  A protocol is proposed to generate singlet states of three logical qubits
constructed by pairs of spins. Single and multiple operations of logical qubits
are studied for the construction of an effective Hamiltonian, with which robust
control fields are derived with invariant-based reverse engineering and optimal
control. Moreover, systematic errors are further compensated by periodic
modulation for better robustness. Furthermore, resistance to decoherence of the
protocol is also shown with numerical simulations. Therefore, the protocol may
provide useful perspectives for generations of logical qubit entanglement in
spin systems.
