Michel Planat, Raymond Aschheim, Marcelo M. Amaral, Klee Irwin (2018)

A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere $S3$. 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 $PSL(2,\mathbb{Z})$ correspond to d-fold $M^3$ – coverings over 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, making use of the catalog of platonic manifolds available on SnapPy. Further connections between POVMs based uqc and $M^3$’s obtained from Dehn fillings are explored.