Quantum Computation and Measurements from an Exotic Space-Time R4
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 2020, 12(5), 736; https://doi.org/10.3390/sym12050736
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
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 of an exotic 4-manifold V, more precisely a ‘small exotic’ (space-time) (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean ). 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 (at index 15), and the combinatorial Grassmannian Gr (at index 28); and (b) it allows the interpretation of MICs measurements as arising from such exotic (space-time) s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of ‘quantum gravity’.