Braiding and fusion of Majorana fermions in minimal Kitaev spin liquid on a single hexagon with $5$ qubits

Abstract

We investigate the minimal Kitaev spin liquid on a single hexagon with three Ising-type exchange interactions proportional to $K_{x}$, $K_{y}$ and $K_{z}$. In the limit $K_{z}=0$, we find 32-fold zero-energy states, leading to 10 free Majorana fermions, and hence, 5 qubits are constructed. These qubits are protected by particle-hole symmetry even for $K_{z}\neq 0$. Braiding of these Majorana fermions is possible by temporally controlling a spin-correlation Hamiltonian. In addition, the fusion is possible by measuring the spin correlation. By switching on the Heisenberg interaction together with magnetic field, only one zero-energy state persists, which can be used as an initialization of qubits. Furthermore, it is shown that $3L+2$ qubits are constructed on the Kitaev spin liquid model on connected $L$ hexagons. All the processes of initialization, operation and readout of qubits are executable in terms of spin operators.