Michel Planat, Raymond Aschheim, Marcelo. M. Amaral, Klee Irwin (2020)

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 $||ψ⟩$ 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 $π1(V)$ 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 $R^4$). Our selected example, due to to S. Akbulut and R.~E. Gompf, has two remarkable properties: (i) it shows the occurence 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) as well as the combinatorial Grassmannian $Gr(2,8)$(at index 28) , (ii) 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’.