Preprint Article Version 1 This version is not peer-reviewed

# Universal Quantum Computing and Three-Manifolds

Version 1 : Received: 8 October 2018 / Approved: 8 October 2018 / Online: 8 October 2018 (16:45:17 CEST)

How to cite: Planat, M.; Aschheim, R.; Amaral, M.; Irwin, K. Universal Quantum Computing and Three-Manifolds. Preprints 2018, 2018100161 (doi: 10.20944/preprints201810.0161.v1). Planat, M.; Aschheim, R.; Amaral, M.; Irwin, K. Universal Quantum Computing and Three-Manifolds. Preprints 2018, 2018100161 (doi: 10.20944/preprints201810.0161.v1).

## 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 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.

## Subject Areas

quantum computation, IC-POVMs, knot theory, three-manifolds, branch coverings, Dehn surgeries.