Local unitary representation of braids and N-qubit entanglements

Masahiro Nozaki A

C.-L. Ho, Tetsuo Deguchi

Using a braid group representation based on the Temperley-Lieb algebra, we construct braid quantum gates that could generate entangled n -partite D -level qudit states. D different sets of D n × D n unitary representation of the braid group generators are presented. With these generators the desired braid quantum gates are obtained. We show that the generalized GHZ states, which are maximally entangled states, can be obtained directly from these braid quantum gates without resorting to further local unitary transformations. We also point out an interesting observation, namely for a general multi-qudit state there exists a unitary braid quantum gate based on the Temperley-Lieb algebra that connects it from one of its component basis states, if the coefficient of the component state is such that the square of its norm is no less than 1/4.

Jennifer F. Vasquez, Zhenghan Wang, Helen Wong

Colleen Delaney, Eric C. Rowell, Zhenghan Wang

We provide an elementary introduction to topological quantum computation based on the Jones representation of the braid group. We first cover the Burau representation and Alexander polynomial. Then we discuss the Jones representation and Jones polynomial and their application to anyonic quantum computation. Finally we outline the approximation of the Jones polynomial and explicit localizations of braid group representations.

Andrzej Veitia

Pramod Padmanabhan, Fumihiko Sugino, Diego Trancanelli

Unitary braiding operators can be used as robust entangling quantum gates. We introduce a solution-generating technique to solve the (d,m,l) -generalized Yang-Baxter equation, for m/2≤l≤m , which allows to systematically construct such braiding operators. This is achieved by using partition algebras, a generalization of the Temperley-Lieb algebra encountered in statistical mechanics. We obtain families of unitary and non-unitary braiding operators that generate the full braid group. Explicit examples are given for a 2-, 3-, and 4-qubit system, including the classification of the entangled states generated by these operators based on Stochastic Local Operations and Classical Communication.