DOI: https://doi.org/10.15446/recolma.v50n2.62211

Local unitary representations of the braid group and their applications to quantum computing

Colleen Delaney¹, Eric C. Rowell², Zhenghan Wang^{1, 3}

¹ University of California Santa Barbara, Santa Barbara, CA, U.S.A. cdelaney@math.ucsb.edu
² Texas A&M University, College Station, TX, U.S.A. rowell@math.tamu.edu
³ Microsoft Station Q, Santa Barbara, CA, U.S.A. zhenghwa@microsoft.com

Abstract

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 by a quantum computer and explicit localizations of braid group representations.

Keywords: topological quantum computation, braid group representations, localizations, quantum algebra.

Mathematics Subject Classification: 81P86, 20F36.

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Recibido: julio de 2016 Aceptado: noviembre de 2016