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A protocol is proposed to generate singlet states of three logical qubitsconstructed by pairs of spins. Single and multiple operations of logical qubitsare studied for the construction of an effective Hamiltonian, with which robustcontrol fields are derived with invariant-based reverse engineering and optimalcontrol. Moreover, systematic errors are further compensated by periodicmodulation for better robustness. Furthermore, resistance to decoherence of theprotocol is also shown with numerical simulations. Therefore, the protocol mayprovide useful perspectives for generations of logical qubit entanglement inspin systems.

Robust generation of logical qubit singlet states with reverse engineering and optimal control with spin qubits