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’.
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