  A quantum computer can perform exponentially faster than its classical
counterpart. It works on the principle of superposition. But due to the
decoherence effect, the superposition of a quantum state gets destroyed by the
interaction with the environment. It is a real challenge to completely isolate
a quantum system to make it free of decoherence. This problem can be
circumvented by the use of topological quantum phases of matter. These phases
have quasiparticles excitations called anyons. The anyons are charge-flux
composites and show exotic fractional statistics. When the order of exchange
matters, then the anyons are called non-Abelian anyons. Majorana fermions in
topological superconductors and quasiparticles in some quantum Hall states are
non-Abelian anyons. Such topological phases of matter have a ground state
degeneracy. The fusion of two or more non-Abelian anyons can result in a
superposition of several anyons. The topological quantum gates are implemented
by braiding and fusion of the non-Abelian anyons. The fault-tolerance is
achieved through the topological degrees of freedom of anyons. Such degrees of
freedom are non-local, hence inaccessible to the local perturbations. In this
paper, the Hilbert space for a topological qubit is discussed. The Ising and
Fibonacci anyonic models for binary gates are briefly given. Ternary logic
gates are more compact than their binary counterparts and naturally arise in a
type of anyonic model called the metaplectic anyons. The mathematical model,
for the fusion and braiding matrices of metaplectic anyons, is the quantum
deformation of the recoupling theory. We proposed that the existing quantum
ternary arithmetic gates can be realized by braiding and topological charge
measurement of the metaplectic anyons.
