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

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 ''>psi 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(V) of an exotic 4-manifold V, more precisely a `small exotic' (space-time) R4 (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean R4). 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) R4s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of `quantum gravity'.