The combinatorial state sum of Turaev and Viro for a compact 3-manifold in terms of quantum 6j-symbols is generalized by introducing observables in the form of coloured graphs. They satisfy braiding relations and allow for surgeries and a discussion of cobordism theory. Application of these techniques gives the dimension and an explicit basis for the vector space of the topological quantum eld ...

In the hexagonal columnar phase of chiral polymers a bias towards cholesteric twist competes with braiding along an average direction. When the chirality is strong, topological defects proliferate, leading to either a tilt grain boundary phase or a new “moiré state” with twisted bond order. This moiré phase can melt leading to a new phase: the chiral hexatic. I will discuss some recent experime...

Certain proposed extended Bose-Hubbard models may exhibit topologically ordered ground states with excitations obeying non-Abelian braid statistics. A sufficient tuning of Hubbard parameters could yield excitation braiding rules allowing implementation of a universal set of topologically protected quantum gates. We discuss potential difficulties in realizing a model with a proposed non-Abelian ...

We consider topological quantum memories for a general class of abelian anyon models defined on spin lattices. These are non-universal for quantum computation when restricting to topological operations alone, such as braiding and fusion. The effects of additional non-topological operations, such as spin measurements, are studied. These are shown to allow universal quantum computation, while sti...

In this text we use the quantum double construction to show that a suitable category of finite dimensional left U -modules is braided, where U is the quantized universal enveloping algebra of sl2(k). We produce the expression of the corresponding braiding as the action of a formal universal R-matrix for U . We indicate the generalization to sln(k) and the relation to Hecke algebras and Temperle...

Braids are ubiquitous in mathematics and physics. The interesting multistrand braiding topology is visualized by a concise acoustic setup. From the measurements of both eigenvalues eigenstates, noncommutative braid relation swappable precisely captured.

Parafermions are ${\mathbb{Z}}_{n}$ generalizations of Majorana quasiparticles, with fractional non-Abelian statistics. They can be used to encode topological qudits and perform Clifford operations by their braiding. Here we investigate the generation quantum gates allowing ${\mathbb{Z}}_{3}$ parafermions interact in order achieve universality. In particular, study form nontopological gate that...