Fault-tolerant quantum secret sharing against collective noise

We present two robust quantum secret sharing protocols against two kinds of collective noise with single logical qubits. Each logical qubit is encoded in two-qubit noiseless states and so it can function over a quantum channel subjected to a collective noise. The safety of transmission is ensured by nonorthogonality of the noiseless states traveling forward and backward on the quantum channel. Moreover, we construct efficient quantum circuits to implement channel encoding and information encoding by means of primitive operations in quantum computation.