Quantum computation and measurements from an exotic space-time R4

• URL: http://arxiv.org/abs/2001.09091v1
• Date: Wed, 22 Jan 2020 15:16:03 GMT
• Title: Quantum computation and measurements from an exotic space-time R4
• Authors: Michel Planat, Raymond Aschheim, Marcelo. M. Amaral and Klee Irwin
• Abstract summary: A valid subgroup $H$ of index $d$ in $G$ leads to a'magic' state $left|psirightrangle$ in $d$-dimensional Hilbert space. A 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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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\left|\psi\right\rangle$ 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 $7$), 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'.

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