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Register a new userQuantum computation and measurements from an exotic space-time R4
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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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In this paper we show how to place Michael Berrys discovery of knotted zeros in the quantum states of hydrogen in the context of general knot theory and in the context of our formulations for quantum knots. Berry gave a time independent wave function for hydrogen, as a map from three space to the complex plane and such that the inverse image of zero in the complex plane contains a knotted curve in three space. We show that for knots in three space this is a generic situation in that every smooth knot K in three space has a smooth classifying map f from three space to the complex plane such that the inverse image of zero is the knot K. This leaves open the question of characterizing just when such f are wave-functions for quantum systems. One can compare this result with the work of Mark Dennis and his collaborators, with the work of Daniel Peralta-Salas and his collaborators, and with the work of Lee Rudolph. Our approach provides great generality to the structure of knotted zeros of a wavefunction and opens up many new avenues for research in the relationships of quantum theory and knot theory. We show how this classifying construction can be related our previous work on two dimensional and three dimensional mosaic and lattice quantum knots.
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Kashaev algebra associated to a surface is a noncommutative deformation of the algebra of rational functions of Kashaev coordinates. For two arbitrary complex numbers, there is a generalized Kashaev algebra. The relationship between the shear coordinates and Kashaev coordinates induces a natural relationship between the quantum Teichmuller space and the generalized Kashaev algebra.
We show that if a closed, oriented 3-manifold M is promised to be homeomorphic to a lens space L(n,k) with n and k unknown, then we can compute both n and k in polynomial time in the size of the triangulation of M. The tricky part is the parameter k. The idea of the algorithm is to calculate Reidemeister torsion using numerical analysis over the complex numbers, rather than working directly in a cyclotomic field.
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