Topological Quantum Computing and 3-Manifolds

Torsten Asselmeyer-Maluga

doi:10.3390/quantum3010009

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.

Highlights

Quantum computing exploits quantum-mechanical phenomena such as superposition and entanglement to perform operations on data, which in many cases, are infeasible to do efficiently on classical computers
Linking and 2-Qubit Operations In the previous section, we described the appearance of the braid group B3 as fundamental group of the trefoil knot complement
We presented some ideas to use 3-manifolds for quantum computing

Summary

Introduction

Quantum computing exploits quantum-mechanical phenomena such as superposition and entanglement to perform operations on data, which in many cases, are infeasible to do efficiently on classical computers. Topological quantum computing seeks to implement a more resilient qubit by utilizing non-Abelian forms of matter to store quantum information. A limiting factor to use topological quantum computing is the usage of non-abelian anyons. The reason for this is the abelian fundamental group of a surface. We discuss the usage of (non-abelian) fundamental groups of 3-manifolds for topological quantum computing. We refer to his book [5] for many relations between knot theory and natural science Note his ideas about topological information [6] (see in [7,8]). Knots are important models in quantum gravity, see, for instance, in [9], and in particle physics [10,11,12]

Some Preliminaries and Motivation

Knot Complement of the Trefoil Knot and the Braid Group B3

Using the Trefoil Knot Complement for Quantum Computing

Knot Group Representations via Berry Phases

Discussion