Universal Quantum Computing and Three-Manifolds

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, we also investigate 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.