Braid Matrices and Quantum Gates for Ising Anyons Topological Quantum Computation

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.

25 pages, 13 figures

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• Thermalization, Error-Correction, and Memory Lifetime for Ising Anyon Systems
• Ternary Logic Design in Topological Quantum Computing
• Exchange and exclusion in the non-abelian anyon gas
• The ZX-calculus as a Language for Topological Quantum Computation
• The construction of a universal quantum gate set for the SU(2)k (k=5,6,7) anyon models via genetic optimized algorithm