Topological Quantum Computing and 3-Manifolds

TL;DR

This paper explores the use of 3D topological structures, specifically knot complements in 3-spheres, to develop quantum computing systems that leverage complex 3-manifold topologies for information encoding and manipulation.

Contribution

It introduces a novel approach to topological quantum computing using 3-manifolds and knot complements, extending beyond traditional 2D systems.

Findings

Quantum states depend on the topology of knot complements.

Quantum operations can be manipulated via the knot group.

Topology influences quantum state phases through non-contractible curves.

Abstract

In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. Today, the basic structure is a 2D system to realize anyons with braiding operations. From the topological point of view, we have to deal with surface topology. However, usual materials are 3D objects. Possible topologies for these objects can be more complex than surfaces. From the topological point of view, Thurston's geometrization theorem gives the main description of 3-dimensional manifolds. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere. The whole system depends strongly on the topology of this complement, which is determined by non-contractible, closed curves. Every curve gives a contribution to the quantum states by a phase (Berry phase). Therefore, the quantum states can be manipulated by using the knot group (fundamental group of the knot complement). The universality of these operations was already showed by M. Planat et al.