Topological Quantum Computing with Read-Rezayi States

In topological quantum computation quantum operations are carried out by moving quasiparticle excitations of certain quantum systems around each other in two space dimensions or equivalently by braiding their world-lines in three-dimensional space-time. Fractional quantum Hall states are among the prime candidates for realizing such quasiparticles. In particular, it has been shown that quasiparticles associated with the so-called Read-Rezayi (RR) states at $k>2, k\neq 4$ can be used for universal quantum computation. In previous work we have shown how to construct two-qubit gates by braiding the so-called Fibonacci anyons --- a class of non-Abelian anyons that are closely related to the quasiparticles of the $k = 3$ RR state. These two-qubit gates then together with single-qubit gates form a universal set of quantum gates. In this talk we point to certain properties of the quasiparticles of the $k = 3$ RR state which are unique to this state and which allow us to construct two-qubit gates in a simple and efficient way. We then present a method that can be used to efficiently construct two-qubit gates for any quasiparticle in the RR sequence at $k>2, k\neq 4$. This work is supported by US DOE.