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 $left|psirightrangle$ 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) $R^4$ (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean $mathbb{R}^4$). 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 $7$), Mermins 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.
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