Michel Planat, Raymond Aschheim, Marcelo M. Amaral, Klee Irwin


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 POVM that one recognizes to be a $3$-manifold $M^3$. E. g., the $d$-dimensional POVMs defined from subgroups of finite index of the modular group $PSL(2,\mathbb{Z})$ in \cite{PlanatModular} correspond to $d$-fold $M^3$- coverings of the trefoil knot. In this paper, one also investigates quantum information on a few \lq universal' knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings \cite{Hilden1987}, making use of the catalog of platonic manifolds available on SnapPy \cite{Fominikh2015}. Further connections between POVMs based uqc and $M^3$'s obtained from Dehn fillings are explored.