Quantum Computation and Measurements from an Exotic Space-Time R⁴

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 $\left|\psi \right.⟩$ 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’.