Universal quantum computing and three-manifolds

TL;DR

This paper explores a novel correspondence between universal quantum computing and 3-manifolds, linking quantum states, POVMs, and knot theory to provide a topological perspective on quantum computation.

Contribution

It introduces a framework connecting POVMs in quantum computing to 3-manifold coverings and knot complements, expanding the topological understanding of quantum information.

Findings

POVMs from subgroups of the modular group correspond to 3-manifold coverings.

Quantum information on universal knots and links is analyzed using SnapPy.

Connections between Dehn fillings and POVMs in UQC are established.

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.