Efficient quantum error correction for fully correlated noise

We investigate an efficient quantum error correction of a fully correlated noise. Suppose the noise is characterized by a quantum channel whose error operators take fully correlated forms given by σx⊗n, σy⊗nand σz⊗n, where n>2 is the number of qubits encoding the codeword. It is proved that (i) n qubits codeword encodes (n-1) data qubits when n is odd and (ii) n qubits codeword implements an error-free encoding, which encode (n-2) data qubits when n is even. Quantum circuits implementing these schemes are constructed.