as the fundamental group ${\pi}_{1}\left(V\right)$ of an exotic 4-manifold V, more precisely a ‘small exotic’ (space-time) ${R}^{4}$ (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean ${\mathbb{R}}^{4}$ ). Our selected example, due to S. Akbulut and R. E. Gompf, has two remarkable properties: (a) it shows the occurrence of standard contextual geometries such as the Fano plane (at index 7), Mermin’s pentagram (at index 10), the two-qubit commutation picture $GQ(2,2)$ (at index 15), and the combinatorial Grassmannian Gr $(2,8)$ (at index 28); and (b) it allows the interpretation of MICs measurements as arising from such exotic (space-time) ${R}^{4}$ s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of ‘quantum gravity’.