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arXiv · 1802.04196

Universal quantum computing and three-manifolds

Michel Planat·Raymond Aschheim·Marcelo M. Amaral·Klee Irwin
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Abstract

A single qubit may be represented on the Bloch sphere or similarly on the $3$-sphere $S^3$. Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of $3$-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a $3$-manifold $M^3$. More precisely, the $d$-dimensional POVMs defined from subgroups of finite index of the modular group $PSL(2,\mathbb{Z})$ correspond to $d$-fold $M^3$- coverings over the trefoil knot. In this paper, one also investigates quantum information on a few "universal" knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on the software SnapPy. Further connections between POVMs based UQC and $M^3$'s obtained from Dehn fillings are explored.

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Michel Planat, Raymond Aschheim, Marcelo M. Amaral, Klee Irwin. 2018-02-12. Universal quantum computing and three-manifolds. https://arxiv.org/abs/1802.04196

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