Eur. Phys. J. B 74, 419-427 (2010)
https://doi.org/10.1140/epjb/e2010-00087-4

Braid matrices and quantum gates for Ising anyons topological quantum computation

¹ Department of Physics, Nanjing University, Nanjing, 210093, P.R. China
² Artificial Intelligence Institute, Computer Science Department, Xiamen University, Xiamen, Fujian Province, P.R. China

Corresponding authors: brucenju@gmail.com profhugodegaris@yahoo.com

Received: 21 August 2009
Revised: 28 January 2010
Published online: 9 March 2010

Abstract

We study various aspects of the topological quantum computation scheme based on the non-Abelian anyons corresponding to fractional quantum hall effect states at filling fraction 5/2 using the Temperley-Lieb recoupling theory. Unitary braiding matrices are obtained by a normalization of the degenerate ground states of a system of anyons, which is equivalent to a modification of the definition of the 3-vertices in the Temperley-Lieb recoupling theory as proposed by Kauffman and Lomonaco. With the braid matrices available, we discuss the problems of encoding of qubit states and construction of quantum gates from the elementary braiding operation matrices for the Ising anyons model. In the encoding scheme where 2 qubits are represented by 8 Ising anyons, we give an alternative proof of the no-entanglement theorem given by Bravyi and compare it to the case of Fibonacci anyons model. In the encoding scheme where 2 qubits are represented by 6 Ising anyons, we construct a set of quantum gates which is equivalent to the construction of Georgiev.